Cover......Page 1
1. Graphs......Page 8
2. Functions and Their Graphs......Page 70
3. Linear and Quadratic Functions......Page 150
4. Polynomial and Rational Functions......Page 206
5. Exponential and Logarithmic Functions......Page 290
6. Trigonometric Functions......Page 406
7. Analytic Trigonometry......Page 500
8. Applications of Trigonometric Functions......Page 578
9. Polar Coordinates; Vectors......Page 634
10. Analytic Geometry......Page 716
11. Systems of Equations and Inequalities......Page 794
12. Sequences; Induction; the Binomial Theorem......Page 898
13. Counting and Probability......Page 948
14. Appendix Review......Page 980
15. Appendix The Limit of a Sequence; Infinite Series......Page 1074
16. Properties, Functions, Formulas, Equations......Page 1082
17. Student's Solutions Manual for Graphs......Page 1086
18. Student's Solutions Manual for Functions and Their Graphs......Page 1130
19. Student's Solutions Manual for Linear and Quadratic Functions......Page 1180
20. Student's Solutions Manual for Polynomial and Rational Functions......Page 1226
21. Student's Solutions Manual for Exponential and Logarithmic Functions......Page 1312
22. Student's Solutions Manual for Trigonometric Functions......Page 1392
23. Student's Solutions Manual for Analytic Trigonometry......Page 1446
24. Student's Solutions Manual for Applications of Trigonometric Functions......Page 1528
25. Student's Solutions Manual for Polar Coordinates; Vectors......Page 1570
26. Student's Solutions Manual for Analytic Geometry......Page 1628
27. Student's Solutions Manual for Systems of Equations and Inequalities......Page 1688
28. Student's Solutions Manual for Sequences; Induction; the Binomial Theorem......Page 1786
29. Student's Solutions Manual for Counting and Probability......Page 1816
30. Student's Solutions Manual for A Preview of Calculus The Limit, Derivative, and Integral of a Function......Page 1828
31. Student's Solutions Manual for Appendix Review......Page 1864
32. Student's Solutions Manual for Appendix The Limit of a Sequence; Infinite Series......Page 1902
B......Page 1906
D......Page 1907
E......Page 1908
F......Page 1909
I......Page 1910
M......Page 1911
N......Page 1912
P......Page 1913
R......Page 1914
S......Page 1915
T......Page 1916
X......Page 1917
Z......Page 1918

##### Citation preview

Precalculus Enhanced with Graphing Utilities Michael Sullivan Mike Sullivan Sixth Edition

Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsoned.co.uk © Pearson Education Limited 2014 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS. All trademarks used herein are the property of their respective owners. The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any afﬁliation with or endorsement of this book by such owners.

ISBN 10: 1-292-02498-4 ISBN 13: 978-1-292-02498-1

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Printed in the United States of America

P

E

A

R

S

O

N

C U

S T O

M

L

I

B

R

A

R Y

1

2. Functions and Their Graphs Michael Sullivan/Michael Sullivan III

63

3. Linear and Quadratic Functions Michael Sullivan/Michael Sullivan III

143

4. Polynomial and Rational Functions Michael Sullivan/Michael Sullivan III

199

5. Exponential and Logarithmic Functions Michael Sullivan/Michael Sullivan III

283

6. Trigonometric Functions Michael Sullivan/Michael Sullivan III

399

7. Analytic Trigonometry Michael Sullivan/Michael Sullivan III

493

8. Applications of Trigonometric Functions Michael Sullivan/Michael Sullivan III

571

9. Polar Coordinates; Vectors Michael Sullivan/Michael Sullivan III

627

10. Analytic Geometry Michael Sullivan/Michael Sullivan III

709

11. Systems of Equations and Inequalities Michael Sullivan/Michael Sullivan III

787

12. Sequences; Induction; the Binomial Theorem Michael Sullivan/Michael Sullivan III

891

13. Counting and Probability Michael Sullivan/Michael Sullivan III

941

I

14. Appendix: Review Michael Sullivan/Michael Sullivan III

973

15. Appendix: The Limit of a Sequence; Infinite Series Michael Sullivan/Michael Sullivan III

1067

16. Properties, Functions, Formulas, Equations Michael Sullivan/Michael Sullivan III

1075

17. Student's Solutions Manual for Graphs Michael Sullivan/Michael Sullivan III

1079

18. Student's Solutions Manual for Functions and Their Graphs Michael Sullivan/Michael Sullivan III

1123

19. Student's Solutions Manual for Linear and Quadratic Functions Michael Sullivan/Michael Sullivan III

1173

20. Student's Solutions Manual for Polynomial and Rational Functions Michael Sullivan/Michael Sullivan III

1219

21. Student's Solutions Manual for Exponential and Logarithmic Functions Michael Sullivan/Michael Sullivan III

1305

22. Student's Solutions Manual for Trigonometric Functions Michael Sullivan/Michael Sullivan III

1385

23. Student's Solutions Manual for Analytic Trigonometry Michael Sullivan/Michael Sullivan III

1439

24. Student's Solutions Manual for Applications of Trigonometric Functions Michael Sullivan/Michael Sullivan III

1521

25. Student's Solutions Manual for Polar Coordinates; Vectors Michael Sullivan/Michael Sullivan III

1563

26. Student's Solutions Manual for Analytic Geometry Michael Sullivan/Michael Sullivan III

1621

27. Student's Solutions Manual for Systems of Equations and Inequalities Michael Sullivan/Michael Sullivan III

1681

28. Student's Solutions Manual for Sequences; Induction; the Binomial Theorem Michael Sullivan/Michael Sullivan III

1779

29. Student's Solutions Manual for Counting and Probability Michael Sullivan/Michael Sullivan III

1809

30. Student's Solutions Manual for A Preview of Calculus: The Limit, Derivative, and Integral of a Function Michael Sullivan/Michael Sullivan III

1821

31. Student's Solutions Manual for Appendix: Review Michael Sullivan/Michael Sullivan III

1857

32. Student's Solutions Manual for Appendix: The Limit of a Sequence; Infinite Series Michael Sullivan/Michael Sullivan III

II

1895

Index

1899

III

Graphs Outline 1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations 2 Intercepts; Symmetry; Graphing Key Equations

3 Solving Equations Using a Graphing Utility 4 Lines 5 Circles

Chapter Review Chapter Test Chapter Project

How to Value a House Two things to consider in valuing a home are, first, how does it compare to similar homes that have sold recently? Is the asking price fair? And second, what value do you place on the advertised features and amenities? Yes, other people might value them highly, but do you? Zestimate home valuation, RealestateABC.com, and Reply.com are among the many algorithmic (generated by a computer model) starting points in figuring out the value of a home. It shows you how the home is priced relative to other homes in the area, but you need to add in all the things that only someone who has seen the house knows. You can do that using My Estimator, and then you create your own estimate and see how it stacks up against the asking price.

Looking at “Comps” Knowing whether an asking price is fair will be important when you’re ready to make an offer on a house. It will be even more important when your mortgage lender hires an appraiser to determine whether the house is worth the loan you’re after. Check with your agent, Zillow.com, propertyshark.com, or other websites to see recent sales of homes in the area that are similar, or comparable, to what you’re looking for. Print them out and keep these “comps” in a three-ring binder; you’ll be referring to them quite a bit. Note that “recent sales” usually means within the last six months. A sales price from a year ago may bear little or no relation to what is going on in your area right now. In fact, some lenders will not accept comps older than three months. Market activity also determines how easy or difficult it is to find accurate comps. In a “hot” or busy market, with sales happening all the time, you’re likely to have lots of comps to choose from. In a less active market, finding reasonable comps becomes harder. And if the home you’re looking at has special design features, finding a comparable property is harder still. It’s also necessary to know what’s going on in a given sub-segment. Maybe large, high-end homes are selling like hotcakes, but owners of smaller houses are staying put, or vice versa.

Andy Dean Photography/ Shutterstock

Source: http://realestate.yahoo.com/Homevalues/How_to_Value_a_House.html

—See the Internet-based Chapter Project—

Here we connect algebra and geometry using the rectangular coordinate system. In the 1600s, algebra had developed sufficiently so that René Descartes (1596 –1650) and Pierre de Fermat (1601–1665) were able to use rectangular coordinates to translate geometry problems into algebra problems, and vice versa. This allowed both geometers and algebraists to gain new insights into their subjects, which were thought to be separate, but now were seen as connected. From Chapter 1 of Precalculus: Enhanced with Graphing Utilities, Sixth Edition. Michael Sullivan, Michael Sullivan, III. Copyright © 2013 by Pearson Education, Inc. All rights reserved.

1

Graphs

1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations PREPARING FOR THIS SECTION Before getting started, review the following: • Algebra Essentials

• Geometry Essentials

Now Work the ‘Are You Prepared?’ problems.

OBJECTIVES 1 Use the Distance Formula 2 3 4 5 6 7

Use the Midpoint Formula Graph Equations by Hand by Plotting Points Graph Equations Using a Graphing Utility Use a Graphing Utility to Create Tables Find Intercepts from a Graph Use a Graphing Utility to Approximate Intercepts

Rectangular Coordinates Figure 1 y 4 2 –4

–2

2

O

4

x

–2 –4

Figure 2 y 4 3 (–3, 1) 1 –4 3 (–2, –3)

2

3

(3, 2) 2

O 3 2

x 4 2 (3, –2)

We locate a point on the real number line by assigning it a single real number, called the coordinate of the point. For work in a two-dimensional plane, points are located by using two numbers. We begin with two real number lines located in the same plane: one horizontal and the other vertical. The horizontal line is called the x-axis, the vertical line the y-axis, and the point of intersection the origin O. See Figure 1. Assign coordinates to every point on these number lines using a convenient scale. Recall that the scale of a number line is the distance between 0 and 1. In mathematics, we usually use the same scale on each axis, but in applications, a different scale is often used. The origin O has a value of 0 on both the x-axis and y-axis. Points on the x-axis to the right of O are associated with positive real numbers, and those to the left of O are associated with negative real numbers. Points on the y-axis above O are associated with positive real numbers, and those below O are associated with negative real numbers. In Figure 1, the x-axis and y-axis are labeled as x and y, respectively, and we have used an arrow at the end of each axis to denote the positive direction. The coordinate system described here is called a rectangular or Cartesian* coordinate system. The plane formed by the x-axis and y-axis is sometimes called the xy-plane, and the x-axis and y-axis are referred to as the coordinate axes. Any point P in the xy-plane can then be located by using an ordered pair 1x, y2 of real numbers. Let x denote the signed distance of P from the y-axis (signed means that, if P is to the right of the y-axis, then x 7 0, and if P is to the left of the y-axis, then x 6 0); and let y denote the signed distance of P from the x-axis. The ordered pair 1x, y2, also called the coordinates of P, then gives us enough information to locate the point P in the plane. For example, to locate the point whose coordinates are 1 -3, 12, go 3 units along the x-axis to the left of O and then go straight up 1 unit. We plot this point by placing a dot at this location. See Figure 2, in which the points with coordinates 1 -3, 12, 1 -2, -32, 13, -22, and 13, 22 are plotted. The origin has coordinates 10, 02. Any point on the x-axis has coordinates of the form 1x, 02, and any point on the y-axis has coordinates of the form 10, y2. If 1x, y2 are the coordinates of a point P, then x is called the x-coordinate, or abscissa, of P and y is the y-coordinate, or ordinate, of P. We identify the point P by its coordinates 1x, y2 by writing P = 1x, y2. Usually, we will simply say “the point 1x, y2 ” rather than “the point whose coordinates are 1x, y2.” *Named after René Descartes (1596–1650), a French mathematician, philosopher, and theologian.

Graphs

Figure 3

The coordinate axes divide the xy-plane into four sections called quadrants, as shown in Figure 3. In quadrant I, both the x-coordinate and the y-coordinate of all points are positive; in quadrant II, x is negative and y is positive; in quadrant III, both x and y are negative; and in quadrant IV, x is positive and y is negative. Points on the coordinate axes belong to no quadrant.

y Quadrant II x < 0, y > 0

Quadrant I x > 0, y > 0

Now Work x Quadrant III x < 0, y < 0

Quadrant IV x > 0, y < 0

PROBLEM

13

Graphing Utilities All graphing utilities (graphing calculators and computer software graphing packages) graph equations by plotting points on a screen. The screen itself actually consists of small rectangles, called pixels. The more pixels the screen has, the better the resolution. Most graphing calculators have 48 pixels per square inch; most computer screens have 32 to 108 pixels per square inch. When a point to be plotted lies inside a pixel, the pixel is turned on (lights up). The graph of an equation is a collection of lighted pixels. Figure 4 shows how the graph of y = 2x looks on a TI-84 Plus graphing calculator. The screen of a graphing utility will display the coordinate axes of a rectangular coordinate system. However, you must set the scale on each axis. You must also include the smallest and largest values of x and y that you want included in the graph. This is called setting the viewing rectangle or viewing window. Figure 5 illustrates a typical viewing window. To select the viewing window, we must give values to the following expressions:

Figure 4 Y = 2X

Figure 5

Xmin: Xmax: Xscl: Ymin: Ymax: Yscl:

the smallest value of x shown on the viewing window the largest value of x shown on the viewing window the number of units per tick mark on the x-axis the smallest value of y shown on the viewing window the largest value of y shown on the viewing window the number of units per tick mark on the y-axis

Figure 6 illustrates these settings and their relation to the Cartesian coordinate system. y

Figure 6

Ymax Yscl

x

Xmin

Xmax Xscl

Ymin

If the scale used on each axis is known, we can determine the minimum and maximum values of x and y shown on the screen by counting the tick marks. Look again at Figure 5. For a scale of 1 on each axis, the minimum and maximum values of x are -10 and 10, respectively; the minimum and maximum values of y are also -10 and 10. If the scale is 2 on each axis, then the minimum and maximum values of x are -20 and 20, respectively; and the minimum and maximum values of y are -20 and 20, respectively. Conversely, if we know the minimum and maximum values of x and y, we can determine the scales being used by counting the tick marks displayed. We shall follow the practice of showing the minimum and maximum values of x and y in our illustrations so that you will know how the window was set. See Figure 7. 4

Figure 7

3

3

4

Xmin = -3 means Xmax = 3 Xscl = 1

Ymin = -4 Ymax = 4 Yscl = 2 3

Graphs

E X A MP L E 1

Finding the Coordinates of a Point Shown on a Graphing Utility Screen Find the coordinates of the point shown in Figure 8. Assume the coordinates are integers.

Figure 8

4

3

3

4

Solution

First note that the viewing window used in Figure 8 is Xmin = -3

Ymin = -4 Ymax = 4 Yscl = 2

Xmax = 3 Xscl = 1

The point shown is 2 tick units to the left on the horizontal axis 1scale = 12 and 1 tick up on the vertical scale 1scale = 22. The coordinates of the point shown are 1 -2, 22.

Now Work

17

PROBLEMS

AND

27

1 Use the Distance Formula If the same units of measurement, such as inches, centimeters, and so on, are used for both the x-axis and y-axis, then all distances in the xy-plane can be measured using this unit of measurement.

E X A MP L E 2

Finding the Distance between Two Points Find the distance d between the points 11, 32 and 15, 62.

Solution

First plot the points 11, 32 and 15, 62 and connect them with a straight line. See Figure 9(a). To find the length d, begin by drawing a horizontal line from 11, 32 to 15, 32 and a vertical line from 15, 32 to 15, 62, forming a right triangle, as in Figure 9(b). One leg of the triangle is of length 4 (since 0 5 - 1 0 = 4 ) and the other is of length 3 (since 0 6 - 3 0 = 3 ). By the Pythagorean Theorem, the square of the distance d that we seek is d 2 = 42 + 32 = 16 + 9 = 25 d = 225 = 5

Figure 9

y 6

(5, 6)

y 6 d

d 3

0

(5, 6)

3 (1, 3) 6 x

3 (a)

0

3

(1, 3) 4 (5, 3) 3

6 x

(b)

The distance formula provides a straightforward method for computing the distance between two points. 4

Graphs

Distance Formula

THEOREM

The distance between two points P1 = 1x1 , y1 2 and P2 = 1x2 , y2 2, denoted by d 1P1 , P2 2, is

In Words To compute the distance between two points, find the difference of the x-coordinates, square it, and add this to the square of the difference of the y-coordinates. The square root of this sum is the distance.

Figure 10 y P2 = (x 2, y2) , d (P 1

P 2)

d 1P1 , P2 2 = 2 1x2 - x1 2 2 + 1y2 - y1 2 2

(1)

Figure 10 illustrates the theorem.

Proof of the Distance Formula Let 1x1 , y1 2 denote the coordinates of point P1 , and let 1x2 , y2 2 denote the coordinates of point P2 . Assume that the line joining P1 and P2 is neither horizontal nor vertical. Refer to Figure 11(a). The coordinates of P3 are 1x2 , y1 2. The horizontal distance from P1 to P3 is the absolute value of the difference of the x-coordinates, 0 x2 - x1 0 . The vertical distance from P3 to P2 is the absolute value of the difference of the y-coordinates, 0 y2 - y1 0 . See Figure 11(b). The distance d 1P1 , P2 2 that we seek is the length of the hypotenuse of the right triangle, so, by the Pythagorean Theorem, it follows that

y2 – y1

3 d 1P1 , P2 2 4 2 = 0 x2 - x1 0 2 + 0 y2 - y1 0 2 P1 = (x1, y1)

x2 – x 1

= 1x2 - x1 2 2 + 1y2 - y1 2 2

x

d 1P1 , P2 2 = 2 1x2 - x1 2 2 + 1y2 - y1 2 2 Figure 11

y

y

y2

P2  (x2, y2)

y1

P1  (x1, y1) x1

P3  (x2, y1) x2

x

P2  (x2, y2)

y2

d(P1, P2)

y1 P1  (x1, y1)

⏐y2  y1⏐

⏐x2  x1⏐

x1

(a)

x2

P3  (x2, y1) x

(b)

Now, if the line joining P1 and P2 is horizontal, then the y-coordinate of P1 equals the y-coordinate of P2; that is, y1 = y2 . Refer to Figure 12(a). In this case, the distance formula (1) still works, because, for y1 = y2 , it reduces to d 1P1 , P2 2 = 2 1x2 - x1 2 2 + 02 = 2 1x2 - x1 2 2 = 0 x2 - x1 0

Figure 12

y

y y2 P1  (x1, y1)

y1

d (P1, P2)

P2  (x2, y1)

⏐y2  y1⏐ d (P1, P2) y1

⏐x2  x1⏐ x2

x1

P2  (x1, y2)

x

(a)

P1  (x1, y1) x1

x

(b)

A similar argument holds if the line joining P1 and P2 is vertical. See Figure 12(b).

E XAMPL E 3

Finding the Length of a Line Segment Find the length of the line segment shown in Figure 13. 5

Graphs

Solution Figure 13 6

The length of the line segment is the distance between the points P1 = 1x1 , y1 2 = 1 -4, 52 and P2 = 1x2, y2 2 = 13, 22. Using the distance formula (1) with x1 = -4, y1 = 5, x2 = 3 , and y2 = 2 , the length d is

(4, 5)

d = 2 1x2 - x1 2 2 + 1y2 - y1 2 2 = 2 3 3 - 1 -42 4 2 + 12 - 52 2 = 272 + 1 -32 2 = 249 + 9 = 258 ⬇ 7.62 (3, 2)

5

Now Work

5

PROBLEM

33

0

The distance between two points P1 = 1x1 , y1 2 and P2 = 1x2 , y2 2 is never a negative number. Furthermore, the distance between two points is 0 only when the points are identical, that is, when x1 = x2 and y1 = y2 . Also, because 1x2 - x1 2 2 = 1x1 - x2 2 2 and 1y2 - y1 2 2 = 1y1 - y2 2 2, it makes no difference whether the distance is computed from P1 to P2 or from P2 to P1; that is, d 1P1 , P2 2 = d 1P2 , P1 2. The introduction to this chapter mentioned that rectangular coordinates enable us to translate geometry problems into algebra problems, and vice versa. The next example shows how algebra (the distance formula) can be used to solve geometry problems.

E X A MP L E 4

Using Algebra to Solve Geometry Problems Consider the three points A = 1 -2, 12, B = 12, 32, and C = 13, 12. (a) (b) (c) (d)

Solution Figure 14 y

3

(a) Figure 14 shows the points A, B, C and the triangle ABC. (b) To find the length of each side of the triangle, we use the distance formula, equation (1). d 1A, B2 = 2 3 2 - 1 -22 4 2 + 13 - 12 2 = 216 + 4 = 220 = 225

B = (2, 3)

A = (–2, 1) –3

Plot each point and form the triangle ABC. Find the length of each side of the triangle. Verify that the triangle is a right triangle. Find the area of the triangle.

d 1B, C2 = 2 13 - 22 2 + 11 - 32 2 = 21 + 4 = 25

C = (3, 1) 3

x

d 1A, C2 = 2 3 3 - 1 -22 4 2 + 11 - 12 2 = 225 + 0 = 5 (c) To show that the triangle is a right triangle, we need to show that the sum of the squares of the lengths of two of the sides equals the square of the length of the third side. (Why is this sufficient?) Looking at Figure 14, it seems reasonable to conjecture that the right angle is at vertex B. We shall check to see whether

3 d 1A, B2 4 2 + 3 d 1B, C2 4 2 = 3 d 1A, C2 4 2 Using the results from part (b),

3 d 1A, B2 4 2 + 3 d 1B, C2 4 2 = 1 225 2 2 + 1 25 2 2 = 20 + 5 = 25 = 3 d 1A, C2 4 2 It follows from the converse of the Pythagorean Theorem that triangle ABC is a right triangle. (d) Because the right angle is at vertex B, the sides AB and BC form the base and height of the triangle. Its area is Area =

1 1 1Base2 1Height2 = 1 225 2 1 25 2 = 5 square units 2 2

Now Work 6

PROBLEM

49

Graphs

2 Use the Midpoint Formula Figure 15 y P2 = (x 2, y2) y2 M = (x, y) y

y – y1

x – x1

y1

P1 = (x1, y1)

y2 – y x2 – x B = (x 2, y)

We now derive a formula for the coordinates of the midpoint of a line segment. Let P1 = 1x1 , y1 2 and P2 = 1x2 , y2 2 be the endpoints of a line segment, and let M = 1x, y2 be the point on the line segment that is the same distance from P1 as it is from P2 . See Figure 15. The triangles P1 AM and MBP2 are congruent. [Do you see why? Angle AP1 M = angle BMP2 ,* angle P1 MA = angle MP2 B, and d 1P1 , M2 = d 1M, P2 2 is given. So, we have angle–side–angle.] Hence, corresponding sides are equal in length. That is, x - x1 = x2 - x

A = (x, y1)

x1

x

x2

y - y1 = y2 - y

2x = x1 + x2

x

x=

THEOREM

and

2y = y1 + y2

x1 + x2 2

y=

y1 + y2 2

Midpoint Formula The midpoint M = 1x, y2 of the line segment from P1 = 1x1 , y1 2 to P2 = 1x2 , y2 2 is

In Words To find the midpoint of a line segment, average the x-coordinates and average the y-coordinates of the endpoints.

M = 1x, y2 = ¢

x1 + x2 y1 + y2 , ≤ 2 2

(2)

Finding the Midpoint of a Line Segment

E XAM PL E 5

Find the midpoint of a line segment from P1 = 1 -5, 52 to P2 = 13, 12. Plot the points P1 and P2 and their midpoint.

Solution

Figure 16 y P1  (–5, 5)

M  (–1, 3) –5

Apply the midpoint formula (2) using x1 = -5, y1 = 5, x2 = 3, and y2 = 1. Then the coordinates 1x, y2 of the midpoint M are x=

5

That is, M = 1 -1, 32. See Figure 16.

P2  (3, 1) 5

y1 + y2 5 + 1 x1 + x2 -5 + 3 = = -1 and y = = =3 2 2 2 2

x

Now Work

PROBLEM

55

3 Graph Equations by Hand by Plotting Points An equation in two variables, say x and y, is a statement in which two expressions involving x and y are equal. The expressions are called the sides of the equation. Since an equation is a statement, it may be true or false, depending on the value of the variables. Any values of x and y that result in a true statement are said to satisfy the equation. For example, the following are all equations in two variables x and y: x2 + y2 = 5

2x - y = 6

y = 2x + 5

x2 = y

The first of these, x2 + y 2 = 5, is satisfied for x = 1, y = 2, since 12 + 22 = 1 + 4 = 5. Other choices of x and y also satisfy this equation. It is not satisfied for x = 2 and y = 3, since 22 + 32 = 4 + 9 = 13 ⬆ 5. *A postulate from geometry states that the transversal P1 P2 forms congruent corresponding angles with the parallel line segments P1 A and MB.

7

Graphs

The graph of an equation in two variables x and y consists of the set of points in the xy-plane whose coordinates 1x, y2 satisfy the equation. Graphs play an important role in helping us to visualize the relationships that exist between two variables or quantities. Figure 17 shows the relation between the level of risk in a stock portfolio and the average annual rate of return. The graph shows that, when 30% of a portfolio of stocks is invested in foreign companies, risk is minimized. Figure 17

18.5

Source: T. Rowe Price Average Annual Returns (%)

18 70%

17.5

(100% foreign)

50% 40%

16 15.5

100%

60%

17 16.5

90%

80%

30% (30% foreign/70% U.S.) 20%

15

10%

14.5

0% (100% U.S.)

14 13.5 13.5 14 14.5 15 15.5 16 16.5 17 17.5 18 18.5 19 19.5 20 Level of Risk (%)

E X A MP L E 6

Determining Whether a Point Is on the Graph of an Equation Determine if the following points are on the graph of the equation 2x - y = 6. (a) 12, 32

Solution

(b) 12, -22

(a) For the point 12, 32, check to see if x = 2, y = 3 satisfies the equation 2x - y = 6. 2x - y = 2122 - 3 = 4 - 3 = 1 ⬆ 6 The equation is not satisfied, so the point 12, 32 is not on the graph. (b) For the point 12, -22, 2x - y = 2122 - 1 -22 = 4 + 2 = 6 The equation is satisfied, so the point 12, -22 is on the graph.

Now Work

E X A MP L E 7

PROBLEM

63

How to Graph an Equation by Hand by Plotting Points Graph the equation: y = -2x + 3

Step-by-Step Solution Step 1 Find some points (x, y) that satisfy the equation. To determine these points, choose values of x and use the equation to find the corresponding values for y. See Table 1. 8

Table 1

x

y   2x  3

(x, y)

-2

-2( -2) + 3 = 7

( -2, 7)

-1

-2( -1) + 3 = 5

( -1, 5)

0

-2(0) + 3 = 3

(0, 3)

1

-2(1) + 3 = 1

(1, 1)

2

-2(2) + 3 = -1

(2, -1)

Graphs

Step 2 Plot the points listed in the table as shown in Figure 18(a). Now connect the points to obtain the graph of the equation (a line), as shown in Figure 18(b).

Figure 18 y 8

y 8 (–2, 7) (–1, 5)

(–2, 7)

6 4

(–1, 5) (0, 3)

2 –4

(1, 1) 2

–2

(0, 3) 2

4 x (2, –1)

–4

(1, 1) 2

–2

4 x (2, –1)

–2

–2 (a)

E XAM PL E 8

6

(b)

Graphing an Equation by Hand by Plotting Points Graph the equation: y = x2

Solution

Table 2 provides several points on the graph. In Figure 19 we plot these points and connect them with a smooth curve to obtain the graph (a parabola). Figure 19

Table 2 x

y  x2

(x, y )

-4

16

( -4, 16)

-3

9

( -3, 9)

-2

4

( -2, 4)

-1

1

( -1, 1)

0

0

(0, 0)

1

1

(1, 1)

2

4

(2, 4)

3

9

(3, 9)

4

16

(4, 16)

y 20 (– 4, 16) (–3, 9)

(4, 16)

15 10

(3, 9)

5 (–2, 4) (2, 4) (1, 1) (–1, 1) (0, 0) –4 4

x

The graphs of the equations shown in Figures 18(b) and 19 do not show all the points that are on the graph. For example, in Figure 18(b), the point 120, -372 is a part of the graph of y = -2x + 3, but it is not shown. Since the graph of y = -2x + 3 could be extended out as far as we please, we use arrows to indicate that the pattern shown continues. It is important when illustrating a graph to present enough of the graph so that any viewer of the illustration will “see” the rest of it as an obvious continuation of what is actually there. This is referred to as a complete graph. One way to obtain a complete graph of an equation is to plot a sufficient number of points on the graph until a pattern becomes evident. Then connect these points with a smooth curve following the suggested pattern. But how many points are sufficient? Sometimes knowledge about the equation tells us. For example, we will learn in Section 4 that, if an equation is of the form y = mx + b, then its graph is a line. In this case, two points would suffice to obtain the graph. One purpose of this text is to investigate the properties of equations in order to decide whether a graph is complete. Sometimes we shall graph equations by plotting a sufficient number of points on the graph until a pattern becomes evident and then connect these points with a smooth curve, following the suggested pattern. (Shortly, we shall investigate various techniques that will enable us to graph an equation without plotting so many points.) Other times we shall graph equations using a graphing utility. 9

Graphs

4 Graph Equations Using a Graphing Utility From Examples 7 and 8, we see that a graph can be obtained by plotting points in a rectangular coordinate system and connecting them. Graphing utilities perform these same steps when graphing an equation. For example, the TI-84 Plus determines 95 evenly spaced input values,* uses the equation to determine the output values, plots these points on the screen, and finally (if in the connected mode) draws a line between consecutive points. To graph an equation in two variables x and y using a graphing utility requires that the equation be written in the form y = 5 expression in x 6 . If the original equation is not in this form, rewrite it using equivalent equations until the form y = 5 expression in x 6 is obtained. In general, there are four ways to obtain equivalent equations.

Procedures That Result in Equivalent Equations 1. Interchange the two sides of the equation: 3x + 5 = y

y = 3x + 5

is equivalent to

2. Simplify the sides of the equation by combining like terms, eliminating parentheses, and so on: 2y + 2 + 6 = 2x + 51x + 12

is equivalent to

2y + 8 = 7x + 5

3. Add or subtract the same expression on both sides of the equation: y + 3x - 5 = 4

is equivalent to

y + 3x - 5 + 5 = 4 + 5

4. Multiply or divide both sides of the equation by the same nonzero expression: 3y = 6 - 2x

E X A MP L E 9

is equivalent to

1 3

# 3y = 1 16 - 2x2 3

Expressing an Equation in the Form y  {expression in x} Solve for y: 2y + 3x - 5 = 4

Solution

We replace the original equation by a succession of equivalent equations. 2y + 3x - 5 = 4 2y + 3x - 5 + 5 = 4 + 5 2y + 3x = 9 2y + 3x - 3x = 9 - 3x 2y = 9 - 3x

WARNING When entering the expression 9 - 3x , be careful. Use parentheses as 2 follows: (9 - 3x)/2. 

2y 9 - 3x = 2 2 9 - 3x y= 2

Add 5 to both sides. Simplify. Subtract 3x from both sides. Simplify. Divide both sides by 2. Simplify.

Now we are ready to graph equations using a graphing utility. *These input values depend on the values of Xmin and Xmax. For example, if Xmin = -10 and Xmax = 10, then the first input value will be - 10 and the next input value will be - 10 + 110 - 1 - 1022 >94 = - 9.7872, and so on.

10

Graphs

E X AMPL E 1 0

How to Graph an Equation Using a Graphing Utility Use a graphing utility to graph the equation: 6x2 + 3y = 36

Step-by-Step Solution Step 1 Solve the equation for y in terms of x.

6x2 + 3y = 36 3y = -6x2 + 36 Subtract 6x2 from both sides of the equation. y = -2x2 + 12 Divide both sides of the equation by 3 and simplify.

Step 2 Enter the equation to be graphed into your graphing utility. Figure 20 shows the equation to be graphed entered on a TI-84 Plus.

Figure 20

Step 3 Choose an initial viewing window. Without any knowledge about the behavior of the graph, it is common to choose the standard viewing window as the initial viewing window. The standard viewing window is

Figure 21

Xmin = -10 Ymin = -10 Xmax = 10 Ymax = 10 Xscl = 1 Yscl = 1 See Figure 21. Step 4 Graph the equation. See Figure 22.

Figure 22

10

10

10

10

Step 5 Adjust the viewing window until a complete graph is obtained.

The graph of y = -2x2 + 12 is not complete. The value of Ymax must be increased so that the top portion of the graph is visible. After increasing the value of Ymax to 12, we obtain the graph in Figure 23. The graph is now complete. Figure 23

NOTE Some graphing utilities have a ZOOM-STANDARD feature that automatically sets the viewing window to the standard viewing window. In addition, some graphing utilities have a ZOOM-FIT feature that determines the appropriate Ymin and Ymax for a given Xmin and Xmax. Consult your owner’s manual for the appropriate keystrokes. 

12

10

10

10

Now Work

PROBLEM

81

11

Graphs

5 Use a Graphing Utility to Create Tables In addition to graphing equations, graphing utilities can also be used to create a table of values that satisfy the equation. This feature is especially useful in determining an appropriate viewing window when graphing an equation.

EX A MP L E 1 1

How to Create a Table Using a Graphing Utility Create a table that displays the points on the graph of 6x2 + 3y = 36 for x = -3, -2, -1, 0, 1, 2, and 3.

Step-by-Step Solution Step 1 Solve the equation for y in terms of x.

We solved the equation for y in terms of x in Example 10 and obtained y = -2x2 + 12.

Step 2 Enter the expression in x following the Y = prompt of the graphing utility.

See Figure 20 on previous page.

Step 3 Set up the table. Graphing utilities typically have two modes for creating tables. In the AUTO mode, the user determines a starting point for the table (TblStart) and  Tbl (pronounced “delta table”). The  Tbl feature determines the increment for x in the table. The ASK mode requires the user to enter values of x, and then the utility determines the corresponding value of y.

Create a table using AUTO mode. The table we wish to create starts at -3, so TblStart = −3. The increment in x is 1, so Tbl = 1. See Figure 24.

Step 4 Create the table. See Table 3.

Table 3

Figure 24

The user can scroll within the table if it is created in AUTO mode. In looking at Table 3, notice that y = 12 when x = 0. This information could have been used to help to create the initial viewing window by letting us know that Ymax needs to be at least 12 in order to get a complete graph.

Figure 25 Graph crosses y-axis

y Graph crosses x-axis

6 Find Intercepts from a Graph x

Graph touches x-axis

Intercepts

EX A MP L E 1 2

The points, if any, at which a graph crosses or touches the coordinate axes are called the intercepts. See Figure 25. The x-coordinate of a point at which the graph crosses or touches the x-axis is an x-intercept, and the y-coordinate of a point at which the graph crosses or touches the y-axis is a y-intercept. For a graph to be complete, all its intercepts must be displayed.

Finding Intercepts from a Graph Find the intercepts of the graph in Figure 26. What are its x-intercepts? What are its y-intercepts?

12

Graphs

Solution

The intercepts of the graph are the points

Figure 26

1 -3, 02,

y 4

(0, 3)

The x-intercepts are -3,

( 3–2 , 0) 4 (3, 0)

10, 32,

(4.5, 0) 5 x

(0, 4–3 )

(0, 3.5)

3 a , 0b , 2

4 a 0, - b , 3

10, -3.52,

14.5, 02

3 4 , and 4.5; the y-intercepts are -3.5, - , and 3. 2 3

In Example 12 notice the following usage: If the type of intercept is not specified (x- versus y-), then report the intercept as an ordered pair. However, if the type of intercept is specified, then report the coordinate of the specified intercept. For x-intercepts, report the x-coordinate of the intercept; for y-intercepts, report the y-coordinate of the intercept.

Now Work

PROBLEM

69

7 Use a Graphing Utility to Approximate Intercepts We can use a graphing utility to approximate the intercepts of the graph of an equation.

Approximating Intercepts Using a Graphing Utility

E X AMPL E 1 3

Use a graphing utility to approximate the intercepts of the equation y = x3 - 16.

Solution

Figure 27(a) shows the graph of y = x3 - 16. The eVALUEate feature of a TI-84 Plus graphing calculator accepts as input a value of x and determines the value of y. If we let x = 0, the y-intercept is found to be -16. See Figure 27(b). The ZERO feature of a TI-84 Plus is used to find the x-intercept(s). See Figure 27(c). Rounded to two decimal places, the x-intercept is 2.52.

Figure 27 10 –5

5

–25 (a)

Now Work

(b) PROBLEM

(c)

91

We discuss finding intercepts algebraically in the next section.

1 Assess Your Understanding ‘Are You Prepared?’

Answers are given at the end of these exercises.

1. On a real number line the origin is assigned the number _________.

5. The area of a triangle whose base is b and whose altitude is h is A = _________.

2. If - 3 and 5 are the coordinates of two points on the real number line, the distance between these points is _________.

6. True or False Two triangles are congruent if two angles and the included side of one equals two angles and the included side of the other.

3. If 3 and 4 are the legs of a right triangle, the hypotenuse is _________. 4. Use the converse of the Pythagorean Theorem to show that a triangle whose sides are of lengths 11, 60, and 61 is a right triangle.

13

Graphs

Concepts and Vocabulary 7. If 1x, y2 are the coordinates of a point P in the xy-plane, then x is called the of P and y is the _____________ of P.

10. True or False The distance between two points is sometimes a negative number. 11. True or False The point 1 - 1, 42 lies in quadrant IV of the Cartesian plane.

8. The coordinate axes divide the xy-plane into four sections called _________.

12. True or False The midpoint of a line segment is found by averaging the x-coordinates and averaging the y-coordinates of the endpoints.

9. If three distinct points P, Q, and R all lie on a line and if d1P, Q2 = d1Q, R2, then Q is called the _________ of the line segment from P to R.

Skill Building In Problems 13 and 14, plot each point in the xy-plane. Tell in which quadrant or on what coordinate axis each point lies. 13.

(a) A = 1 - 3, 22

(d) D = 16, 52

(a) A = 11, 42

(d) D = 14, 12

(b) B = 16, 02

(e) E = 10, - 32

(b) B = 1 - 3, - 42

(e) E = 10, 12

(c) C = 1 - 2, - 22

(f ) F = 16, - 32

(c) C = 1 - 3, 42

(f ) F = 1 - 3, 02

14.

15. Plot the points 12, 02, 12, - 32, 12, 42, 12, 12, and 12, - 12. Describe the set of all points of the form 12, y2, where y is a real number. 16. Plot the points 10, 32, 11, 32, 1 - 2, 32, 15, 32, and 1 - 4, 32. Describe the set of all points of the form 1x, 32, where x is a real number. In Problems 17–20, determine the coordinates of the points shown. Tell in which quadrant each point lies. Assume the coordinates are integers. 17.

18.

10

5

19.

10

5

5

5

5

10

20.

5

10

10

10

5

10

5

10

In Problems 21–26, select a setting so that each given point will lie within the viewing window. 21. 1 - 10, 52, 13, - 22, 14, - 12

22. 15, 02, 16, 82, 1 - 2, - 32

23. 140, 202, 1 - 20, - 802, 110, 402

24. 1 - 80, 602, 120, - 302, 1 - 20, - 402

25. 10, 02, 1100, 52, 15, 1502

26. 10, - 12, 1100, 502, 1 - 10, 302

In Problems 27–32, determine the viewing window used. 27.

28.

4

6

29.

2

3

6

3

3 6

4

30.

2

31.

4 9

6 1

32.

10

8

9

3

12

22

9

10 4

2

In Problems 33–44, find the distance d1P1 , P2 2 between the points P1 and P2 . 33.

34.

y 2 P = (2, 1) 2 P1 = (0, 0) –2

14

–1

2

x

35.

y P2 = (–2, 1) 2 P = (0, 0) 1 –2

–1

2

x

36. P2  (–2, 2)

–2

y

y 2

P1  (1, 1) 2 x

P1 = (–1, 1) 2 –2

–1

P2 = (2, 2)

2

x

Graphs

37. P1 = 13, - 42;

P2 = 15, 42

39. P1 = 1 - 5, - 32; 41. P1 = 14, - 32; 43. P1 = 1a, b2;

P2 = 111, 92 P2 = 16, 42

38. P1 = 1 - 1, 02;

P2 = 12, 42

40. P1 = 12, - 32;

P2 = 110, 32

42. P1 = 1 - 4, - 32;

P2 = 10, 02

44. P1 = 1a, a2;

P2 = 16, 22

P2 = 10, 02

In Problems 45–48, find the length of the line segment. Assume that the endpoints of each line segment have integer coordinates. 18

45.

6

46.

6

12

6

12

18

12

47.

12

48.

12

6

12

12

6

6

12

In Problems 49–54, plot each point and form the triangle ABC. Verify that the triangle is a right triangle. Find its area. 49. A = 1 - 2, 52;

B = 11, 32;

C = 1 - 1, 02

50. A = 1 - 2, 52;

B = 112, 32;

C = 110, - 112

51. A = 1 - 5, 32;

B = 16, 02;

C = 15, 52

52. A = 1 - 6, 32;

B = 13, - 52;

C = 1 - 1, 52

53. A = 14, - 32;

B = 10, - 32;

54. A = 14, - 32;

B = 14, 12;

C = 14, 22

C = 12, 12

In Problems 55–62, find the midpoint of the line segment joining the points P1 and P2 . 55. P1 = 13, - 42;

P2 = 15, 42

57. P1 = 1 - 5, - 32; 59. P1 = 14, - 32; 61. P1 = 1a, b2;

P2 = 111, 92 P2 = 16, 12

56. P1 = 1 - 2, 02;

P2 = 12, 42

58. P1 = 12, - 32;

P2 = 110, 32

60. P1 = 1 - 4, - 32;

P2 = 10, 02

62. P1 = 1a, a2;

P2 = 12, 22

P2 = 10, 02

In Problems 63–68, tell whether the given points are on the graph of the equation. 63. Equation:

y = x 4 - 1x

64. Equation:

10, 02; 11, 12; 1 - 1, 02

Points:

66. Equation:

67. Equation:

11, 22; 10, 12; 1 - 1, 02

Points:

65. Equation:

10, 02; 11, 12; 11, - 12

Points:

y3 = x + 1

y = x 3 - 2 1x

Points:

x2 + y2 = 4

10, 32; 13, 02; 1 - 3, 02

68. Equation:

10, 22; 1- 2, 22; 1 22 , 22 2

Points:

Points:

y2 = x2 + 9 x 2 + 4y 2 = 4

1 10, 12; 12, 02; a2, b 2

In Problems 69–76, the graph of an equation is given. List the intercepts of the graph. 69.

70.

y 3

3x

–3

74.

3x –3

y 4

 –  – –– 2



–1

–– 2

x

3 x

3 4

–3

y 3

–3

72.

y 1

3x

–3

–3

73.

71.

y 3

75.

y 3

76.

y

y

8

5

3x

–3

8 x

–8

5 x

–5

–3 –8

–5

15

Graphs

In Problems 77–88, graph each equation by hand by plotting points. Verify your results using a graphing utility. 77. y = x + 2

78. y = x - 6

79. y = 2x + 8

80. y = 3x - 9

81. y = x - 1

82. y = x - 9

83. y = - x + 4

84. y = - x 2 + 1

85. 2x + 3y = 6

86. 5x + 2y = 10

87. 9x 2 + 4y = 36

88. 4x 2 + y = 4

2

2

2

In Problems 89–96, graph each equation using a graphing utility. Use a graphing utility to approximate the intercepts rounded to two decimal places. Use the TABLE feature to help to establish the viewing window. 89. y = 2x - 13

90. y = - 3x + 14

91. y = 2x 2 - 15

92. y = - 3x 2 + 19

93. 3x - 2y = 43

94. 4x + 5y = 82

95. 5x 2 + 3y = 37

96. 2x 2 - 3y = 35

97. If the point (2, 5) is shifted 3 units right and 2 units down, what are its new coordinates? 98. If the point ( - 1, 6) is shifted 2 units left and 4 units up, what are its new coordinates?

Applications and Extensions 99. The medians of a triangle are the line segments from each vertex to the midpoint of the opposite side (see the figure). Find the lengths of the medians of the triangle with vertices at A = 10, 02, B = 16, 02, and C = 14, 42.

100. An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are 10, 42 and 10, 02, find the third vertex. How many of these triangles are possible?

C Median

s

s

Midpoint s A

B

In Problems 101–104, find the length of each side of the triangle determined by the three points P1 , P2 , and P3 . State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. (An isosceles triangle is one in which at least two of the sides are of equal length.) 101. P1 = 12, 12;

P2 = 1 - 4, 12;

103. P1 = 1 - 2, - 12;

P2 = 10, 72;

P3 = 1 - 4, - 32 P3 = 13, 22

105. Baseball A major league baseball “diamond” is actually a square, 90 feet on a side (see the figure). What is the distance directly from home plate to second base (the diagonal of the square)?

2nd base

90 ft

3rd base

Pitching rubber 1st base

90 ft

Home plate

106. Little League Baseball The layout of a Little League playing field is a square, 60 feet on a side. How far is it directly from home plate to second base (the diagonal of the square)? Source: Little League Baseball, Official Regulations and Playing Rules, 2007. 107. Baseball Refer to Problem 105. Overlay a rectangular coordinate system on a major league baseball diamond so that the origin is at home plate, the positive x-axis lies in

16

102. P1 = 1 - 1, 42; 104. P1 = 17, 22;

P2 = 16, 22; P2 = 1 - 4, 02;

P3 = 14, - 52 P3 = 14, 62

the direction from home plate to first base, and the positive y-axis lies in the direction from home plate to third base. (a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement. (b) If the right fielder is located at 1310, 152, how far is it from there to second base? (c) If the center fielder is located at 1300, 3002, how far is it from there to third base? 108. Little League Baseball Refer to Problem 106. Overlay a rectangular coordinate system on a Little League baseball diamond so that the origin is at home plate, the positive x-axis lies in the direction from home plate to first base, and the positive y-axis lies in the direction from home plate to third base. (a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement. (b) If the right fielder is located at 1180, 202, how far is it from there to second base? (c) If the center fielder is located at 1220, 2202, how far is it from there to third base? 109. Distance between Moving Objects A Ford Focus and a Mack truck leave an intersection at the same time. The Focus heads east at an average speed of 30 miles per hour, while the truck heads south at an average speed of 40 miles per hour. Find an expression for their distance apart d (in miles) at the end of t hours.

Graphs

110. Distance of a Moving Object from a Fixed Point A hot-air balloon, headed due east at an average speed of 15 miles per hour at a constant altitude of 100 feet, passes over an intersection (see the figure). Find an expression for its distance d (measured in feet) from the intersection t seconds later.

112. Net Sales The figure illustrates how net sales of Wal-Mart Stores, Inc., have grown from 2006 through 2010. Use the midpoint formula to estimate the net sales of Wal-Mart Stores, Inc., in 2008. How does your result compare to the reported value of \$374 billion? Source: Wal-Mart Stores, Inc., 2010 Annual Report

Net sales (\$ billions)

Wal-Mart Stores, Inc. Net sales (in \$ billions)

15 mph 100 ft

2006

111. Drafting Error When a draftsman draws three lines that are to intersect at one point, the lines may not intersect as intended and subsequently will form an error triangle. If this error triangle is long and thin, one estimate for the location of the desired point is the midpoint of the shortest side. The figure shows one such error triangle. (a) Find an estimate for the desired intersection point. (b) Find the length of the median for the midpoint found in part (a). See Problem 99. Source: www.uwgb.edu/dutchs/STRUCTGE/s100.htm y

1.3

2007

2008 Year

2009

2010

113. Poverty Threshold Poverty thresholds are determined by the U.S. Census Bureau. A poverty threshold represents the minimum annual household income for a family not to be considered poor. In 2001, the poverty threshold for a family of four with two children under the age of 18 years was \$17,960. In 2011, the poverty threshold for a family of four with two children under the age of 18 years was \$22,350. Assuming poverty thresholds increase in a straight-line fashion, use the midpoint formula to estimate the poverty threshold of a family of four with two children under the age of 18 in 2006. How does your result compare to the actual poverty threshold in 2006 of \$20,444? Source: U.S. Census Bureau

(2.7, 1.7)

114. Completing a Line Segment Plot the points A = ( - 1, 8) and M = (2, 3) in the xy -plane. If M is the midpoint of a line segment AB , find the coordinates of B.

1.7 1.5

405

400 350 309 300 250 200 150 100 50 0

(2.6, 1.5) (1.4, 1.3) 1.4

2.6 2.7

x

Explaining Concepts: Discussion and Writing In Problem 115, you may use a graphing utility, but it is not required. 115. (a) Graph y = 2x 2 , y = x, y = 0 x 0 , and y = 1 1x2 2, noting which graphs are the same. (b) Explain why the graphs of y = 2x 2 and y = 0 x 0 are the same. (c) Explain why the graphs of y = x and y = 1 1x2 2 are not the same. (d) Explain why the graphs of y = 2x 2 and y = x are not the same. 116. Make up an equation satisfied by the ordered pairs 12, 02, 14, 02, and 10, 12. Compare your equation with a friend’s equation. Comment on any similarities.

117. Draw a graph that contains the points 1 - 2, - 12, 10, 12, 11, 32, and 13, 52. Compare your graph with those of other students. Are most of the graphs almost straight lines? How many are “curved”? Discuss the various ways that these points might be connected. 118. Explain what is meant by a complete graph. 119. Write a paragraph that describes a Cartesian plane. Then write a second paragraph that describes how to plot points in the Cartesian plane. Your paragraphs should include the terms “coordinate axes,” “ordered pair,” “coordinates,” “plot,” “ x -coordinate,” and “ y -coordinate.”

‘Are You Prepared?’ Answers 1. 0

2. 8

3. 5

4. 112 + 602 = 612

5.

1 bh 2

6. True

17

Graphs

2 Intercepts; Symmetry; Graphing Key Equations PREPARING FOR THIS SECTION Before getting started, review the following: • Solve Linear Equations

Now Work the ’Are You Prepared?’ problems.

OBJECTIVES 1 Find Intercepts Algebraically from an Equation 2 Test an Equation for Symmetry 3 Know How to Graph Key Equations

1 Find Intercepts Algebraically from an Equation Figure 28 y 5

(4, 0)

(4, 0) 5

(1, 0)

5

x

In Section 1, we discussed how to find intercepts from a graph and how to approximate intercepts from an equation using a graphing utility. Now we discuss how to find intercepts from an equation algebraically. To help understand the procedure, we present Figure 28. From the graph, we can see that the intercepts are (-4, 0) , ( -1, 0), (4, 0), and (0, -3) . The x-intercepts are -4, -1 , and 4. The y-intercept is -3 . Notice that x-intercepts have y-coordinates that equal 0; y-intercepts have x-coordinates that equal 0. This leads to the following procedure for finding intercepts.

(0, 3)

Procedure for Finding Intercepts

5

1. To find the x-intercept(s), if any, of the graph of an equation, let y = 0 in the equation and solve for x, where x is a real number. 2. To find the y-intercept(s), if any, of the graph of an equation, let x = 0 in the equation and solve for y, where y is a real number.

E X A MP L E 1

Finding Intercepts from an Equation Find the x-intercept(s) and the y-intercept(s) of the graph of y = x2 - 4. Then graph y = x2 - 4 by plotting points.

Solution

To find the x-intercept(s), let y = 0 and obtain the equation x2 - 4 = 0 (x + 2)(x - 2) = 0 x+2=0

or

x = -2 or

y = x2 - 4 with y = 0 Factor.

x - 2 = 0 Zero-Product Property x = 2 Solve.

The equation has two solutions, -2 and 2. The x-intercepts are -2 and 2. To find the y-intercept(s), let x = 0 in the equation. y = x2 - 4 = 02 - 4 = -4 The y-intercept is -4. Since x2 Ú 0 (the square of any real number is greater than or equal to 0) for all x, we conclude from the equation y = x2 - 4 that y Ú -4 for all x. This information, the intercepts, and the points from Table 4 enable us to graph y = x2 - 4 by hand. See Figure 29.

18

Graphs

Table 4

Figure 29

x

y  x2  4

-3

( -3) - 4 = 5

-1

-3

( -1, -3)

1

-3

(1, -3)

3

5

(3, 5)

(x, y)

y 5

(– 3, 5)

(3, 5)

( -3, 5)

2

(2, 0)

(– 2, 0)

5 x

–5 (– 1, – 3)

(1, – 3) (0, – 4) –5

Barrett & MacKay/Glow Images

Now Work

PROBLEM

15

2 Test an Equation for Symmetry Another helpful tool for graphing equations by hand involves symmetry, particularly symmetry with respect to the x-axis, the y-axis, and the origin. Symmetry often occurs in nature. Consider the picture of the butterfly. Do you see the symmetry? A graph is said to be symmetric with respect to the x-axis if, for every point (x, y) on the graph, the point (x, -y) is also on the graph. A graph is said to be symmetric with respect to the y-axis if, for every point (x, y) on the graph, the point ( -x, y) is also on the graph. A graph is said to be symmetric with respect to the origin if, for every point (x, y) on the graph, the point ( -x, -y) is also on the graph.

DEFINITION

Figure 30 illustrates the definition. Notice that, when a graph is symmetric with respect to the x-axis, the part of the graph above the x-axis is a reflection or mirror image of the part below it, and vice versa. When a graph is symmetric with respect to the y-axis, the part of the graph to the right of the y-axis is a reflection of the part to the left of it, and vice versa. Symmetry with respect to the origin may be viewed in two ways: 1. As a reflection about the y-axis, followed by a reflection about the x-axis 2. As a projection along a line through the origin so that the distances from the origin are equal Figure 30

y

y

y (–x1, y1)

(x2, y2) (x1, y1)

(x3, y3)

(x1, –y1)

x (x3, –y3)

(–x2, y2)

(x2, y2)

(x2, –y2)

Symmetry with respect to the x-axis

E XAM PL E 2

(x1, y1)

(x1, y1)

x

(x2, y2)

x

(–x2, –y2) (–x1, –y1)

Symmetry with respect to the y-axis

Symmetry with respect to the origin

Symmetric Points (a) If a graph is symmetric with respect to the x-axis and the point (4, 2) is on the graph, then the point (4, -2) is also on the graph. (b) If a graph is symmetric with respect to the y-axis and the point (4, 2) is on the graph, then the point ( -4, 2) is also on the graph. (c) If a graph is symmetric with respect to the origin and the point (4, 2) is on the graph, then the point ( -4, -2) is also on the graph.

Now Work

PROBLEM

23

19

Graphs

When the graph of an equation is symmetric with respect to the x-axis, the y-axis, or the origin, the number of points that you need to plot is reduced. For example, if the graph of an equation is symmetric with respect to the y-axis, then, once points to the right of the y-axis are plotted, an equal number of points on the graph can be obtained by reflecting them about the y-axis. Because of this, before we graph an equation, we first want to determine whether it has any symmetry. The following tests are used for this purpose.

Tests for Symmetry To test the graph of an equation for symmetry with respect to the x-Axis Replace y by -y in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the x-axis. y-Axis Replace x by -x in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the y-axis. Origin Replace x by -x and y by -y in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the origin.

E X A MP L E 3

Finding Intercepts and Testing an Equation for Symmetry For the equation y =

Solution

x2 - 4 : (a) find the intercepts and (b) test for symmetry. x2 + 1

(a) To obtain the x-intercept(s), let y = 0 in the equation and solve for x. x2 x2 + x2 x = -2 or

4 = 0 Let y = 0 1 4 = 0 Multiply both sides by x2 + 1. x = 2 Factor and use the Zero-Product Property.

To obtain the y-intercept(s), let x = 0 in the equation and solve for y. y=

x2 - 4 02 - 4 -4 = 2 = = -4 2 1 x +1 0 +1

The x-intercepts are -2 and 2; the y-intercept is -4. (b) We now test the equation for symmetry with respect to the x-axis, the y-axis, and the origin. x-Axis: To test for symmetry with respect to the x-axis, replace y by - y. x2 - 4 x2 + 1

-y =

is not equivalent to

y=

x2 - 4 x2 + 1

The graph of the equation is not symmetric with respect to the x-axis. y-Axis: To test for symmetry with respect to the y-axis, replace x by - x. y=

( -x)2 - 4 ( -x) + 1 2

=

x2 - 4 x2 + 1

is equivalent to

y=

x2 - 4 x2 + 1

The graph of the equation is symmetric with respect to the y-axis. Origin: To test for symmetry with respect to the origin, replace x by -x and y by -y. -y = -y = 20

( -x)2 - 4 ( -x)2 + 1 x2 - 4 x2 + 1

Replace x by -x and y by -y. Simplify.

Graphs

Since the result is not equivalent to the original equation, the graph of the x2 - 4 is not symmetric with respect to the origin. equation y = 2 x +1 x2 - 4 using a graphing utility. Do you see the symmetry with x2 + 1 respect to the y-axis? Also, did you notice that the point (2, 0) is on the graph along with (-2, 0)? How could we have found the second x-intercept using symmetry?

Seeing the Concept Figure 31

Figure 31 shows the graph of y =

2

5

5

5

Now Work

PROBLEM

49

3 Know How to Graph Key Equations There are certain equations whose graphs we should be able to easily visualize in our mind’s eye. For example, you should know the graph of y = x2 discussed in Example 8 in Section 1. The next three examples use intercepts, symmetry, and point plotting to obtain the graphs of additional key equations. It is important to know the graphs of these key equations.

Graphing the Equation y  x 3 by Finding Intercepts and Checking for Symmetry

E XAM PL E 4

Graph the equation y = x3 by hand by plotting points. Find any intercepts and check for symmetry first.

Solution Figure 32

Replace y by -y. Since -y = x3 is not equivalent to y = x3, the graph is not symmetric with respect to the x-axis. y-Axis: Replace x by -x. Since y = ( -x)3 = -x3 is not equivalent to y = x3, the graph is not symmetric with respect to the y-axis. Origin: Replace x by -x and y by -y. Since -y = ( -x)3 = -x3 is equivalent to y = x3 (multiply both sides by -1 ), the graph is symmetric with respect to the origin. x-Axis:

y 8

(0, 0)

6

(1, 1)

(2, 8)

8

First, find the intercepts of y = x3 . When x = 0, then y = 0; and when y = 0, then x = 0. The origin (0, 0) is the only intercept. Now test y = x3 for symmetry.

(2, 8)

(1, 1) 6

x

To graph by hand, use the equation to obtain several points on the graph. Because of the symmetry with respect to the origin, we only need to locate points on the graph for which x Ú 0. See Table 5. Points on the graph could also be obtained using the TABLE feature on a graphing utility. See Table 6. Do you see the symmetry with respect to the origin from the table? Figure 32 shows the graph.

Table 5

Table 6

x

y  x3

(x, y)

0

0

(0, 0)

1

1

(1, 1)

2

8

(2, 8)

3

27

(3, 27)

21

Graphs

E X A MP L E 5

Graphing the Equation x  y 2 (a) Graph the equation x = y 2. Find any intercepts and check for symmetry first. (b) Graph x = y 2 where y Ú 0 .

Solution Table 7 y

x  y2

(x, y)

0

0

(0, 0)

1

1

(1, 1)

2

4

(4, 2)

3

9

(9, 3)

Figure 33 y 6

(a) The lone intercept is (0, 0). The graph is symmetric with respect to the x-axis since x = ( - y)2 is equivalent to x = y 2. The graph is not symmetric with respect to the y-axis or the origin. To graph x = y 2 by hand, use the equation to obtain several points on the graph. Because the equation is solved for x, it is easier to assign values to y and use the equation to determine the corresponding values of x. Because of the symmetry, we can restrict ourselves to points whose y-coordinates are non-negative. Then use the symmetry to find additional points on the graph. See Table 7. For example, since (1, 1) is on the graph, so is (1, -1). Since (4, 2) is on the graph, so is (4, -2), and so on. Plot these points and connect them with a smooth curve to obtain Figure 33. To graph the equation x = y 2 using a graphing utility, write the equation in the form y = {expression in x}. We proceed to solve for y.

(9, 3) (1, 1)

x = y2 y2 = x y = { 1x Square Root Method

(4, 2)

(0, 0)

2 (1, 1)

5 (4, 2)

10 x

To graph x = y 2, graph both Y1 = 1x and Y2 = - 1x on the same screen. Figure 34 shows the result. Table 8 shows various values of y for a given value of x when Y1 = 1x and Y2 = - 1x. Notice that when x 6 0 we get an error. Can you explain why?

(9, 3)

Figure 34

Y1  x

6

2

Table 8

10

6

Y2   x

(b) If we restrict y so that y Ú 0, the equation x = y 2, y Ú 0, may be written as y = 1x. The portion of the graph of x = y 2 in quadrant I plus the origin is the graph of y = 1x. See Figure 35. y 6

Figure 35 6

Y1  x

(1, 1)

(4, 2)

(9, 3)

(0, 0)

2

10

2

5

10 x

6

E X A MP L E 6

Solution 22

Graphing the Equation y 

1 x

1 Graph the equation y = . Find any intercepts and check for symmetry first. x Check for intercepts first. If we let x = 0, we obtain a 0 in the denominator, which is not defined. We conclude that there is no y-intercept. If we let y = 0, we get the

Graphs

1 = 0, which has no solution. We conclude that there is no x-intercept. The x 1 graph of y = does not cross or touch the coordinate axes. x Next check for symmetry. equation

Table 9 x 1 10 1 3

y  10

1 x

(x, y) a

1 , 10b 10

3

1 a , 3b 3

2

1 a , 2b 2

1

1

(1, 1)

2

1 2

1 a2, b 2

3

1 3

1 a3, b 3

10

1 10

1 2

1 1 , which is not equivalent to y = . x x 1 1 1 y-Axis: Replacing x by - x yields y = = - , which is not equivalent to y = . -x x x 1 Origin: Replacing x by - x and y by - y yields -y = - , which is equivalent x 1 to y = . The graph is symmetric only with respect to the origin. x Use the equation to form Table 9 and obtain some points on the graph. Because of symmetry, we only find points (x, y) for which x is positive. From 1 Table 9 we infer that, if x is a large and positive number, then y = is a positive x number close to 0. We also infer that if x is a positive number close to 0 then 1 y = is a large and positive number. Armed with this information, we can graph x 1 the equation. Figure 36 illustrates some of these points and the graph of y = . x Observe how the absence of intercepts and the existence of symmetry with respect to the origin were utilized. Figure 37 confirms our algebraic analysis using a TI-84 Plus. x-Axis:

a10,

1 b 10

Replacing y by -y yields -y =

Figure 36

Figure 37

y

Y1  1x

3

4

(––12 , 2) (1, 1) 3

(2, ––12 ) 3

3

3

x

(2,  ––12 )

(1, 1)

( ––12 , 2)

4 3

2 Assess Your Understanding ’Are You Prepared?’

Answers are given at the end of these exercises.

1. Solve: 2 1x + 32 - 1 = - 7

2. Solve: x 2 - 4x - 12 = 0

Concepts and Vocabulary 3. The points, if any, at which a graph crosses or touches the coordinate axes are called ___________. 4. The x-intercepts of the graph of an equation are those x-values for which ___________.

7. If the graph of an equation is symmetric with respect to the origin and (3, - 4) is a point on the graph, then ________ is also a point on the graph. 8. True or False To find the y-intercepts of the graph of an equation, let x = 0 and solve for y.

5. If for every point (x, y) on the graph of an equation the point ( - x, y) is also on the graph, then the graph is symmetric with respect to the ___________.

9. True or False The y-coordinate of a point at which the graph crosses or touches the x-axis is an x-intercept.

6. If the graph of an equation is symmetric with respect to the y-axis and - 4 is an x-intercept of this graph, then ________ is also an x-intercept.

10. True or False If a graph is symmetric with respect to the x-axis, then it cannot be symmetric with respect to the y-axis.

23

Graphs

Skill Building In Problems 11–22, find the intercepts and graph each equation by plotting points. Be sure to label the intercepts. 11. y = x + 2

12. y = x - 6

13. y = 2x + 8

14. y = 3x - 9

15. y = x - 1

16. y = x - 9

17. y = - x + 4

18. y = - x 2 + 1

19. 2x + 3y = 6

20. 5x + 2y = 10

21. 9x + 4y = 36

22. 4x 2 + y = 4

2

2

2

2

In Problems 23–32, plot each point. Then plot the point that is symmetric to it with respect to (a) the x-axis; (b) the y-axis; (c) the origin. 23. (3, 4)

24. (5, 3)

25. ( - 2, 1)

26. (4, - 2)

27. (5, - 2)

28. ( - 1, - 1)

29. ( - 3, - 4)

30. (4, 0)

31. (0, - 3)

32. ( - 3, 0)

In Problems 33–40, the graph of an equation is given. (a) Find the intercepts. (b) Indicate whether the graph is symmetric with respect to the x-axis, the y-axis, or the origin. 33.

34.

y 3

3x

–3

36.

y

y 4

1    ––

3x

3

2



–– 2

1

x

3 x

3 4

3

–3

37.

35.

y 3

38.

y 3

3x

–3

39.

y 3

–3

y 40

x

3x

3

3x

–3

40.

y 6

6

6

6

–3

40

In Problems 41–44, draw a complete graph so that it has the type of symmetry indicated. 41. y-axis

42. x-axis

y

43. Origin

y 5

9

(0, 2)

4

(5, 3)

(–4, 0)

–9

(0, 0) 5x

–5

9x (2, –5)

–9

44. y-axis y (0, 4) 4

y

–5

 , 2) (–– 2

(2, 2)

(, 0) x

(0, 0)

–2

–4

(0, –9)

3x

–3

In Problems 45–60, list the intercepts and test for symmetry. 45. y 2 = x + 4

46. y 2 = x + 9

3 47. y = 1 x

5 48. y = 1 x

49. y = x 4 - 8x 2 - 9

50. y = x 4 - 2x 2 - 8

51. 9x 2 + 4y 2 = 36

52. 4x 2 + y 2 = 4

53. y = x 3 - 27

54. y = x 4 - 1

55. y = x 2 - 3x - 4

56. y = x 2 + 4

57. y =

3x x + 9 2

58. y =

x2 - 4 2x

59. y =

- x3 x - 9 2

60. y =

x4 + 1 2x 5

In Problems 61–64, draw a quick sketch of each equation.

65. If (3, b) is a point on the graph of y = 4x + 1, what is b?

1 x 66. If ( - 2, b) is a point on the graph of 2x + 3y = 2, what is b?

67. If (a, 4) is a point on the graph of y = x 2 + 3x, what is a?

68. If (a, - 5) is a point on the graph of y = x 2 + 6x, what is a?

61. y = x 3

24

62. x = y 2

63. y = 1x

64. y =

Graphs

Mixed Practice In Problems 69–76, (a) find the intercepts of each equation, (b) test each equation for symmetry with respect to the x-axis, the y-axis, and the origin, and (c) graph each equation by hand by plotting points. Be sure to label the intercepts on the graph and use any symmetry to assist in drawing the graph. Verify your results using a graphing utility. 69. y = x 2 - 5

70. y = x 2 - 8

71. x - y 2 = - 9

72. x + y 2 = 4

73. x 2 + y 2 = 9

74. x 2 + y 2 = 16

75. y = x 3 - 4x

76. y = x 3 - x

Applications and Extensions

78. If the graph of an equation is symmetric with respect to the y-axis and 6 is an x-intercept of this graph, name another x-intercept.

82. Solar Energy The solar electric generating systems at Kramer Junction, California, use parabolic troughs to heat a heat-transfer fluid to a high temperature. This fluid is used to generate steam that drives a power conversion system to produce electricity. For troughs 7.5 feet wide, an equation for the cross-section is 16y 2 = 120x - 225 . Department of Energy (DOE) Digital Photo Archive

77. Given that the point (1, 2) is on the graph of an equation that is symmetric with respect to the origin, what other point is on the graph?

79. If the graph of an equation is symmetric with respect to the origin and - 4 is an x-intercept of this graph, name another x-intercept. 80. If the graph of an equation is symmetric with respect to the x-axis and 2 is a y-intercept, name another y-intercept. 81. Microphones In studios and on stages, cardioid microphones are often preferred for the richness they add to voices and for their ability to reduce the level of sound from the sides and rear of the microphone. Suppose one such cardioid pattern is given by the equation (x 2 + y 2 - x)2 = x 2 + y 2. (a) Find the intercepts of the graph of the equation. (b) Test for symmetry with respect to the x-axis, y-axis, and origin. Source: www.notaviva.com

(a) Find the intercepts of the graph of the equation. (b) Test for symmetry with respect to the x-axis, y-axis, and origin. Source: U.S. Department of Energy

Explaining Concepts: Discussion and Writing 83. Draw a graph of an equation that contains two x-intercepts; at one the graph crosses the x-axis, and at the other the graph touches the x-axis. 84. Make up an equation with the intercepts (2, 0), (4, 0) , and (0, 1) . Compare your equation with a friend’s equation. Comment on any similarities. 85. Draw a graph that contains the points (0, 1), (1, 3) , and (3, 5) and is symmetric with respect to the x-axis. Compare your graph with those of other students. Are most of the graphs almost straight lines? How many are “curved”? Discuss the various ways that these points might be connected.

86. An equation is being tested for symmetry with respect to the x-axis, the y-axis, and the origin. Explain why, if two of these symmetries are present, the remaining one must also be present. 87. Draw a graph that contains the points ( - 2, 5) , ( - 1, 3) , and (0, 2) that is symmetric with respect to the y-axis. Compare your graph with those of other students; comment on any similarities. Can a graph contain these points and be symmetric with respect to the x-axis? The origin? Why or why not?

Interactive Exercises Ask your instructor if the applets below are of interest to you. 88. y-axis Symmetry Open the y-axis symmetry applet. Move point A around the Cartesian plane with your mouse. How are the coordinates of point A and the coordinates of point B related? 89. x-axis Symmetry Open the x-axis symmetry applet. Move point A around the Cartesian plane with your mouse. How

are the coordinates of point A and the coordinates of point B related? 90. Origin Symmetry Open the origin symmetry applet. Move point A around the Cartesian plane with your mouse. How are the coordinates of point A and the coordinates of point B related?

’Are You Prepared?’ Answers 1. 5 - 6 6

2. 5 - 2, 6 6

25

Graphs

3 Solving Equations Using a Graphing Utility PREPARING FOR THIS SECTION Before getting started, review the following: • Solve Linear, Quadratic, and Rational Equations Now Work the ‘Are You Prepared?’ problems.

OBJECTIVE 1 Solve Equations Using a Graphing Utility

In this text, we present two methods for solving equations: algebraic and graphical. Some equations can be solved using algebraic techniques that result in exact solutions. For other equations, however, there are no algebraic techniques that lead to an exact solution. For such equations, a graphing utility can often be used to investigate possible solutions. When a graphing utility is used to solve an equation, usually approximate solutions are obtained. One goal of this text is to determine when equations can be solved algebraically. If an algebraic method for solving an equation exists, we shall use it to obtain an exact solution. A graphing utility can then be used to support the algebraic result. However, if no algebraic techniques are available to solve an equation, a graphing utility will be used to obtain approximate solutions.

1 Solve Equations Using a Graphing Utility When a graphing utility is used to solve an equation, usually approximate solutions are obtained. Unless otherwise stated, we shall follow the practice of giving approximate solutions as decimals rounded to two decimal places. The ZERO (or ROOT) feature of a graphing utility can be used to find the solutions of an equation when one side of the equation is 0. In using this feature to solve equations, we make use of the fact that when the graph of an equation in two variables, x and y, crosses or touches the x-axis then y = 0. For this reason, any value of x for which y = 0 will be a solution to the equation. That is, solving an equation for x when one side of the equation is 0 is equivalent to finding where the graph of the corresponding equation in two variables crosses or touches the x-axis.

E X A M PL E 1

Using ZERO (or ROOT) to Approximate Solutions of an Equation Find the solution(s) of the equation x3 - x + 1 = 0. Round answers to two decimal places.

Solution

26

The solutions of the equation x3 - x + 1 = 0 are the same as the x -intercepts of the graph of Y1 = x3 - x + 1 . Begin by graphing Y1 . Figure 38 shows the graph. From the graph there appears to be one x-intercept (solution to the equation) between -2 and -1. Using the ZERO (or ROOT) feature of a graphing utility, we determine that the x-intercept, and thus the solution to the equation, is x = -1.32 rounded to two decimal places. See Figure 39.

Graphs

NOTE Graphing utilities use a process in which they search for a solution until the answer is found within a certain tolerance level (such as within 0.0001).Therefore, the y-coordinate may sometimes be a nonzero value such as 1.1527E@8, which is 1.1527 * 10 - 8 , very close to zero. 

Figure 38

Figure 39

10

–3

3

10

–3

–5

Now Work

PROBLEM

3

–5

5

A second method for solving equations using a graphing utility involves the INTERSECT feature of the graphing utility. This feature is used most effectively when neither side of the equation is 0.

E XAM PL E 2

Using INTERSECT to Approximate Solutions of an Equation Find the solution(s) to the equation 4x4 - 3 = 2x + 1. Round answers to two decimal places.

Solution

Begin by graphing each side of the equation as follows: graph Y1 = 4x4 - 3 and Y2 = 2x + 1. See Figure 40. At a point of intersection of the graphs, the value of the y-coordinate is the same for Y1 and Y2. Thus, the x-coordinate of the point of intersection represents a solution to the equation. Do you see why? The INTERSECT feature on a graphing utility determines a point of intersection of the graphs. Using this feature, we find that the graphs intersect at 1 -0.87, -0.732 and 11.12, 3.232 rounded to two decimal places. See Figure 41(a) and (b). The solutions of the equation are x = -0.87 and x = 1.12 rounded to two decimal places. Figure 41

Figure 40

10

10

–4

4

10

–4

4 –4

–10 (a)

–10

Now Work

PROBLEM

4

–10 (b)

7

SUMMARY Steps for Approximating Solutions of Equations Using ZERO (or ROOT) STEP 1: Write the equation in the form 5 expression in x 6 = 0. STEP 2: Graph Y1 = 5 expression in x 6 . STEP 3: Use ZERO (or ROOT) to determine each x-intercept of the graph.

Steps for Approximating Solutions of Equations Using INTERSECT STEP 1: Graph Y1 = 5 expression in x on the left side of the equation 6 Y2 = 5 expression in x on the right side of the equation 6 STEP 2: Use INTERSECT to determine the x-coordinate of each point of intersection.

27

Graphs

E X A MP L E 3

Solving an Equation Algebraically and Graphically Solve the equation: 31x - 22 = 51x - 12

Algebraic Solution 31x - 22 3x - 6 3x - 6 - 5x -2x - 6 -2x - 6 + 6 -2x -2 x -2

Graphing Solution Graph Y1 = 31x - 22 and Y2 = 51x - 12. Using INTERSECT, we find the point of intersection to be 1 -0.5, -7.52. See Figure 42. The solution of the equation is x = -0.5.

= = = = = =

51x - 12 5x - 5 Distribute. 5x - 5 - 5x Subtract 5x from each side. Simplify. -5 -5 + 6 Add 6 to each side. 1 Simplify. 1 Divide each side by - 2. = -2 1 Simplify. x= 2 1 Check: Let x = - in the expression in x on the left side of the 2 1 equation and simplify. Let x = - in the expression in x 2 on the right side of the equation and simplify. If the two expressions are equal, the solution checks. 31x - 22 = 3a -

Figure 42

5 –5

5

–15

1 5 15 - 2b = 3a - b = 2 2 2

1 3 15 - 1b = 5a - b = 2 2 2 Since the two expressions are equal, the solution 1 x = - checks. 2 51x - 12 = 5a -

Now Work

PROBLEM

19

3 Assess Your Understanding ‘Are You Prepared?’

Answers are given at the end of these exercises.

1. Solve the equation 2x + 5x + 2 = 0.

2. Solve the equation 2x + 3 = 4 1x - 12 + 1.

2

Concepts and Vocabulary 3. To solve an equation of the form 5 expression in x 6 = 0 using a graphing utility, we graph Y1 = 5 expression in x 6 and use ________ to determine each x-intercept of the graph.

4. True or False In using a graphing utility to solve an equation, exact solutions are always obtained.

Skill Building In Problems 5–16, use a graphing utility to approximate the real solutions, if any, of each equation rounded to two decimal places. All solutions lie between - 10 and 10. 5. x 3 - 4x + 2 = 0 8. - x 4 + 1 = 2x 2 - 3 5 7 11. - x 3 - x 2 + x + 2 = 0 3 2 1 3 1 2 14. x - 5x = x - 4 4 5

6. x 3 - 8x + 1 = 0 9. x 4 - 2x 3 + 3x - 1 = 0 7 15 12. - x 4 + 3x 3 + x 2 x + 2 = 0 3 2

7. - 2x 4 + 5 = 3x - 2 10. 3x 4 - x 3 + 4x 2 - 5 = 0 5 1 2 2 13. - x 4 - 2x 3 + x = - x 2 + 3 2 3 2

15. x 4 - 5x 2 + 2x + 11 = 0

16. - 3x 4 + 8x 2 - 2x - 9 = 0

In Problems 17–36, solve each equation algebraically. Verify your solution using a graphing utility. 17. 2 13 + 2x2 = 3 1x - 42

28

18. 3 12 - x2 = 2x - 1

19. 8x - 12x + 12 = 3x - 13

Graphs

20. 5 - 12x - 12 = 10 - x 23.

5 4 + = 3 y y

21.

x + 1 x + 2 + = 5 3 7

22.

24.

4 18 - 5 = y 2y

25. 1x + 72 1x - 12 = 1x + 12 2

27. x 2 - 3x - 28 = 0

26. 1x + 22 1x - 32 = 1x - 32 2

2x + 1 + 16 = 3x 3

28. x 2 - 7x - 18 = 0

29. 3x 2 = 4x + 4

30. 5x 2 = 13x + 6

31. x 3 + x 2 - 4x - 4 = 0

33. 2x + 1 = 4

34. 2x - 2 = 3

35.

2 3 -8 + = x + 2 x - 1 5

32. x 3 + 2x 2 - 9x - 18 = 0 36.

1 5 21 = x + 1 x - 4 4

‘Are You Prepared?’ Answers 1 1. e - 2, - f 2

2. 5 3 6

4 Lines OBJECTIVES 1 Calculate and Interpret the Slope of a Line 2 3 4 5 6 7 8 9 10

Graph Lines Given a Point and the Slope Find the Equation of a Vertical Line Use the Point–Slope Form of a Line; Identify Horizontal Lines Find the Equation of a Line Given Two Points Write the Equation of a Line in Slope–Intercept Form Identify the Slope and y-Intercept of a Line from Its Equation Graph Lines Written in General Form Using Intercepts Find Equations of Parallel Lines Find Equations of Perpendicular Lines

In this section we study a certain type of equation that contains two variables, called a linear equation, and its graph, a line.

1 Calculate and Interpret the Slope of a Line Figure 43 Line

Rise Run

DEFINITION

Consider the staircase illustrated in Figure 43. Each step contains exactly the same horizontal run and the same vertical rise. The ratio of the rise to the run, called the slope, is a numerical measure of the steepness of the staircase. For example, if the run is increased and the rise remains the same, the staircase becomes less steep. If the run is kept the same, but the rise is increased, the staircase becomes more steep. This important characteristic of a line is best defined using rectangular coordinates. Let P = (x1 , y1) and Q = (x2 , y2) be two distinct points. If x1 ⬆ x2 , the slope m of the nonvertical line L containing P and Q is defined by the formula m=

y2 - y1 x2 - x1

x1 ⬆ x2

(1)

If x1 = x2 , L is a vertical line and the slope m of L is undefined (since this results in division by 0).

Figure 44(a) on the next page provides an illustration of the slope of a nonvertical line; Figure 44(b) illustrates a vertical line. 29

Graphs

Figure 44

L L

Q = (x 2, y2) y2

Q = (x 1, y2)

y1

P = (x 1, y1)

Rise = y2 – y1

P = (x 1, y1) y1

y2

Run = x2 – x1 x1

x2

(a) Slope of L is m =

x1

y2 – y1 _______ x2 – x1

(b) Slope is undefined; L is vertical

As Figure 44(a) illustrates, the slope m of a nonvertical line may be viewed as

In Words

The symbol  is the Greek letter delta. In mathematics, we read y is read,  as “change in,” so x “change in y divided by change in x.”

m=

y2 - y1 Rise = x2 - x1 Run

y2 - y1 Change in y y = = x2 - x1 Change in x x

or m =

That is, the slope m of a nonvertical line measures the amount y changes when x y changes from x1 to x2 . The expression is called the average rate of change of y, x with respect to x. Two comments about computing the slope of a nonvertical line may prove helpful: 1. Any two distinct points on the line can be used to compute the slope of the line. (See Figure 45 for justification.) Figure 45 Triangles ABC and PQR are similar (equal angles), so ratios of corresponding sides are equal. Then y2 - y1 Slope using P and Q = = x2 - x1 d(B, C) = Slope using A and B d(A, C) A

y

Q = (x 2, y2) y2 – y1

P = (x 1, y1) B

x2 – x1

R x

C

Since any two distinct points can be used to compute the slope of a line, the average rate of change of a line is always the same number. 2. The slope of a line may be computed from P = (x1 , y1) to Q = (x2 , y2) or from Q to P because y2 - y1 y1 - y2 = x2 - x1 x1 - x2

E X A MP L E 1

Finding and Interpreting the Slope of a Line Given Two Points The slope m of the line containing the points (1, 2) and (5, -3) may be computed as m=

-3 - 2 -5 5 = = 5-1 4 4

or as

m=

2 - ( -3) 5 5 = = 1-5 -4 4

For every 4-unit change in x, y will change by -5 units . That is, if x increases by 4 units, then y will decrease by 5 units. The average rate of change of y with respect 5 to x is - . 4

Now Work 30

PROBLEMS

11

AND

17

Graphs

Figure 46

Square Screens 4

(4, 4)

4

4

4

(4, 4)

To get an undistorted view of slope, the same scale must be used on each axis. However, most graphing utilities have a rectangular screen. Because of this, using the same interval for both x and y will result in a distorted view. For example, Figure 46 shows the graph of the line y = x connecting the points ( -4, -4) and (4, 4). We expect the line to bisect the first and third quadrants, but it doesn’t. We need to adjust the selections for Xmin, Xmax, Ymin, and Ymax so that a square screen results. On most graphing utilities, this is accomplished by setting the ratio of x to y at 3 : 2. * Figure 47 shows the graph of the line y = x on a square screen using a TI-84 Plus. Notice that the line now bisects the first and third quadrants. Compare this illustration to Figure 46. Figure 47 4

6

(4, 4)

6

(4, 4)

4

To get a better idea of the meaning of the slope m of a line, consider the following:

Exploration

On the same square screen, graph the following equations:

Figure 48

Y1 = 0

Y6  6x Y5  2x Y4  x Y3  12 x Y2  14 x

2

3

3 Y1  0

Y2 =

1 x 4

Slope of line is

1 . 4

Y3 =

1 x 2

Slope of line is

1 . 2

Y4 = x

Slope of line is 1.

Y5 = 2x

Slope of line is 2.

Y6 = 6x

Slope of line is 6.

See Figure 48.

2

Exploration

On the same square screen, graph the following equations:

Figure 49 Y6  6x Y5  2x Y4  x Y3   12 x Y2   14 x

Slope of line is 0.

2

3

3

Y1  0 2

Y1 = 0

Slope of line is 0.

1 Y2 = - x 4

Slope of line is -

1 . 4

1 Y3 = - x 2

Slope of line is -

1 . 2

Y4 = - x

Slope of line is - 1.

Y5 = - 2x

Slope of line is - 2.

Y6 = - 6x

Slope of line is - 6.

See Figure 49.

*Most graphing utilities have a feature that automatically squares the viewing window. Consult your owner’s manual for the appropriate keystrokes.

31

Graphs

Figures 48 and 49 illustrate the following facts: 1. When the slope of a line is positive, the line slants upward from left to right. 2. When the slope of a line is negative, the line slants downward from left to right. 3. When the slope is 0, the line is horizontal. Figures 48 and 49 also illustrate that the closer the line is to the vertical position, the greater the magnitude of the slope. So, a line with slope 6 is steeper than a line whose slope is 3.

2 Graph Lines Given a Point and the Slope E X A M PL E 2

Graphing a Line Given a Point and a Slope Draw a graph of the line that contains the point (3, 2) and has a slope of: 3 4 (a) (b) 4 5

Solution Figure 50 y 6 (7, 5) Rise = 3 (3, 2)

Rise 3 . The fact that the slope is means that for every horizontal Run 4 movement (run) of 4 units to the right there will be a vertical movement (rise) of 3 units. If we start at the given point (3, 2) and move 4 units to the right and 3 units up, we reach the point (7, 5). By drawing the line through this point and the point (3, 2), we have the graph. See Figure 50.

(a) Slope =

(b) The fact that the slope is

Run = 4 –2

-

10 x

5

means that for every horizontal movement of 5 units to the right there will be a corresponding vertical movement of -4 units (a downward movement of 4 units). If we start at the given point (3, 2) and move 5 units to the right and then 4 units down, we arrive at the point (8, -2). By drawing the line through these points, we have the graph. See Figure 51. Alternatively, we can set

Figure 51 y (–2, 6) 6 Rise = 4

-

(3, 2) Run = 5 Run = –5

Rise = – 4 10 x

–2 –2

4 -4 Rise = = 5 5 Run

(8, –2)

4 4 Rise = = 5 -5 Run

so that for every horizontal movement of -5 units (a movement to the left of 5 units) there will be a corresponding vertical movement of 4 units (upward). This approach brings us to the point ( -2, 6), which is also on the graph shown in Figure 51.

Now Work

PROBLEM

23

3 Find the Equation of a Vertical Line E X A M PL E 3

Graphing a Line Graph the equation: x = 3

Solution

32

To graph x = 3 by hand, find all points (x, y) in the plane for which x = 3. No matter what y-coordinate is used, the corresponding x-coordinate always equals 3. Consequently, the graph of the equation x = 3 is a vertical line with x-intercept 3 and undefined slope. See Figure 52(a).

Graphs

To use a graphing utility, we need to express the equation in the form y = 5expression in x6. But x = 3 cannot be put into this form, so an alternative method must be used. Consult your manual to determine the methodology required to draw vertical lines. Figure 52(b) shows the graph that you should obtain.

y

Figure 52 x= 3

4 (3, 3) x3

5

(3, 2) (3, 1) 1 1

2

5 x

(3, 0) (3, 1)

6

5 (b)

(a)

As suggested by Example 3, we have the following result:

THEOREM

Equation of a Vertical Line A vertical line is given by an equation of the form x=a where a is the x-intercept.

Figure 53

4 Use the Point–Slope Form of a Line; Identify Horizontal Lines

y

L (x, y) y – y1

(x 1, y1)

Let L be a nonvertical line with slope m and containing the point (x1, y1) . See Figure 53. For any other point (x, y) on L, we have

x – x1

m = x

THEOREM

y - y1 x - x1

or

y - y1 = m(x - x1)

Point–Slope Form of an Equation of a Line An equation of a nonvertical line with slope m that contains the point (x1, y1) is y - y1 = m(x - x1)

E XAM PL E 4 Figure 54 y 6

Using the Point–Slope Form of a Line An equation of the line with slope 4 and containing the point (1, 2) can be found by using the point–slope form with m = 4, x1 = 1 , and y1 = 2 .

(2, 6)

y - y1 = m(x - x1) Rise  4

y - 2 = 4(x - 1)

(1, 2)

y = 4x - 2

Run  1 2

(2)

5

10 x

m = 4, x1 = 1, y1 = 2 Solve for y.

See Figure 54 for the graph.

Now Work

PROBLEM

45

33

Graphs

E X A MP L E 5

Finding the Equation of a Horizontal Line Find an equation of the horizontal line containing the point (3, 2) .

Solution Figure 55

y - y1 = m(x - x1)

y

y - 2 = 0 # (x - 3)

4 (3, 2)

–1

Because all the y-values are equal on a horizontal line, the slope of a horizontal line is 0. To get an equation, use the point–slope form with m = 0, x1 = 3 , and y1 = 2 .

1

m = 0, x1 = 3, and y1 = 2

y-2=0 y=2

5 x

3

See Figure 55 for the graph. As suggested by Example 5, we have the following result:

THEOREM

Equation of a Horizontal Line A horizontal line is given by an equation of the form y=b where b is the y-intercept.

5 Find the Equation of a Line Given Two Points E X A MP L E 6

Finding an Equation of a Line Given Two Points Find an equation of the line containing the points (2, 3) and (-4, 5) . Graph the line.

Solution

First compute the slope of the line. m=

5-3 2 1 = = -4 - 2 -6 3

1 Use the point (2, 3) and the slope m = - to get the point–slope form of the 3 equation of the line.

Figure 56 y (–4, 5)

1 y - 3 = - (x - 2) 3

(2, 3) 2 –4

–2

10

x

See Figure 56 for the graph. In the solution to Example 6, we could have used the other point, ( -4, 5) , instead of the point (2, 3) . The equation that results, although it looks different, is equivalent to the equation obtained in the example. (Try it for yourself.)

Now Work

PROBLEM

37

6 Write the Equation of a Line in Slope–Intercept Form Another useful equation of a line is obtained when the slope m and y-intercept b are known. In this event, we know both the slope m of the line and the point (0, b) on the line; then use the point–slope form, equation (2), to obtain the following equation: y - b = m(x - 0) or y = mx + b 34

Graphs

THEOREM

Slope–Intercept Form of an Equation of a Line An equation of a line with slope m and y-intercept b is y = mx + b

Now Work

PROBLEM

51 (EXPRESS

(3)

IN SLOPE–INTERCEPT FORM)

Figure 57

y = mx + 2

Y5  3x  2 Y3  x  2 4

Seeing the Concept To see the role that the slope m plays, graph the following lines on the same screen.

Y4  3x  2 Y2  x  2

Y1 = 2 Y2 = x + 2

Y1  2 6

Y3 = - x + 2

6

Y4 = 3x + 2 Y5 = - 3x + 2

Figure 58

4

See Figure 57. What do you conclude about the lines y = mx + 2?

y = 2x + b

Seeing the Concept

Y4  2x  4 4

Y2  2x  1 Y1  2x Y3  2x  1 Y5  2x  4

To see the role of the y-intercept b, graph the following lines on the same screen.

Y1 = 2x Y2 = 2x + 1 Y3 = 2x - 1 Y4 = 2x + 4

6

6

Y5 = 2x - 4 See Figure 58. What do you conclude about the lines y = 2x + b?

4

7 Identify the Slope and y-Intercept of a Line from Its Equation When the equation of a line is written in slope–intercept form, it is easy to find the slope m and y-intercept b of the line. For example, suppose that the equation of a line is y = -2x + 7 Compare it to y = mx + b . y = -2x + 7

c

c

y = mx + b The slope of this line is -2 and its y-intercept is 7.

Now Work

E XAM PL E 7

PROBLEM

71

Finding the Slope and y-Intercept Find the slope m and y-intercept b of the equation 2x + 4y = 8 . Graph the equation.

Solution

To obtain the slope and y-intercept, write the equation in slope–intercept form by solving for y. 35

Graphs

2x + 4y = 8

Figure 59

4y = -2x + 8 1 y= - x+2 2

y 4 2

(0, 2)

1 (2, 1)

–3

x

3

y = mx + b

1 The coefficient of x, - , is the slope, and the y-intercept is 2. Graph the line using the 2 1 fact that the y-intercept is 2 and the slope is - . Then, starting at the point (0, 2), go to 2 the right 2 units and then down 1 unit to the point (2, 1) . See Figure 59.

Now Work

PROBLEM

77

8 Graph Lines Written in General Form Using Intercepts Refer to Example 7. The form of the equation of the line 2x + 4y = 8 is called the general form.

DEFINITION

The equation of a line is in general form* when it is written as Ax + By = C

(4)

where A, B, and C are real numbers and A and B are not both 0. If B = 0 in (4), then A ⬆ 0 and the graph of the equation is a vertical line: C x = . If B ⬆ 0 in (4), then we can solve the equation for y and write the equation A in slope–intercept form as we did in Example 7. Another approach to graphing equation (4) would be to find its intercepts. Remember, the intercepts of the graph of an equation are the points where the graph crosses or touches a coordinate axis.

E X A MP L E 8

Graphing an Equation in General Form Using Its Intercepts Graph the equation 2x + 4y = 8 by finding its intercepts.

Solution

To obtain the x-intercept, let y = 0 in the equation and solve for x. 2x + 4y = 8 2x + 4(0) = 8

Let y = 0.

2x = 8 x=4

Divide both sides by 2.

The x-intercept is 4 and the point (4, 0) is on the graph of the equation. To obtain the y-intercept, let x = 0 in the equation and solve for y. 2x + 4y = 8 2(0) + 4y = 8

Let x = 0.

4y = 8 y=2

Figure 60 y 4

The y-intercept is 2 and the point (0, 2) is on the graph of the equation. Plot the points (4, 0) and (0, 2) and draw the line through the points. See Figure 60.

(0, 2) (4, 0) –3

3

Now Work x

PROBLEM

91

*Some books use the term standard form.

36

Divide both sides by 4.

Graphs

Every line has an equation that is equivalent to an equation written in general form. For example, a vertical line whose equation is x=a can be written in the general form

1#x + 0#y = a

A = 1, B = 0, C = a

A horizontal line whose equation is y=b can be written in the general form

0 #x + 1#y = b

A = 0, B = 1, C = b

Lines that are neither vertical nor horizontal have general equations of the form Ax + By = C

A ⬆ 0 and B ⬆ 0

Because the equation of every line can be written in general form, any equation equivalent to equation (4) is called a linear equation.

Figure 61 y

9 Find Equations of Parallel Lines Rise Run

Rise Run x

THEOREM

When two lines (in the plane) do not intersect (that is, they have no points in common), they are said to be parallel. Look at Figure 61. There we have drawn two parallel lines and have constructed two right triangles by drawing sides parallel to the coordinate axes. The right triangles are similar. (Do you see why? Two angles are equal.) Because the triangles are similar, the ratios of corresponding sides are equal.

Criterion for Parallel Lines Two nonvertical lines are parallel if and only if their slopes are equal and they have different y-intercepts. The use of the words “if and only if” in the preceding theorem means that actually two statements are being made, one the converse of the other. If two nonvertical lines are parallel, then their slopes are equal and they have different y-intercepts. If two nonvertical lines have equal slopes and they have different y-intercepts, then they are parallel.

E XAM PL E 9

Showing That Two Lines Are Parallel Show that the lines given by the following equations are parallel: L1 : 2x + 3y = 6,

Solution

L2 : 4x + 6y = 0

To determine whether these lines have equal slopes and different y-intercepts, write each equation in slope–intercept form: L 1 : 2x + 3y = 6 3y = -2x + 6 2 y= - x+2 3 2 Slope = - ; y@intercept = 2 3

L 2 : 4x + 6y = 0 6y = -4x 2 y= - x 3 2 Slope = - ; y@intercept = 0 3 37

Graphs

2 Because these lines have the same slope, - , but different y-intercepts, the lines are 3 parallel. See Figure 62. Figure 62 Parallel lines

y 5 4

6

6 L1

4

EX A MP L E 1 0

5 x L1

5

L2

L2

Finding a Line That Is Parallel to a Given Line Find an equation for the line that contains the point (2, -3) and is parallel to the line 2x + y = 6 .

Solution

Since the two lines are to be parallel, the slope of the line that we seek equals the slope of the line 2x + y = 6 . Begin by writing the equation of the line 2x + y = 6 in slope–intercept form. 2x + y = 6

Figure 63 y

y = -2x + 6

6

The slope is -2 . Since the line that we seek also has slope -2 and contains the point (2, -3) , use the point–slope form to obtain its equation. y - y1 = m(x - x1) y - ( -3) = -2(x - 2)

6 x

6

y + 3 = -2x + 4

2x  y  6

y = -2x + 1

(2, 3) 5

2x + y = 1

2x  y  1

Point–slope form m = - 2, x1 = 2, y1 = - 3 Simplify. Slope–intercept form General form

This line is parallel to the line 2x + y = 6 and contains the point (2, -3) . See Figure 63. Figure 64

Now Work

y

PROBLEM

59

10 Find Equations of Perpendicular Lines 90° x

THEOREM

When two lines intersect at a right angle (90°), they are said to be perpendicular. See Figure 64. The following result gives a condition, in terms of their slopes, for two lines to be perpendicular.

Criterion for Perpendicular Lines Two nonvertical lines are perpendicular if and only if the product of their slopes is - 1 . Here we shall prove the “only if ” part of the statement: If two nonvertical lines are perpendicular, then the product of their slopes is - 1 .

38

Graphs

In Problem 128 you are asked to prove the “if” part of the theorem; that is: If two nonvertical lines have slopes whose product is -1 , then the lines are perpendicular. Figure 65 y Slope m2 A = (1, m2) Slope m1

Rise = m 2 x

Run = 1 O

1

Rise = m1

B = (1, m1)

Proof Let m1 and m2 denote the slopes of the two lines. There is no loss in generality (that is, neither the angle nor the slopes are affected) if we situate the lines so that they meet at the origin. See Figure 65. The point A = (1, m2) is on the line having slope m 2, and the point B = (1, m1) is on the line having slope m1 . (Do you see why this must be true?) Suppose that the lines are perpendicular. Then triangle OAB is a right triangle. As a result of the Pythagorean Theorem, it follows that

3 d(O, A) 4 2 + 3 d(O, B) 4 2 = 3 d(A, B) 4 2

(5)

Using the distance formula, the squares of these distances are

3 d(O, A) 4 2 = (1 - 0)2 + (m2 - 0)2 = 1 + m22 3 d(O, B) 4 2 = (1 - 0)2 + (m1 - 0)2 = 1 + m21 3 d(A, B) 4 2 = (1 - 1)2 + (m2 - m1)2 = m22 - 2m1 m2 + m21 Using these facts in equation (5), we get

11

+ m22 2 +

11

+ m21 2 = m22 - 2m1 m2 + m21

which, upon simplification, can be written as m1 m2 = -1 If the lines are perpendicular, the product of their slopes is -1. You may find it easier to remember the condition for two nonvertical lines to be perpendicular by observing that the equality m1 m2 = -1 means that m1 and m2 are 1 1 negative reciprocals of each other; that is, either m1 = or m2 = . m2 m1

E X AM PL E 11

Finding the Slope of a Line Perpendicular to Another Line 3 2 If a line has slope , any line having slope - is perpendicular to it. 2 3

E X AM PL E 12

Finding the Equation of a Line Perpendicular to a Given Line Find an equation of the line that contains the point (1, -2) and is perpendicular to the line x + 3y = 6 . Graph the two lines.

Solution

First write the equation of the given line in slope–intercept form to find its slope. x + 3y = 6 3y = -x + 6

Proceed to solve for y.

1 y = - x + 2 Place in the form y = mx + b. 3 1 The given line has slope - . Any line perpendicular to this line will have slope 3. 3 Because we require the point (1, -2) to be on this line with slope 3, use the point– slope form of the equation of a line. y - y1 = m(x - x1) Point–slope form y - ( -2) = 3(x - 1)

m = 3, x1 = 1, y1 = -2 39

Graphs

To obtain other forms of the equation, proceed as follows: y + 2 = 3(x - 1) y + 2 = 3x - 3

Simplify.

y = 3x - 5 3x - y = 5

Slope–intercept form General form

Figure 66 shows the graphs.

Figure 66

y Y2 

WARNING Be sure to use a square screen when you graph perpendicular lines. Otherwise, the angle between the two lines will appear distorted. 

 13 x

y  3x  5

6 2 6

Y1  3x  5

x  3y  6

4 2

6

x

12

2

2 2

4

6

(1, 2)

4

6

Now Work

PROBLEM

65

4 Assess Your Understanding Concepts and Vocabulary 1. The slope of a vertical line is _____________ ; the slope of a horizontal line is ________.

7. Two nonvertical lines have slopes m1 and m 2, respectively. The lines are parallel if _____________ and the _____________ are unequal; the lines are perpendicular if _____________.

2. For the line 2x + 3y = 6 , the x-intercept is ________ and the y-intercept is ________.

8. The lines y = 2x + 3 and y = ax + 5 are parallel if a = ________.

3. A horizontal line is given by an equation of the form _________, where b is the _____________. 4. True or slope.

False

Vertical

lines

have

an

9. The lines y = 2x - 1 and y = ax + 2 are perpendicular if a = ________.

undefined

10. True or False Perpendicular lines have slopes that are reciprocals of one another.

5. True or False The slope of the line 2y = 3x + 5 is 3. 6. True or False The point (1, 2) is on the line 2x + y = 4.

Skill Building In Problems 11–14, (a) find the slope of the line and (b) interpret the slope. 11.

12.

y 2

(–2, 1) 2

(2, 1)

(0, 0) –2

–1

13.

y

14.

y (–2, 2)

2

(1, 1)

y (–1, 1)

2

(2, 2)

(0, 0) 2

x

–2

–1

2

x

–2

–1

2

x

–2

–1

2

In Problems 15–22, plot each pair of points and determine the slope of the line containing them. Graph the line by hand. 15. (2, 3); (4, 0)

16. (4, 2); (3, 4)

17. ( - 2, 3); (2, 1)

18. ( - 1, 1); (2, 3)

19. ( - 3, - 1); (2, - 1)

20. (4, 2); ( - 5, 2)

21. ( - 1, 2); ( - 1, - 2)

22. (2, 0); (2, 2)

40

x

Graphs

In Problems 23–30, graph the line containing the point P and having slope m. 23. P = (1, 2); m = 3

24. P = (2, 1); m = 4

27. P = ( - 1, 3); m = 0

28. P = (2, - 4); m = 0

3 4 29. P = (0, 3) ; slope undefined 25. P = (2, 4); m = -

2 5 30. P = ( - 2, 0) ; slope undefined 26. P = (1, 3); m = -

In Problems 31–36, the slope and a point on a line are given. Use this information to locate three additional points on the line. Answers may vary. [Hint: It is not necessary to find the equation of the line. See Example 2.] 31. Slope 4; point (1, 2)

32. Slope 2; point ( - 2, 3)

3 33. Slope - ; point (2, - 4) 2

4 34. Slope ; point ( - 3, 2) 3

35. Slope - 2; point ( - 2, - 3)

36. Slope - 1; point (4, 1)

In Problems 37–44, find an equation of the line L. y

37.

38.

2

L

(2, 1)

(–2, 1) 2

(0, 0) –2

41.

39.

y

y

L (–1, 3)

3

2

x

–2

3

3

–2

L

y

42.

y

–2

x

2

–1

(–1, 1)

43. (1, 2)

–1

x

2

y 3

y = 2x

3 x

L is parallel to y = 2x

–1

2

–1

x

y 3

44.

L

L L

L

(2, 2)

(1, 2)

(3, 3)

–1

2

(1, 1)

(0, 0)

–1

y

40.

3 x y = –x

L is parallel to y = –x

–1 y = 2x

3 x

L is perpendicular to y = 2x

(–1, 1) –3 L

1 x y = –x

L is perpendicular to y = –x

In Problems 45–70, find an equation for the line with the given properties. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer. 45. Slope = 3; containing the point (- 2, 3)

46. Slope = 2; containing the point (4, - 3)

2 47. Slope = - ; containing the point (1, - 1) 3

48. Slope =

49. Containing the points (1, 3) and ( - 1, 2)

50. Containing the points ( - 3, 4) and (2, 5)

51. Slope = - 3; y@intercept = 3

52. Slope = - 2; y@intercept = - 2

53. x@intercept = 2; y@intercept = - 1

54. x@intercept = - 4; y@intercept = 4

55. Slope undefined; containing the point (2, 4)

56. Slope undefined; containing the point (3, 8)

57. Horizontal; containing the point ( - 3, 2)

58. Vertical; containing the point (4, - 5)

59. Parallel to the line y = 2x; containing the point ( - 1, 2)

60. Parallel to the line y = - 3x; containing the point ( - 1, 2)

61. Parallel to the line 2x - y = - 2; containing the point (0, 0)

62. Parallel to the line x - 2y = - 5; containing the point (0, 0)

63. Parallel to the line x = 5; containing the point (4, 2)

64. Parallel to the line y = 5; containing the point (4, 2)

1 65. Perpendicular to the line y = x + 4; containing the point 2 (1, - 2)

66. Perpendicular to the line y = 2x - 3; containing the point (1, - 2)

67. Perpendicular to the line 2x + y = 2; containing the point ( - 3, 0)

68. Perpendicular to the line x - 2y = - 5; containing the point (0, 4)

69. Perpendicular to the line x = 8; containing the point (3, 4)

70. Perpendicular to the line y = 8; containing the point (3, 4)

1 ; containing the point (3, 1) 2

41

Graphs

In Problems 71–90, find the slope and y-intercept of each line. Graph the line by hand. Verify your graph using a graphing utility. 71. y = 2x + 3

1 y = x - 1 2

1 x + y = 2 3

73.

77. x + 2y = 4

78. - x + 3y = 6

79. 2x - 3y = 6

80. 3x + 2y = 6

81. x + y = 1

82. x - y = 2

83. x = - 4

84. y = - 1

85. y = 5

86. x = 2

87. y - x = 0

88. x + y = 0

89. 2y - 3x = 0

90. 3x + 2y = 0

76. y = 2x +

1 2

74.

75. y =

1 x + 2 2

72. y = - 3x + 4

In Problems 91–100, (a) find the intercepts of the graph of each equation and (b) graph the equation. 91. 2x + 3y = 6

92. 3x - 2y = 6

95. 7x + 2y = 21

96. 5x + 3y = 18

93. - 4x + 5y = 40 1 1 97. x + y = 1 2 3 100. - 0.3x + 0.4y = 1.2

99. 0.2x - 0.5y = 1

94. 6x - 4y = 24 2 98. x - y = 4 3

102. Find an equation of the y-axis.

101. Find an equation of the x-axis.

In Problems 103–106, the equations of two lines are given. Determine if the lines are parallel, perpendicular, or neither. 1 106. y = - 2x + 3 103. y = 2x - 3 105. y = 4x + 5 104. y = x - 3 2 1 y = 2x + 4 y = - 4x + 2 y = - 2x + 4 y = - x + 2 2 In Problems 107–110, write an equation of each line. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer. 107.

108.

4

6

6

4

109.

2

3

3

2

110.

2

3

3

2

2

3

3

2

Applications and Extensions 111. Geometry Use slopes to show that the triangle whose vertices are ( - 2, 5) , (1, 3) , and ( - 1, 0) is a right triangle. 112. Geometry Use slopes to show that the quadrilateral whose vertices are (1, - 1) , (4, 1) , (2, 2) , and (5, 4) is a parallelogram. 113. Geometry Use slopes to show that the quadrilateral whose vertices are ( - 1, 0) , (2, 3) , (1, - 2) , and (4, 1) is a rectangle. 114. Geometry Use slopes and the distance formula to show that the quadrilateral whose vertices are (0, 0) , (1, 3) , (4, 2) , and (3, - 1) is a square. 115. Truck Rentals A truck rental company rents a moving truck for one day by charging \$29 plus \$0.20 per mile. Write a linear equation that relates the cost C, in dollars, of renting the truck to the number x of miles driven. What is the cost of renting the truck if the truck is driven 110 miles? 230 miles? 116. Cost Equation The fixed costs of operating a business are the costs incurred regardless of the level of production. Fixed costs include rent, fixed salaries, and costs of leasing machinery. The variable costs of operating a business are the costs that change with the level of output. Variable costs include raw materials, hourly wages,

42

and electricity. Suppose that a manufacturer of jeans has fixed daily costs of \$500 and variable costs of \$8 for each pair of jeans manufactured. Write a linear equation that relates the daily cost C, in dollars, of manufacturing the jeans to the number x of jeans manufactured. What is the cost of manufacturing 400 pairs of jeans? 740 pairs? 117. Cost of Driving a Car The annual fixed costs for owning a small sedan are \$1289, assuming the car is completely paid for. The cost to drive the car is approximately \$0.15 per mile. Write a linear equation that relates the cost C and the number x of miles driven annually. Source: www.pacebus.com 118. Wages of a Car Salesperson Dan receives \$375 per week for selling new and used cars at a car dealership in Oak Lawn, Illinois. In addition, he receives 5% of the profit on any sales that he generates. Write a linear equation that represents Dan’s weekly salary S when he has sales that generate a profit of x dollars. 119. Electricity Rates in Illinois Commonwealth Edison Company supplies electricity to residential customers for a monthly customer charge of \$11.47 plus 11 cents per kilowatt-hour for up to 600 kilowatt-hours.

Graphs

y Platform 30"

Ramp

Tetra Images/Alamy

x

(a) Write a linear equation that relates the height y of the ramp above the floor to the horizontal distance x from the platform. (b) Find and interpret the x-intercept of the graph of your equation. (c) Design requirements stipulate that the maximum run be 30 feet and that the maximum slope be a drop of 1 inch for each 12 inches of run. Will this ramp meet the requirements? Explain. (d) What slopes could be used to obtain the 30-inch rise and still meet design requirements? Source: www.adaptiveaccess.com/wood_ramps.php

(a) Write a linear equation that relates the monthly charge C, in dollars, to the number x of kilowatt-hours used in a month, 0 … x … 600. (b) Graph this equation. (c) What is the monthly charge for using 200 kilowatthours? (d) What is the monthly charge for using 500 kilowatthours? (e) Interpret the slope of the line. Source: Commonwealth Edison Company, March, 2011. 120. Electricity Rates in Florida Florida Power & Light Company supplies electricity to residential customers for a monthly customer charge of \$5.90 plus 8.81 cents per kilowatt-hour for up to 1000 kilowatt-hours. (a) Write a linear equation that relates the monthly charge C, in dollars, to the number x of kilowatt-hours used in a month, 0 … x … 1000 . (b) Graph this equation. (c) What is the monthly charge for using 200 kilowatthours? (d) What is the monthly charge for using 500 kilowatthours? (e) Interpret the slope of the line. Source: Florida Power & Light Company, March, 2011. 121. Measuring Temperature The relationship between Celsius (°C) and Fahrenheit (°F) degrees of measuring temperature is linear. Find a linear equation relating °C and °F if 0°C corresponds to 32°F and 100°C corresponds to 212°F. Use the equation to find the Celsius measure of 70°F. 122. Measuring Temperature The Kelvin (K) scale for measuring temperature is obtained by adding 273 to the Celsius temperature. (a) Write a linear equation relating K and °C. (b) Write a linear equation relating K and °F (see Problem 121). 123. Access Ramp A wooden access ramp is being built to reach a platform that sits 30 inches above the floor. The ramp drops 2 inches for every 25-inch run.

124. Cigarette Use A report in the Child Trends DataBase indicated that, in 2000, 20.6% of twelfth grade students reported daily use of cigarettes. In 2009, 11.2% of twelfth grade students reported daily use of cigarettes. (a) Write a linear equation that relates the percent y of twelfth grade students who smoke cigarettes daily to the number x of years after 2000. (b) Find the intercepts of the graph of your equation. (c) Do the intercepts have any meaningful interpretation? (d) Use your equation to predict the percent for the year 2025. Is this result reasonable? Source: www.childtrendsdatabank.org 125. Product Promotion A cereal company finds that the number of people who will buy one of its products in the first month that it is introduced is linearly related to the amount of money it spends on advertising. If it spends \$40,000 on advertising, then 100,000 boxes of cereal will be sold, and if it spends \$60,000, then 200,000 boxes will be sold. (a) Write a linear equation that relates the amount A spent on advertising to the number x of boxes the company aims to sell. (b) How much advertising is needed to sell 300,000 boxes of cereal? (c) Interpret the slope. 126. Show that the line containing the points (a, b) and (b, a) , a ⬆ b , is perpendicular to the line y = x . Also show that the midpoint of (a, b) and (b, a) lies on the line y = x. 127. The equation 2x - y = C defines a family of lines, one line for each value of C. On one set of coordinate axes, graph the members of the family when C = - 4, C = 0 , and C = 2 . Can you draw a conclusion from the graph about each member of the family? 128. Prove that if two nonvertical lines have slopes whose product is - 1 then the lines are perpendicular. [Hint: Refer to Figure 65 and use the converse of the Pythagorean Theorem.]

43

Graphs

Explaining Concepts: Discussion and Writing 129. Which of the following equations might have the graph shown? (More than one answer is possible.) (a) 2x + 3y = 6 y (b) - 2x + 3y = 6 (c) 3x - 4y = - 12 (d) x - y = 1 (e) x - y = - 1 x (f) y = 3x - 5 (g) y = 2x + 3 (h) y = - 3x + 3

133. m is for Slope The accepted symbol used to denote the slope of a line is the letter m. Investigate the origin of this symbolism. Begin by consulting a French dictionary and looking up the French word monter. Write a brief essay on your findings. 134. Grade of a Road The term grade is used to describe the inclination of a road. How does this term relate to the notion of slope of a line? Is a 4% grade very steep? Investigate the grades of some mountainous roads and determine their slopes. Write a brief essay on your findings.

130. Which of the following equations might have the graph shown? (More than one answer is possible.) (a) 2x + 3y = 6 (b) 2x - 3y = 6 y (c) 3x + 4y = 12 (d) x - y = 1 x (e) x - y = - 1 (f) y = - 2x - 1 1 (g) y = - x + 10 2 (h) y = x + 4 131. The figure shows the graph of two parallel lines. Which of the following pairs of equations might have such a graph? (a) x - 2y = 3 x + 2y = 7 y (b) x + y = 2 x + y = -1 (c) x - y = - 2 x x - y = 1 (d) x - y = - 2 2x - 2y = - 4 (e) x + 2y = 2 x + 2y = - 1 132. The figure shows the graph of two perpendicular lines. Which of the following pairs of equations might have such a graph? (a) y - 2x = 2 y + 2x = - 1 y (b) y - 2x = 0 2y + x = 0 (c) 2y - x = 2 x 2y + x = - 2 (d) y - 2x = 2 x + 2y = - 1 (e) 2x + y = - 2 2y + x = - 2

135. Carpentry Carpenters use the term pitch to describe the steepness of staircases and roofs. How does pitch relate to slope? Investigate typical pitches used for stairs and for roofs. Write a brief essay on your findings. 136. Can the equation of every line be written in slope– intercept form? Why? 137. Does every line have exactly one x-intercept and one y-intercept? Are there any lines that have no intercepts? 138. What can you say about two lines that have equal slopes and equal y-intercepts? 139. What can you say about two lines with the same x-intercept and the same y-intercept? Assume that the x-intercept is not 0. 140. If two distinct lines have the same slope, but different x-intercepts, can they have the same y-intercept? 141. If two distinct lines have the same y-intercept, but different slopes, can they have the same x-intercept? 142. Which form of the equation of a line do you prefer to use? Justify your position with an example that shows that your choice is better than another. Have reasons. 143. What Went Wrong? A student is asked to find the slope of the line joining ( - 3, 2) and (1, - 4) . He states that the 3 slope is . Is he correct? If not, what went wrong? 2

Interactive Exercise Ask your instructor if the applet below is of interest to you. 144. Slope Open the slope applet. Move point B around the Cartesian plane with your mouse. (a) Move B to the point whose coordinates are ( 2, 7) . What is the slope of the line? (b) Move B to the point whose coordinates are ( 3, 6) . What is the slope of the line?

44

Graphs

(c) (d) (e) (f) (g) (h)

Move B to the point whose coordinates are (4, 5) . What is the slope of the line? Move B to the point whose coordinates are (4, 4) . What is the slope of the line? Move B to the point whose coordinates are (4, 1) . What is the slope of the line? Move B to the point whose coordinates are (3, - 2) . What is the slope of the line? Slowly move B to a point whose x-coordinate is 1. What happens to the value of the slope as the x-coordinate approaches 1? What can be said about a line whose slope is positive? What can be said about a line whose slope is negative? What can be said about a line whose slope is 0? (i) Consider the results of parts (a)–(c). What can be said about the steepness of a line with positive slope as its slope increases? ( j) Move B to the point whose coordinates are (3, 5) . What is the slope of the line? Move B to the point whose coordinates are (5, 6) . What is the slope of the line? Move B to the point whose coordinates are (- 1, 3) . What is the slope of the line?

5 Circles PREPARING FOR THIS SECTION Before getting started, review the following: • Completing the Square

• Square Root Method

Now Work the ’Are You Prepared?’ problems.

OBJECTIVES 1 Write the Standard Form of the Equation of a Circle 2 Graph a Circle by Hand and by Using a Graphing Utility 3 Work with the General Form of the Equation of a Circle

1 Write the Standard Form of the Equation of a Circle One advantage of a coordinate system is that it enables us to translate a geometric statement into an algebraic statement, and vice versa. Consider, for example, the following geometric statement that defines a circle.

DEFINITION

Figure 67 y

A circle is a set of points in the xy-plane that are a fixed distance r from a fixed point (h, k) . The fixed distance r is called the radius, and the fixed point (h, k) is called the center of the circle.

Figure 67 shows the graph of a circle. To find the equation, let (x, y) represent the coordinates of any point on a circle with radius r and center (h, k) . Then the distance between the points (x, y) and (h, k) must always equal r. That is, by the distance formula

(x, y) r (h, k)

21x - h2 2 + 1y - k2 2 = r x

or, equivalently, 1x - h2 2 + 1y - k2 2 = r 2

DEFINITION

The standard form of an equation of a circle with radius r and center (h, k) is 1x - h2 2 + 1y - k2 2 = r 2

(1)

45

Graphs

THEOREM

The standard form of an equation of a circle of radius r with center at the origin 10, 02 is x2 + y2 = r 2

DEFINITION

If the radius r = 1 , the circle whose center is at the origin is called the unit circle and has the equation x2 + y2 = 1

See Figure 68. Notice that the graph of the unit circle is symmetric with respect to the x-axis, the y-axis, and the origin. Figure 68 Unit circle x 2 + y 2 = 1

y 1

1

(0,0)

1

x

1

E X A MP L E 1

Writing the Standard Form of the Equation of a Circle Write the standard form of the equation of the circle with radius 5 and center 1 -3, 62 .

Solution

Using equation (1) and substituting the values r = 5, h = -3 , and k = 6 , we have 1x - h2 2 + 1y - k2 2 = r 2 1x - 1 -322 2 + 1y - 62 2 = 52 1x + 32 2 + 1y - 62 2 = 25

Now Work

PROBLEM

7

2 Graph a Circle by Hand and by Using a Graphing Utility E X A MP L E 2

Graphing a Circle by Hand and by Using a Graphing Utility Graph the equation: 1x + 32 2 + 1y - 22 2 = 16

Solution Figure 69 (–3, 6)

y

1x + 32 2 + 1y - 22 2 = 16

6

1x - 1 -322 2 + 1y - 22 2 = 42

4 (–7, 2) –10

(–3, 2)

(–3, –2)

46

(1, 2) 2 x

–5

Since the equation is in the form of equation (1), its graph is a circle. To graph the equation by hand, compare the given equation to the standard form of the equation of a circle. The comparison yields information about the circle.

c c c 2 2 1x - h2 + 1y - k2 = r 2 We see that h = -3, k = 2, and r = 4. The circle has center 1 -3, 22 and a radius of 4 units. To graph this circle, first plot the center 1 -3, 22. Since the radius is 4, locate four points on the circle by plotting points 4 units to the left, to the right, up, and down from the center. These four points can then be used as guides to obtain the graph. See Figure 69.

Graphs

To graph a circle on a graphing utility, we must write the equation in the form y = 5 expression involving x 6 . * We must solve for y in the equation

In Words The symbol { is read “plus or minus.” It means to add and subtract the quantity following the { symbol. For example, 5 { 2 means 5 - 2 = 3 or 5 + 2 = 7.

1x + 32 2 + 1y - 22 2 = 16 Subtract (x + 3)2 from both sides.

1y - 22 2 = 16 - 1x + 32 2 y - 2 = { 216 - 1x + 32 2

Use the Square Root Method.

y = 2 { 216 - 1x + 32 2 Add 2 to both sides. Figure 70

To graph the circle, we graph the top half 6

Y1 = 2 + 216 - 1x + 32 2

Y1

and the bottom half Y2 = 2 - 216 - 1x + 32 2 –9

3 Y2 –2

Also, be sure to use a square screen. Otherwise, the circle will appear distorted. Figure 70 shows the graph on a TI-84 Plus. The graph is “disconnected” due to the resolution of the calculator.

Now Work

E XAM PL E 3

PROBLEMS

23(a)

AND

(b)

Finding the Intercepts of a Circle For the circle 1x + 32 2 + 1y - 22 2 = 16, find the intercepts, if any, of its graph.

Solution

This is the equation discussed and graphed in Example 2. To find the x-intercepts, if any, let y = 0 and solve for x. Then 1x + 32 2 + 1y - 22 2 = 16 1x + 32 2 + 10 - 22 2 = 16 1x + 32 + 4 = 16 2

1x + 32 2 = 12 x + 3 = { 212

y=0 Simplify. Subtract 4 from both sides. Apply the Square Root Method.

x = -3 { 223 Solve for x. The x-intercepts are -3 - 223 ⬇ -6.46 and -3 + 223 ⬇ 0.46. To find the y-intercepts, if any, let x = 0 and solve for y. Then 1x + 32 2 + 1y - 22 2 = 16 10 + 32 2 + 1y - 22 2 = 16

x=0

9 + 1y - 22 = 16 2

1y - 22 2 = 7 y - 2 = { 27 Apply the Square Root Method. y = 2 { 27 Solve for y. The y-intercepts are 2 - 27 ⬇ -0.65 and 2 + 27 ⬇ 4.65. Look back at Figure 69 to verify the approximate locations of the intercepts.

Now Work

PROBLEM

23 (c)

*Some graphing utilities (e.g., TI-83, TI-84, and TI-86) have a CIRCLE function that allows the user to enter only the coordinates of the center of the circle and its radius to graph the circle.

47

Graphs

3 Work with the General Form of the Equation of a Circle If we eliminate the parentheses from the standard form of the equation of the circle given in Example 3, we get 1x + 32 2 + 1y - 22 2 = 16 x2 + 6x + 9 + y 2 - 4y + 4 = 16 which, upon simplifying, is equivalent to x2 + y 2 + 6x - 4y - 3 = 0 It can be shown that any equation of the form x2 + y 2 + ax + by + c = 0 has a graph that is a circle or a point, or has no graph at all. For example, the graph of the equation x2 + y 2 = 0 is the single point 10, 02. The equation x2 + y 2 + 5 = 0, or x2 + y 2 = -5, has no graph, because sums of squares of real numbers are never negative.

DEFINITION

When its graph is a circle, the equation x2 + y 2 + ax + by + c = 0 is referred to as the general form of the equation of a circle.

If an equation of a circle is in the general form, we use the method of completing the square to put the equation in standard form so that we can identify its center and radius.

E X A MP L E 4

Graphing a Circle Whose Equation Is in General Form Graph the equation x2 + y 2 + 4x - 6y + 12 = 0 .

Solution

Group the expression involving x, group the expression involving y, and put the constant on the right side of the equation. The result is 1x2 + 4x2 + 1y 2 - 6y2 = -12 Next, complete the square of each expression in parentheses. Remember that any number added on the left side of the equation must be added on the right. (x2  4x  4)  (y2  6y  9)  12  4  9

( 42 )

2

2

(6 ) 2

4

9

(x  2)  (y  3)2  1 2

Factor

This equation is the standard form of the equation of a circle with radius 1 and center 1 -2, 32. To graph the equation by hand, use the center 1 -2, 32 and the radius 1. See Figure 71(a). To graph the equation using a graphing utility, solve for y. 1y - 32 2 = 1 - 1x + 22 2 y - 3 = {21 - 1x + 22 2 y = 3 {21 - 1x + 22 48

Use the Square Root Method. 2

Graphs

Figure 71(b) illustrates the graph on a TI-84 Plus graphing calculator. Figure 71

y

(2, 4) 4

1 (3, 3)

Y1  3  1  (x  2)2 5

(1, 3)

(2, 3) (2, 2) 4.5

1 x

3

1.5

Y2  3  1  (x  2)2 (b)

(a)

Now Work

E XAM PL E 5

1

PROBLEM

27

Finding the General Equation of a Circle Find the general equation of the circle whose center is 11, -22 and whose graph contains the point 14, -22.

Solution Figure 72 y 3

To find the equation of a circle, we need to know its center and its radius. Here, the center is 11, -22. Since the point 14, -22 is on the graph, the radius r will equal the distance from 14, -22 to the center 11, -22. See Figure 72. Thus, r = 2 14 - 12 2 + 3 -2 - 1 -22 4 2

r (1, 2)

5

= 29 = 3

x

(4, 2)

The standard form of the equation of the circle is 1x - 12 2 + 1y + 22 2 = 9

5

Eliminate the parentheses and rearrange the terms to get the general equation x2 + y 2 - 2x + 4y - 4 = 0

Now Work

PROBLEM

13

Overview The discussion in Sections 4 and 5 about lines and circles dealt with two main types of problems that can be generalized as follows: 1. Given an equation, classify it and graph it. 2. Given a graph, or information about a graph, find its equation. This text deals with both types of problems. You may study various equations, classify them, and graph them. The second type of problem is usually more difficult to solve than the first. In many instances a graphing utility can be used to solve problems when information about the problem (such as data) is given.

5 Assess Your Understanding ‘Are You Prepared?’

Answers are given at the end of these exercises.

1. To complete the square of x 2 + 10x , you would (add/ subtract) the number ______.

2. Use the Square Root Method to solve the equation 1x - 22 2 = 9 .

49

Graphs

Concepts and Vocabulary 3. True or False

Every equation of the form

5. True or False The radius of the circle x 2 + y 2 = 9 is 3.

x + y + ax + by + c = 0

6. True or False The center of the circle

2

2

1x + 32 2 + 1y - 22 2 = 13

has a circle as its graph. is (3, - 2) .

4. For a circle, the ______ is the distance from the center to any point on the circle.

Skill Building In Problems 7–10, find the center and radius of each circle. Write the standard form of the equation. 7. y

8.

9. y

y

10. y

(4, 2)

(2, 3) (1, 2) (0, 1)

(2, 1) (0, 1) (1, 2)

x

x

x

(1, 0)

x

In Problems 11–20, write the standard form of the equation and the general form of the equation of each circle of radius r and center 1h, k2 . Graph each circle. 11. r = 2;

1h, k2 = 10, 02

12. r = 3;

1h, k2 = 10, 02

13. r = 2;

1h, k2 = 10, 22

14. r = 3;

15. r = 5;

1h, k2 = 14, - 32

16. r = 4;

1h, k2 = 12, - 32

17. r = 4;

1h, k2 = 1 - 2, 12

18. r = 7; 1h, k2 = 1- 5, - 22

1 ; 2

1 1h, k2 = a , 0 b 2

19. r =

20. r =

1 ; 2

1h, k2 = 11, 02

1 1h, k2 = a0, - b 2

In Problems 21–34, (a) find the center 1h, k2 and radius r of each circle; (b) graph each circle; (c) find the intercepts, if any. 21. x 2 + y 2 = 4

22. x 2 + 1y - 12 2 = 1

23. 2 1x - 32 2 + 2y 2 = 8

24. 3 1x + 12 2 + 3 1y - 12 2 = 6

25. x 2 + y 2 - 2x - 4y - 4 = 0

26. x 2 + y 2 + 4x + 2y - 20 = 0

27. x 2 + y 2 + 4x - 4y - 1 = 0 1 30. x 2 + y 2 + x + y - = 0 2 33. 2x 2 + 8x + 2y 2 = 0

28. x 2 + y 2 - 6x + 2y + 9 = 0

29. x 2 + y 2 - x + 2y + 1 = 0

31. 2x 2 + 2y 2 - 12x + 8y - 24 = 0

32. 2x 2 + 2y 2 + 8x + 7 = 0

34. 3x 2 + 3y 2 - 12y = 0

In Problems 35–42, find the standard form of the equation of each circle. 35. Center at the origin and containing the point 1 - 2, 32

36. Center 11, 02 and containing the point 1 - 3, 22

37. Center 12, 32 and tangent to the x-axis

38. Center 1 - 3, 12 and tangent to the y-axis

39. With endpoints of a diameter at 11, 42 and 1 - 3, 22

40. With endpoints of a diameter at 14, 32 and 10, 12

41. Center 1 - 1, 32 and tangent to the line y = 2

42. Center 14, - 22 and tangent to the line x = 1

In Problems 43–46, match each graph with the correct equation. (a) 1x - 32 2 + 1y + 32 2 = 9

(b) 1x + 12 2 + 1y - 22 2 = 4

43.

44.

4

50

46.

9

9

6

6

4

6

6

(d) 1x + 32 2 + 1y - 32 2 = 9

45.

4

6

(c) 1x - 12 2 + 1y + 22 2 = 4

6

6

4

9

9

6

Graphs

Applications and Extensions 47. Find the area of the square in the figure. y

51. Weather Satellites Earth is represented on a map of a portion of the solar system so that its surface is the circle with equation x 2 + y 2 + 2x + 4y - 4091 = 0 . A weather satellite circles 0.6 unit above Earth with the center of its circular orbit at the center of Earth. Find the equation for the orbit of the satellite on this map.

x2  y2  9

x

r

48. Find the area of the blue shaded region in the figure, assuming the quadrilateral inside the circle is a square. y

52. The tangent line to a circle may be defined as the line that intersects the circle in a single point, called the point of tangency. See the figure.

x 2  y 2  36

y x

r x

49. Ferris Wheel The original Ferris wheel was built in 1893 by Pittsburgh, Pennsylvania, bridge builder George W. Ferris. The Ferris wheel was originally built for the 1893 World’s Fair in Chicago and was later reconstructed for the 1904 World’s Fair in St. Louis. It had a maximum height of 264 feet and a wheel diameter of 250 feet. Find an equation for the wheel if the center of the wheel is on the y-axis. Source: inventors.about.com

50. Ferris Wheel In 2008, the Singapore Flyer opened as the world’s largest Ferris wheel. It has a maximum height of 165 meters and a diameter of 150 meters, with one full rotation taking approximately 30 minutes. Find an equation for the wheel if the center of the wheel is on the y-axis. Source: Wikipedia

If the equation of the circle is x 2 + y 2 = r 2 and the equation of the tangent line is y = mx + b , show that: (a) r 2 11 + m 2 2 = b2 [Hint: The quadratic equation x 2 + 1mx + b2 2 = r 2 has exactly one solution.] -r2 m r2 (b) The point of tangency is ¢ , ≤. b b (c) The tangent line is perpendicular to the line containing the center of the circle and the point of tangency. 53. The Greek Method The Greek method for finding the equation of the tangent line to a circle uses the fact that at any point on a circle the lines containing the center and the tangent line are perpendicular (see Problem 52). Use this method to find an equation of the tangent line to the circle x 2 + y 2 = 9 at the point 11, 2 122 . 54. Use the Greek method described in Problem 53 to find an equation of the tangent line to the circle x 2 + y 2 - 4x + 6y + 4 = 0 at the point 13, 2 12 - 32.

Jasonleehl/Shutterstock

55. Refer to Problem 52. The line x - 2y + 4 = 0 is tangent to a circle at 10, 22 . The line y = 2x - 7 is tangent to the same circle at 13, - 12 . Find the center of the circle. 56. Find an equation of the line containing the centers of the two circles x 2 + y 2 - 4x + 6y + 4 = 0 and x 2 + y 2 + 6x + 4y + 9 = 0 57. If a circle of radius 2 is made to roll along the x-axis, what is an equation for the path of the center of the circle? 58. If the circumference of a circle is 6p , what is its radius?

51

Graphs

Explaining Concepts: Discussion and Writing 59. Which of the following equations might have the graph shown? (More than one answer is possible.) y (a) 1x - 22 2 + 1y + 32 2 = 13 (b) 1x - 22 2 + 1y - 22 2 = 8 (c) 1x - 22 2 + 1y - 32 2 = 13 (d) 1x + 22 2 + 1y - 22 2 = 8 (e) x 2 + y 2 - 4x - 9y = 0 x (f) x 2 + y 2 + 4x - 2y = 0 2 2 (g) x + y - 9x - 4y = 0 (h) x 2 + y 2 - 4x - 4y = 4

60. Which of the following equations might have the graph shown? (More than one answer is possible.) (a) 1x - 22 2 + y 2 = 3 y (b) 1x + 22 2 + y 2 = 3 2 2 (c) x + 1y - 22 = 3 (d) 1x + 22 2 + y 2 = 4 (e) x 2 + y 2 + 10x + 16 = 0 x (f) x 2 + y 2 + 10x - 2y = 1 (g) x 2 + y 2 + 9x + 10 = 0 (h) x 2 + y 2 - 9x - 10 = 0

61. Explain how the center and radius of a circle can be used to graph the circle. 62. What Went Wrong? A student stated that the center and radius of the graph whose equation is (x + 3)2 + (y - 2)2 = 16 are (3, - 2) and 4, respectively. Why is this incorrect?

Interactive Exercises Ask your instructor if the applets below are of interest to you. 63. Center of a Circle Open the “Circle: the role of the center” applet. Place the cursor on the center of the circle and hold the mouse button. Drag the center around the Cartesian plane and note how the equation of the circle changes. (a) What is the radius of the circle? (b) Draw a circle whose center is at 11, 32 . What is the equation of the circle? (c) Draw a circle whose center is at 1 - 1, 32 . What is the equation of the circle? (d) Draw a circle whose center is at 1 - 1, - 32 . What is the equation of the circle? (e) Draw a circle whose center is at 11, - 32 . What is the equation of the circle? (f) Write a few sentences explaining the role the center of the circle plays in the equation of the circle. 64. Radius of a Circle Open the “Circle: the role of the radius” applet. Place the cursor on point B, press and hold the mouse button. Drag B around the Cartesian plane.

(a) What is the center of the circle? (b) Move B to a point in the Cartesian plane directly above the center such that the radius of the circle is 5. (c) Move B to a point in the Cartesian plane such that the radius of the circle is 4. (d) Move B to a point in the Cartesian plane such that the radius of the circle is 3. (e) Find the coordinates of two points with integer coordinates in the fourth quadrant on the circle that result in a circle of radius 5 with center equal to that found in part (a). (f) Use the concept of symmetry about the center, vertical line through the center of the circle, and horizontal line through the center of the circle to find three other points with integer coordinates in the other three quadrants that lie on the circle of radius 5 with center equal to that found in part (a).

2. 5 - 1, 5 6

CHAPTER REVIEW Things to Know Formulas Distance formula Midpoint formula Slope Parallel lines Perpendicular lines

52

d = 21x2 - x1 2 2 + 1y2 - y1 2 2 x1 + x2 y 1 + y 2 , ≤ 2 2 y2 - y1 if x1 ⬆ x2; undefined if x1 = x2 m= x2 - x1 Equal slopes 1m 1 = m 2 2 and different y-intercepts 1b1 ⬆ b2 2 1x, y2 = ¢

Product of slopes is - 1 1m 1 # m 2 = - 12 ; slopes are negative reciprocals of 1 each other am 1 = b m2

Graphs

Equations of Lines and Circles Vertical line

x = a ; a is the x-intercept

Horizontal line

y = b ; b is the y-intercept

Point–slope form of the equation of a line

y - y1 = m(x - x1); m is the slope of the line, (x1, y1) is a point on the line

Slope–intercept form of the equation of a line

y = mx + b; m is the slope of the line, b is the y-intercept

General form of the equation of a line

Ax + By = C; A, B not both 0

Standard form of the equation of a circle

(x - h)2 + (y - k)2 = r 2; r is the radius of the circle, (h, k) is the center of the circle

Equation of the unit circle

x2 + y2 = 1

General form of the equation of a circle

x 2 + y 2 + ax + by + c = 0 , with restrictions on a, b , and c

Objectives Section 1

You should be able to . . . 1 2 3 4 5 6 7

Example(s)

Use the distance formula Use the midpoint formula Graph equations by hand by plotting points Graph equations using a graphing utility Use a graphing utility to create tables Find intercepts from a graph Use a graphing utility to approximate intercepts

2, 3, 4 5 6, 7, 8 9, 10 11 12 13

Review Exercises 1(a)–4(a), 32(a), 34–36 1(b)–4(b), 35 7–9 6, 7–9 6 5 6

3

Find intercepts algebraically from an equation Test an equation for symmetry Know how to graph key equations

1 2, 3 4–6

7–9 10–14 15

3

1

Solve equations using a graphing utility

1–3

16, 17

4

1

1 2 3 4, 5 6

1(c)–4(c), 1(d)–4(d), 32(b), 33 37 20 18, 19 21–23 18, 19, 21–25 26–27 7 24 25

2

1 2

Calculate and interpret the slope of a line Graph lines given a point and the slope 3 Find the equation of a vertical line 4 Use the point–slope form of a line; identify horizontal lines 5 Find the equation of a line given two points 6 Write the equation of a line in slope–intercept form 7 Identify the slope and y-intercept of a line from its equation 8 Graph lines written in general form using intercepts 9 Find equations of parallel lines 10 Find equations of perpendicular lines 2

5

1 2 3

Write the standard form of the equation of a circle Graph a circle by hand and by using a graphing utility Work with the general form of the equation of a circle

7 8 9, 10 11, 12 1 2, 3 4, 5

28, 29 30, 31 30, 31, 35

Review Exercises In Problems 1– 4, find the following for each pair of points: (a) The distance between the points. (b) The midpoint of the line segment connecting the points. (c) The slope of the line containing the points. (d) Then interpret the slope found in part (c). 1. 10, 02; 14, 22

3. 14, - 42; 14, 82

2. 11, - 12; 1 - 2, 32

4. 1 - 2, - 12; 13, - 12

53

Graphs

6. Graph y = - x 2 + 15 using a graphing utility. Create a table of values to determine a good initial viewing window. Use a graphing utility to approximate the intercepts.

5. List the intercepts of the following graph. y

2 4

4

x

2

In Problems 7–9, determine the intercepts and graph each equation by hand by plotting points. Verify your results using a graphing utility. Label the intercepts on the graph. 7. 2x - 3y = 6

8. y = x 2 - 9

9. x 2 + 2y = 16

In Problems 10–14, test each equation for symmetry with respect to the x-axis, the y-axis, and the origin. 10. 2x = 3y 2

11. x 2 + 4y 2 = 16

13. y = x - x

14. x + x + y + 2y = 0

3

2

12. y = x 4 - 3x 2 - 4

2

15. Sketch a graph of y = x 3. In Problems 16 and 17, use a graphing utility to approximate the solutions of each equation rounded to two decimal places. All solutions lie between - 10 and 10. 16. x 3 - 5x + 3 = 0

17. x 4 - 3 = 2x + 1

In Problems 18–25, find an equation of the line having the given characteristics. Express your answer using either the general form or the slope–intercept form of the equation of a line, whichever you prefer. Graph the line. 18. Slope = - 2; containing the point 13, - 12

19. Slope = 0; containing the point 1 - 5, 42

20. Slope undefined; containing the point 1 - 3, 42

21. x@intercept = 2; containing the point 14, - 52

22. y@intercept = - 2; containing the point 15, - 32

23. Containing the points 13, - 42 and 12, 12

24. Parallel to the line 2x - 3y = - 4; containing the point 1 - 5, 32 25. Perpendicular to the line 3x - y = - 4; containing the point 1 - 2, 42 In Problems 26 and 27, find the slope and y-intercept of each line. 26. 4x + 6y = 36

27.

1 5 x + y = 10 2 2

In Problems 28 and 29, find the standard form of the equation of the circle whose center and radius are given. 28. 1h, k2 = 1 - 2, 32; r = 4

29. 1h, k2 = 1 - 1, - 22; r = 1

In Problems 30 and 31, find the center and radius of each circle. Graph each circle by hand. Determine the intercepts of the graph of each circle. 30. x 2 + y 2 - 2x + 4y - 4 = 0

31. 3x 2 + 3y 2 - 6x + 12y = 0

32. Show that the points A = 1 - 2, 02, B = 1 - 4, 42, and C = 18, 52 are the vertices of a right triangle in two ways: (a) By using the converse of the Pythagorean Theorem (b) By using the slopes of the lines joining the vertices

36. Find two numbers y such that the distance from 1 - 3, 22 to 15, y2 is 10. 2 37. Graph the line with slope containing the point 11, 22. 3 38. Make up four problems that you might be asked to do given the two points 1 - 3, 42 and 16, 12. Each problem should involve a different concept. Be sure that your directions are clearly stated. 39. Describe each of the following graphs in the xy-plane. Give justification. (a) x = 0 (b) y = 0 (c) x + y = 0 (d) xy = 0

33. Show that the points A = 12, 52, B = 16, 12, C = 18, - 12 lie on a straight line by using slopes.

and

34. Show that the points A = 11, 52, B = 12, 42, and C = 1 - 3, 52 lie on a circle with center 1 - 1, 22. What is the radius of this circle? 35. The endpoints of the diameter of a circle are 1 - 3, 22 and 15, - 62. Find the center and radius of the circle. Write the general equation of this circle.

54

(e) x 2 + y 2 = 0

Graphs

The Chapter Test Prep Videos are step-by-step test solutions available in Channel. , or on this text’s

CHAPTER TEST

1. Suppose the points 1 - 2, - 32 and 14, 52 are the endpoints of a line segment. (a) Find the distance between the two points. (b) Find the midpoint of the line segment connecting the two points.

7. Use P1 = 1 - 1, 32 and P2 = 15, - 12 . (a) Find the slope of the line containing P1 and P2 . (b) Interpret this slope. 8. Sketch the graph of y 2 = x . 9. List the intercepts and test for symmetry: x 2 + y = 9 .

In Problems 2 and 3, graph each equation by hand by plotting points. Use a graphing utility to approximate the intercepts and label them on the graph. 2. 2x - 7y = 21

10. Write the slope–intercept form of the line with slope - 2 containing the point 13, - 42 . Graph the line. 11. Write the general form of the circle with center 14, - 32 and radius 5.

3. y = x 2 - 5

In Problems 4–6, use a graphing utility to approximate the real solutions of each equation rounded to two decimal places. All solutions lie between - 10 and 10.

12. Find the center and radius of the circle x 2 + y 2 + 4x - 2y - 4 = 0 . Graph this circle. 13. For the line 2x + 3y = 6 , find a line parallel to it containing the point 11, - 12 . Also find a line perpendicular to it containing the point (0, 3).

4. 2x 3 - x 2 - 2x + 1 = 0 5. x 4 - 5x 2 - 8 = 0 6. - x 3 + 7x - 2 = x 2 + 3x - 3

CHAPTER PROJECT Sale Price (000s of dollars)

291.5

268

320

305

371.5

375

303.5

283

351.5

350

314

275

332.5

356

295

300

Andy Dean Photography/Shutterstock

Zestimate (000s of dollars)

285

368

385

The graph below, called a scatter diagram, shows the points (291.5, 268), (320, 305), . . . , (368, 385) in a Cartesian plane. From the graph, it appears that the data follow a linear relation. Zestimate vs. Sale Price in Oak Park, IL Sale Price (thousands of dollars)

Internet-based Project Determining the Selling Price of a Home Determining how much to pay for a home is one of the more difficult decisions that must be made when purchasing a home. There are many factors that play a role in a home’s value. Location, size, number of bedrooms, number of bathrooms, lot size, and building materials are just a few. Fortunately, the website Zillow.com has developed its own formula for predicting the selling price of a home. This information is a great tool for predicting the actual sale price. For example, the data to the right show the “zestimate”—the selling price of a home as predicted by the folks at Zillow and the actual selling price of the home for homes in Oak Park, Illinois.

313

380 360 340 320 300 280 300 320 340 360 Zestimate (thousands of dollars)

55

Graphs

6. Choose a location in which you would like to live. Go to www.zillow.com and randomly select at least ten homes that have recently sold. (a) Draw a scatter diagram of your data. (b) Select two points from the scatter diagram and find the equation of the line through the points. (c) Interpret the slope. (d) Find a home from the Zillow website that interests you under the “Make Me Move” option for which a zestimate is available. Use your equation to predict the sale price based on the zestimate.

1. Imagine drawing a line through the data that appears to fit the data well. Do you believe the slope of the line would be positive, negative, or close to zero? Why? 2. Pick two points from the scatter diagram. Treat the zestimate as the value of x and treat the sale price as the corresponding value of y. Find the equation of the line through the two points you selected. 3. Interpret the slope of the line. 4. Use your equation to predict the selling price of a home whose zestimate is \$335,000. 5. Do you believe it would be a good idea to use the equation you found in part 2 if the zestimate is \$950,000? Why or why not?

15. The points will be on a vertical line that is 2 units to the right of the y-axis.

y 6 A  (3, 2) C  (2, 2)

(2, 1) (2, 0)

B  (6, 0)

5 x (2, 3)

(2, 1)

F  (6, 3)

29. Xmin = - 6, Xmax = 6, Xscl = 2, Ymin = - 1, Ymax = 3, Yscl = 1 33. 15

35. 110

49. d(A, B) d(B, C) d(A, C) ( 113)2

= = = +

53. d(A, B) d(B, C) d(A, C) 42 + 52 Area

= = = = =

37. 2 117

113 113 126 ( 113)2 = ( 126)2 13 Area = square units 2

77.

y 5

4 141 5 ( 141)2 10 square units yx2

41. 153

39. 20

y A  (2, 5) 5

y 5

B  (0, 3)

2

5 x

= = = + =

A  (4, 3)

73. (0, 2), (1, 0), (0, - 2)

y  2x  8

81.

y 5

(4, 0) 10 x

87.

(0, 2)

y 10 (0, 9) (2, 0)

5 (3, 0)

x

(2, 0) 10 x 9x 2  4y  36

56

B  (6, 0) 10 x

75. (- 4 , 0), (- 1, 0), (0, - 3), (4, 0)

(0, 8)

2x  3y  6

C  (5, 5)

59. (5, –1)

(1, 0) (0, 1)

y 5

A  (5, 3)

67. (0, 2) and 1 12, 122 are on the graph. 69. (- 1 , 0), (1, 0)

5 x

85.

y 10

a b 61. a , b 2 2 63. (0, 0) is on the graph. 65. (0, 3) is on the graph.

5 x

y 10

12. T

47. 2 165

1130 126 2 126 (2 126)2 = ( 1130)2 26 square units

55. (4, 0) 57. 13, 32 C  (4, 2)

79.

11. F

31. Xmin = 3, Xmax = 9, Xscl = 1, Ymin = 2, Ymax = 10, Yscl = 2

45. 4 110

51. d(A, B) d(B, C) d(A, C) ( 126)2 Area

B  (1, 3)

C  (1, 0)

(0, 2)

(2, 0)

43. 2a + b 2

10. F

17. (–1, 4); Quadrant II 19. (3, 1); Quadrant I 21. Xmin = - 11, Xmax = 5, Xscl = 1, Ymin = - 3, Ymax = 6, Yscl = 1 23. Xmin = - 30, Xmax = 50, Xscl = 10, Ymin = - 90, Ymax = 50, Yscl = 10 25. Xmin = - 10, Xmax = 110, Xscl = 10, Ymin = - 10, Ymax = 160, Yscl = 10 27. Xmin = - 6, Xmax = 6, Xscl = 2, Ymin = - 4, Ymax = 4, Yscl = 2

(2, 4)

y 5

D  (6, 5)

7 x

E  (0, 3)

9. midpoint

71. a -

p p , 0 b , 10, 12, a , 0 b 2 2

83.

y 5 (0, 4)

y  x2  1 (1, 0) 5 x

(2, 0) y  x 2  4

(2, 0) 5 x

Graphs 91. x-intercepts: - 2.74, 2.74 y-intercept: - 15

89. x-intercept: 6.5 y-intercept: - 13

15

10

5

95. x-intercepts: - 2.72, 2.72 y-intercept: 12.33

93. x-intercept: 14.33 y-intercept: - 21.5

15

10 4

10 4

20

4

5

15

20

5

10

25

97. (5, 3) 99. 117; 2 15; 129 101. d(P1, P2) = 6; d(P2, P3) = 4; d(P1, P3) = 2 113 ; right triangle 103. d1P1, P2 2 = 2 117; d1P2, P3 2 = 134; d1P1, P3 2 = 134; isosceles right triangle 105. 90 12 ⬇ 127.28 ft 107. (a) (90, 0), (90, 90), (0, 90) (b) 5 12161 ⬇ 232.43 ft (c) 30 1149 ⬇ 366.20 ft 109. d = 50t 111. (a) (2.65, 1.6) (b) ⬇ 1.285 units 113. \$20,155 115 (a) y = 2x 2 and y = |x| have the same graph. (b) 2x 2 = |x| (c) x Ú 0 for y = ( 1x)2 , while x can be any real number for y = x . (d) y Ú 0 for y = 2x 2

2 Assess Your Understanding 3. intercepts 4. y = 0 11. ( - 2, 0), (0, 2)

y 10

9. F

10. F

15. ( - 1, 0), (1, 0), (0, - 1) y 5

y  2x  8 (0, 8)

(0, 2)

(2, 0)

8. T

13. ( - 4, 0), (0, 8)

yx2

y 5

7. ( - 3, 4)

5. y-axis 6. 4

(4, 0) 5 x

10 x

17. ( - 2, 0), (2, 0), (0, 4)

y  x2  1 (2, 0)

(1, 0) 5 x

(1, 0)

23.

y 10 (0, 9)

5 x

(2, 0) 10 x

(2, 0)

(c)  (3, 4)

9x  4y  36 2

31.

5 x

41.

43.

y 4 5 x

(2, 5) (0, 9)

61.

(, 0)

(2, 5)

  , 2 2

63.

y 5

(1, 1)

(, 0) (0, 0) 5 x

(1, 1) 5 x (0, 0)

y (1, 1) 5 (4, 2) (0, 0)

y 5 (2, 1) (a)  (2, 1)

27. (b)  (2, 1) 5 x (c)  (2, 1)

(c)  (5, 2)

y 5

29. (a)  (5, 2)

x  y  9 or y  x9 (9, 0)

y 5

(b)  (5, 2)

79. 4

x

(c)  (3, 4) 5 x

(5, 2) (3, 4)

(b)  (3, 4)

37. (a) (0, 0) (b) Symmetric with respect to the x-axis

45. ( - 4, 0), (0, - 2), (0, 2); symmetric with respect to the x-axis 47. (0, 0); symmetric with respect to the origin 49. (0, - 9), (3, 0), ( - 3, 0); symmetric with respect to the y-axis 51. ( - 2, 0), (2, 0), (0, - 3), (0, 3); symmetric with respect to the x-axis, y-axis, and origin 53. (0, - 27), (3, 0) ; no symmetry 55. (0, - 4), (4, 0), ( - 1, 0) ; no symmetry 57. (0, 0) ; symmetric with respect to the origin 59. (0, 0) ; symmetric with respect to the origin 65. b = 13 67. a = - 4 or a = 1 69. (a) 10, - 52, 1 - 25, 02, 1 25, 02 (b) symmetric with respect to the y-axis

5 x

73. (a) 10, 32,10, - 32, 1 - 3, 02, 13, 02 (b) symmetric with respect to the x-axis, y-axis, and origin (c) y x2  y2  9

(0, 3) 1 x

5 (0, 3)

(3, 0)

(3, 0) 5 x

(0, 3)

77. ( - 1, - 2)

y 5 (a)  (3, 4)

5 x

(c)

y  x2  5 y 2

( 5, 0)

(2, 1) (1, 4)

(2, 1)

( 5, 0)

(0, 5)

71. (a) 1 - 9, 02, 10, - 32, 10, 32 (b) symmetric with respect to the x-axis (c) 2

5 (3, 0)

y  x  4

(a)  (3, 4)

y 5 , 2 2

2x  3y  6

(0, 2)

(2, 0) 5 x

33. (a) ( - 1, 0), (1, 0) p p 35. (a) a - , 0 b , (0, 1), a , 0 b (b) Symmetric with respect to 2 2 the x-axis, the y-axis, and the (b) Symmetric with respect to the origin y-axis 39. (a) ( - 2, 0), (0, 0), (2, 0) (b) Symmetric with respect to the origin

y 5 (a)  (0, 3) (c)  (0, 3) (0, 3) (b)  (0, 3)

25.

y 5 (b)  (3, 4) (3, 4)

y 5

2

(0, 1)

21. ( - 2, 0), (2, 0), (0, 9)

19. (3, 0), (0, 2)

y 5 (0, 4)

81. (a) (0, 0), (2, 0), (0, 1), (0, - 1)

5 x (1, 4)

75. (a) 10, 02, 1 - 2, 02, 12, 02 (b) symmetric with respect to the origin (c) y (1, 3) (2, 0)

5

y  x 3  4x

(2, 0) 5 x (1, 3)

(0, 3)

(b) x-axis symmetry

57

Graphs

3 Assess Your Understanding 3. ZERO

5. 5 - 2.21, 0.54, 1.68 6

4. F

15. No real solutions 17. 5 - 18 6 31. 5 - 2, - 1, 2 6

33. 5 15 6

7. 5 - 1.55, 1.15 6

19. 5 - 4 6

21. e

46 f 5

9. 5 - 1.12, 0.36 6 23. 5 3 6

11. 5 - 2.69, - 0.49, 1.51 6

25. 5 2 6

13. 5 - 2.86, - 1.34, 0.20, 1.00 6

2 29. e - , 2 f 3

27. 5 - 4, 7 6

1 35. e - 4, - f 8

4 Assess Your Understanding 1 2. 3; 2 3. y = b; y-intercept 4. T 5. F 6. T 7. m 1 = m 2; y-intercepts; m1m2 = - 1 8. 2 9. 10. F 2 1 1 (b) If x increases by 2 units, y will increase by 1 unit. 13. (a) Slope = - (b) If x increases by 3 units, y will decrease by 1 unit. 11. (a) Slope = 2 3 3 1 15. Slope = 17. Slope = 19. Slope = 0 21. Slope undefined 2 2 1. undefined; 0

y 5

y 8

y 5 5

8 x

25.

y 8 (2, 5) 3

P  (1, 2)

(3, 1) (2,1)

27.

y 8 P  (2, 4)

3

4

1 5 x

(1, 2)

5 x

(4, 0)

23.

5

(2, 1)

(2, 3)

(2, 3)

y

x

5 x

(1, 2)

29.

y 5

y 5 P  (0, 3)

P  (1, 3) 5 x

(6, 1) 7 x

5 x

1 35. ( - 1, - 5); (0, - 7); (1, - 9) 37. x - 2y = 0 or y = x 2 1 5 39. x + y = 2 or y = - x + 2 41. 2x - y = 3 or y = 2x - 3 43. x + 2y = 5 or y = - x + 45. 3x - y = - 9 or y = 3x + 9 2 2 2 1 1 5 1 47. 2x + 3y = - 1 or y = - x 49. x - 2y = - 5 or y = x + 51. 3x + y = 3 or y = - 3x + 3 53. x - 2y = 2 or y = x - 1 55. x = 2; 3 3 2 2 2 no slope–intercept form 57. y = 2 59. 2x - y = - 4 or y = 2x + 4 61. 2x - y = 0 or y = 2x 63. x = 4; no slope–intercept form 1 3 65. 2x + y = 0 or y = - 2x 67. x - 2y = - 3 or y = x + 69. y = 4 2 2 1 1 71. Slope = 2; y@intercept = 3 73. Slope = 2; y@intercept = - 2 77. Slope = - ; y@intercept = 2 75. Slope = ; y@intercept = 2 2 2 31. (2, 6); (3, 10); (4, 14)

33. (4, - 7); (6, - 10); (8, - 13)

y 5

y 8 (1, 5)

(0, 3)

(1, 0) 5 x

(0, 2)

y 5 (0, 2)

y 5 (0, 2)

(2, 3)

5 5 x

x

(4, 0)

5 x

79. Slope =

2 ; y@intercept = - 2 3

81. Slope = - 1; y@intercept = 1

y 5

y 5

y 2.5 (3, 0)

y 8 (0, 5)

(0, 1)

(1, 0) 2.5 x

5 x

(0, 2)

85. Slope = 0; y@intercept = 5

83. Slope undefined; no y-intercept

(4, 0) 5 x 5 x

87. Slope = 1; y@intercept = 0 y 5 (0, 0)

3 ; y@intercept = 0 2

y 5 (2, 2) (1, 1)

58

89. Slope =

5 x

91. (a) x-intercept: 3; y-intercept: 2 (b) y 5

(2, 3) (0, 0) 5 x

93. (a) x-intercept: - 10 ; y-intercept: 8 (b) y 10 (0, 8)

(0, 2)

(10, 0)

5 (3, 0)

x

8 x

Graphs 97. (a) x-intercept: 2; y-intercept: 3

95. (a) x-intercept: 3; 21 y-intercept: 2 (b) y 0,

21 2

(b)

101. 103. 105. 107.

(b) y 5

12

y 5

(0, 3)

(3, 0) 5 x

y = 0 Parallel Neither x - y = - 2 or y = x + 2 1 109. x + 3y = 3 or y = - x + 1 3

99. (a) x-intercept: 5; y-intercept: - 2

(2, 0) 5 x

(5, 0) 6 x (0, 2)

2 3 111. P1 = ( - 2, 5), P2 = (1, 3), m 1 = - ; P2 = (1, 3), P3 = ( - 1, 0), m 2 = ; because m 1m 2 = - 1 , the lines are perpendicular and the 3 2 points ( - 2, 5), (1, 3), and ( - 1, 0) are the vertices of a right triangle. 113. P1 = ( - 1, 0), P2 = (2, 3), m = 1; P3 = (1, - 2), P4 = (4, 1) , m = 1; P1 = ( - 1, 0), P3 = (1, - 2), m = - 1 ; P2 = (2, 3), P4 = (4, 1), m = - 1 ; opposite sides are parallel, and adjacent sides are perpendicular; the points are the vertices of a rectangle. 115. C = 0.20x + 29; +51.00; +75.00 117. C = 0.15x + 1289 5 2 (F - 32); approximately 21.1C 123. (a) y = - x + 30 (b) x-intercept: 375; 9 25 The ramp meets the floor 375 in. (31.25 ft) from the base of the platform. (c) The ramp does not meet design requirements. It has a run of 31.25 ft long. (d) The only slope possible for the ramp to comply with the requirement is for it to drop 1 in. for every 12-in. run. 1 125. (a) A = x + 20,000 (b) +80,000 (c) Each additional box sold requires an additional +0.20 in 5 advertising. 127. All have the same slope, 2; the lines are parallel.

121. C =

119. (a) C = 0.11x + 11.47, 0 … x … 600 (b) C  0.11x  11.47 0 ⱕ x ⱕ 600

Cost (dollars)

C

100 80 60 (0, 11.47) 40 20 0 100

(600, 77.47)

300 500 x kW-hr

y 2x  y  0 6 (0, 4) (0, 0) 2x  y  2

(c) +33.47 (d) +66.47 (e) Each additional kW-hr used adds +0.11 to the bill.

5 x (0, 2)

2x  y  4

129. (b), (c), (e), (g) 131. (c) 137. No; no 139. They are the same line. 141. Yes, if the y-intercept is 0.

6. F 7. Center (2, 1); radius = 2; (x - 2)2 + (y - 1)2 = 4

11. x 2 + y 2 = 4; x2 + y2 - 4 = 0

13. x 2 + (y - 2)2 = 4; x 2 + y 2 - 4y = 0

y 5

y 5

5 3 5 2 9 9. Center a , 2 b ; radius = ; a x - b + (y - 2)2 = 2 2 2 4

15. (x - 4)2 + (y + 3)2 = 25; x 2 + y 2 - 8x + 6y = 0 y 2

(0, 2)

y (2, 1) 6

(0, 0)

9 x

5 x

(4, 3)

5 x

1 2 1 b + y2 = ; 2 4 x2 + y2 - x = 0

19. a x -

21. (a) (h, k) = (0, 0); r = 2 (b) y 5

y 2.5

4 x

23. (a) (h, k) = (3, 0); r = 2 (b) y 5

25. (a) (h, k) = (1, 2); r = 3 (b) y 5

(3, 0)

(0, 0)

5 x (1, 2)

6 x

5 x 1, 0 2

17. (x + 2)2 + (y - 1)2 = 16; x 2 + y 2 + 4x - 2y - 11 = 0

2.5 x

(c) ({ 2, 0); (0, { 2) 27. (a) (h, k) = ( - 2, 2); r = 3 (b) y (2, 2) 6

1 1 29. (a) (h, k) = a , - 1 b ; r = 2 2 (b) y 1, 1 2

(c) ( - 2 { 15, 0); (0, 2 { 15)

(c) (0, - 1)

31. (a) (h, k) = (3, - 2); r = 5 (b) y 3

2.5

8 x 2.5 x

4 x

(c) (1 { 15, 0); (0, 2 { 2 12)

(c) (1, 0); (5, 0)

(3, 2)

(c) (3 { 121, 0); (0, - 6), (0, 2)

33. (a) (h, k) = ( - 2, 0); r = 2 (b) y (2, 0)

5

5 x

(c) (0, 0), ( - 4, 0)

59

Graphs 35. x 2 + y 2 = 13

37. (x - 2)2 + (y - 3)2 = 9

49. x + (y - 139) = 15,625 2

2

39. (x + 1)2 + (y - 3)2 = 5

51. x + y + 2x + 4y - 4168.16 = 0 2

2

41. (x + 1)2 + (y - 3)2 = 1

53. 12x + 4y - 9 12 = 0

55. (1, 0)

43. (c)

47. 18 units2

45. (b)

57. y = 2

59. (b), (c), (e), (g)

Review Exercises 1 1 4 (d) For each run of 2, there is a rise of 1. 2. (a) 5 (b) a - , 1 b (c) 2 2 3 (d) For each run of 3, there is a rise of - 4. 3. (a) 12 (b) (4, 2) (c) Undefined (d) No change in x 4. (a) 5 1. (a) 2 15

(b) (2, 1) (c)

(d) No change in y 6.

1 (b) a , - 1 b 2

(c) 0

5. (−4, 0), (0, 2), (0, 0), (0, −2), (2, 0) ; −3.87, 3.87

20

7.

8.

y 5

(3, 0)

(3, 0) 5 x (0, –2)

5

–5

9.

y 10

(3, 0)

y 10 (0, 8) (4, 0)

(4, 0)

10 x

10 x

(0, 9)

–5

10. x-axis

11. x-axis, y-axis, origin

15.

y 10

y  x3

12. y-axis

13. Origin

14. no symmetry

16. { - 2.49, 0.66, 1.83 } 17. { - 1.14, 1.64 }

y 5

y

(1, 1) (0, 0) 5 x

(1, 1)

19. y = 4

18. 2x + y = 5 or y = - 2x + 5

10

(5, 4) 3 x

5 x (3, 1)

20. x = - 3; no slope–intercept form

(3, 4)

5 21. 5x + 2y = 10 or y = - x + 5 2

y 5

1 22. x + 5y = - 10 or y = - x - 2 5

y 4

y 5 5 x

5 x

6 x

(4, 5)

23. 5x + y = 11 or y = - 5x + 11

24. 2x - 3y = - 19 or y =

(5, 3)

2 19 x + 3 3

1 10 25. x + 3y = 10 or y = - x + 3 3

y 8

y 5

y 5

(5, 3)

(2, 1)

(2, 4)

5 x (3, 4)

2 26. Slope = - , y-intercept = 6 3

1 27. Slope = - , y-intercept = 4 5

30. Center (1, −2); radius = 3

31. Center (1, −2); radius = 15

y 5

5 x

2 x

28. 1x + 22 2 + 1y - 32 2 = 16

29. 1x + 12 2 + 1y + 22 2 = 1

y 5

5 x

5 x

(1, 2)

(1, 2)

1 0, - 2 {2 22 2 , 1 1 { 25, 0 2

(0, 0), (0, −4), (2, 0)

32. (a) d(A, B) = 2 15; d(B, C) = 1145; d(A, C) = 5 15; Since d(A, B)2 + d1A, C2 2 = d1B, C2 2 , by the converse of the Pythagorean Theorem, the points A, B, and C are vertices of a right triangle. 1 1 1 (b) Slope of AB = −2; Slope of BC = ; Slope of AC = ; Since 1 - 22 a b = - 1 , the lines AB and AC are perpendicular and hence form a right 12 2 2 angle.

60

Graphs 33. m AB = - 1; m BC = - 1 34. Let D = (−1, 2). Then d(A, D) = d(B, D) = d(C, D) = 113. A, B, and C lie on the circle (x + 1)2 + 1y - 22 2 = 13 which has radius 113. 35. Center (1, −2); radius = 4 12 ; x 2 + y 2 - 2x + 4y - 27 = 0 36. - 4 and 8

37.

y 5 (4, 4) (1, 2) 5 x

Chapter Test 1. (a) 10 (b) (1, 1) 2.

3.

y 8 10 (7, 5)

7.

(10.5, 0)

(2.24, 0)

x (7, 1) (0, 3)

(2, 1) (1, 4)

2 3 (b) For every 3-unit change in x, y will change by −2 units.

(a) m = −

8.

y 5 (1, 1) (0, 0) (1, 1)

10. y = - 2x + 2 y 5 (0, 2)

11. x 2 + y 2 - 8x + 6y = 0

(2.24, 0) 5 x (2, 1) (1, 4) (0, 5)

(4, 2)

9. Intercepts: (−3, 0), (3, 0), (0, 9); symmetric with respect to the y-axis

(9, 3)

10 x (4, 2)

(9, 3)

12. Center: (−2, 1); radius: 3 (2, 1)

(1, 0) 5 x

4. 5 - 1, 0.5, 1 6 5. 5 - 2.50, 2.50 6 6. 5 - 2.46, - 0.24, 1.70 6

y 5

y 5

2 1 13. Parallel line: y = - x - ; 3 3 3 perpendicular line: y = x + 3 2

4 x

(3, 4)

61

Functions and Their Graphs Outline 1 2 3 4

Functions The Graph of a Function Properties of Functions Library of Functions; Piecewise-defined Functions

5 Graphing Techniques: Transformations 6 Mathematical Models: Building Functions Chapter Review

Chapter Test Chapter Projects

Choosing a Cellular Telephone Plan Most consumers choose a cellular telephone provider first, and then select an appropriate plan from that provider. The choice as to the type of plan selected depends upon your use of the phone. For example, is text messaging important? How many minutes do you plan to use the phone? Do you desire a data plan to browse the Web? The mathematics learned in this chapter can help you decide the plan best-suited for your particular needs.

—See the Internet-based Chapter Project—

Stephen Coburn/Shutterstock

In this chapter, we look at a special type of equation involving two variables called a function. This chapter deals with what a function is, how to graph functions, properties of functions, and how functions are used in applications. The word function apparently was introduced by René Descartes in 1637. For him, a function simply meant any positive integral power of a variable x. Gottfried Wilhelm Leibniz (1646–1716), who always emphasized the geometric side of mathematics, used the word function to denote any quantity associated with a curve, such as the coordinates of a point on the curve. Leonhard Euler (1707–1783) employed the word to mean any equation or formula involving variables and constants. His idea of a function is similar to the one most often seen in courses that precede calculus. Later, the use of functions in investigating heat flow equations led to a very broad definition, due to Lejeune Dirichlet (1805–1859), which describes a function as a correspondence between two sets. It is his definition that we use here.

1

63

Functions and Their Graphs

1 Functions PREPARING FOR THIS SECTION Before getting started, review the following: • Intervals • Solving Inequalities

• Evaluating Algebraic Expressions, Domain of a Variable

Now Work the ‘Are You Prepared?’ problems.

OBJECTIVES 1 Determine Whether a Relation Represents a Function 2 Find the Value of a Function 3 Find the Domain of a Function Defined by an Equation 4 Form the Sum, Difference, Product, and Quotient of Two Functions

1 Determine Whether a Relation Represents a Function

Figure 1 y 5 y  3x 1 (1, 2) 4

2 (0, 1)

2

4

x

5

E X A MP L E 1

Often there are situations where one variable is somehow linked to the value of another variable. For example, an individual’s level of education is linked to annual income. Engine size is linked to gas mileage. When the value of one variable is related to the value of a second variable, we have a relation. A relation is a correspondence between two sets. If x and y are two elements in these sets and if a relation exists between x and y, then we say that x corresponds to y or that y depends on x, and we write x S y. There are a number of ways to express relations between two sets. For example, the equation y = 3x - 1 shows a relation between x and y. It says that if we take some number x, multiply it by 3, and then subtract 1 we obtain the corresponding value of y. In this sense, x serves as the input to the relation and y is the output of the relation. We can also express this relation as a graph as shown in Figure 1. Not only can a relation be expressed through an equation or graph, but we can also express a relation through a technique called mapping. A map illustrates a relation by using a set of inputs and drawing arrows to the corresponding element in the set of outputs. Ordered pairs can be used to represent x S y as (x, y).

Maps and Ordered Pairs as Relations Figure 2 shows a relation between states and the number of representatives each state has in the House of Representatives. The relation might be named “number of representatives.”

Number of Representatives 1 7 9 27 53

In this relation, Alaska corresponds to 1, Arizona corresponds to 9, and so on. Using ordered pairs, this relation would be expressed as

5 (Alaska, 1), (Arizona, 9), (California, 53), (Colorado, 7), (Florida, 27), (North Dakota, 1) 6 One of the most important concepts in algebra is the function. A function is a special type of relation. To understand the idea behind a function, let’s revisit the relation presented in Example 1. If we were to ask, “How many representatives 64

Functions and Their Graphs

Figure 3 Person

Phone number

Dan

555 – 2345

Gizmo

549 – 9402 930 – 3956

Colleen

555 – 8294

Phoebe

839 – 9013

does Alaska have?,” you would respond “1.” In fact, each input state corresponds to a single output number of representatives. Let’s consider a second relation where we have a correspondence between four people and their phone numbers. See Figure 3. Notice that Colleen has two telephone numbers. If asked, “What is Colleen’s phone number?” you cannot assign a single number to her. Let’s look at one more relation. Figure 4 is a relation that shows a correspondence between animals and life expectancy. If asked to determine the life expectancy of a dog, we would all respond “11 years.” If asked to determine the life expectancy of a rabbit, we would all respond “7 years.” Figure 4 Animal

Life Expectancy

Dog

11

Duck

10

Rabbit

7

Notice that the relations presented in Figures 2 and 4 have something in common. What is it? The common link between these two relations is that each input corresponds to exactly one output. This leads to the definition of a function.

DEFINITION Figure 5

y

x

Range

X Domain

Y

E XAM PL E 2

Let X and Y be two nonempty sets.* A function from X into Y is a relation that associates with each element of X exactly one element of Y. The set X is called the domain of the function. For each element x in X, the corresponding element y in Y is called the value of the function at x, or the image of x. The set of all images of the elements in the domain is called the range of the function. See Figure 5. Since there may be some elements in Y that are not the image of some x in X, it follows that the range of a function may be a subset of Y, as shown in Figure 5. For example, consider the function y = x 2 . Since x2 Ú 0 for all real numbers x, the range of y = x2 is 5 y  y Ú 0 6, which is a subset of the set of all real numbers , Y. Not all relations between two sets are functions. The next example shows how to determine whether a relation is a function.

Determining Whether a Relation Represents a Function Determine which of the following relations represent a function. If the relation is a function, then state its domain and range. (a) See Figure 6. For this relation, the domain represents the number of calories in a sandwich from a fast-food restaurant and the range represents the fat content (in grams).

Figure 6

Calories

Fat

(Wendy's Single) 470

21

(Burger King Whopper) 670

40

(Burger King Chicken Sandwich) 630

39

(McDonald's Big Mac) 540

29

(McDonald's McChicken) 360

16

Source: Each company’s Web site.

* The sets X and Y will usually be sets of real numbers, in which case a (real) function results. The two sets can also be sets of complex numbers, and then we have defined a complex function. In the broad definition (due to Lejeune Dirichlet), X and Y can be any two sets.

65

Functions and Their Graphs

(b) See Figure 7. For this relation, the domain represents gasoline stations in Collier County, Florida, and the range represents the price per gallon of regular unleaded gasoline in July, 2011. (c) See Figure 8. For this relation, the domain represents the weight (in carats) of pear-cut diamonds and the range represents the price (in dollars). Figure 8

Figure 7

Source: Used with permission of Diamonds.com Price per gallon

Carats

Price

\$3.71

0.70

\$1529

Shell

\$3.72

0.71

\$1575

Sunoco

\$3.69

0.75

\$1765

0.78

\$1798

Gas Station Mobil

7-Eleven

Solution

\$1952

(a) The relation in Figure 6 is a function because each element in the domain corresponds to exactly one element in the range. The domain of the function is 5 470, 670, 630, 540, 360 6, and the range of the function is 5 21, 40, 39, 29, 16 6. (b) The relation in Figure 7 is a function because each element in the domain corresponds to exactly one element in the range. The domain of the function is 5 Mobil, Shell, Sunoco, 7@Eleven 6. The range of the function is 5 3.69, 3.71, 3.72 6. Notice that it is okay for more than one element in the domain to correspond to the same element in the range (Shell and 7-Eleven each sell gas for \$3.72 a gallon). (c) The relation in Figure 8 is not a function because each element in the domain does not correspond to exactly one element in the range. If a 0.71-carat diamond is chosen from the domain, a single price cannot be assigned to it.

Now Work

In Words For a function, no input has more than one output. The domain of a function is the set of all inputs; the range is the set of all outputs.

E X A MP L E 3

PROBLEM

15

The idea behind a function is its predictability. If the input is known, we can use the function to determine the output. With “nonfunctions,” we don’t have this predictability. Look back at Figure 6. If asked, “How many grams of fat are in a 470-calorie sandwich?” we can use the correspondence to answer “21.” Now consider Figure 8. If asked, “What is the price of a 0.71-carat diamond?” we could not give a single response because two outputs result from the single input “0.71.” For this reason, the relation in Figure 8 is not a function. We may also think of a function as a set of ordered pairs (x, y) in which no ordered pairs have the same first element and different second elements. The set of all first elements x is the domain of the function, and the set of all second elements y is its range. Each element x in the domain corresponds to exactly one element y in the range.

Determining Whether a Relation Represents a Function Determine whether each relation represents a function. If it is a function, state the domain and range.

Solution

66

(a) 5 (1, 4), (2, 5), (3, 6), (4, 7) 6 (b) 5 (1, 4), (2, 4), (3, 5), (6, 10) 6 (c) 5 ( -3, 9), ( -2, 4), (0, 0), (1, 1), ( -3, 8) 6

(a) This relation is a function because there are no ordered pairs with the same first element and different second elements. The domain of this function is 5 1, 2, 3, 4 6, and its range is 5 4, 5, 6, 7 6 .

Functions and Their Graphs

(b) This relation is a function because there are no ordered pairs with the same first element and different second elements. The domain of this function is 5 1, 2, 3, 6 6 , and its range is 5 4, 5, 10 6 . (c) This relation is not a function because there are two ordered pairs, ( -3, 9) and ( -3, 8), that have the same first element and different second elements. In Example 3(b), notice that 1 and 2 in the domain each have the same image in the range. This does not violate the definition of a function; two different first elements can have the same second element. A violation of the definition occurs when two ordered pairs have the same first element and different second elements, as in Example 3(c).

Now Work

PROBLEM

19

Up to now we have shown how to identify when a relation is a function for relations defined by mappings (Example 2) and ordered pairs (Example 3). But relations can also be expressed as equations. The circumstances under which equations are functions are discussed next. To determine whether an equation, where y depends on x, is a function, it is often easiest to solve the equation for y. If any value of x in the domain corresponds to more than one y, the equation does not define a function; otherwise, it does define a function.

E XAM PL E 4

Determining Whether an Equation Is a Function Determine if the equation y = 2x - 5 defines y as a function of x.

Solution

E XAM PL E 5

The equation tells us to take an input x, multiply it by 2, and then subtract 5. For any input x, these operations yield only one output y. For example, if x = 1, then y = 2(1) - 5 = -3. If x = 3, then y = 2(3) - 5 = 1. For this reason, the equation is a function.

Determining Whether an Equation Is a Function Determine if the equation x2 + y 2 = 1 defines y as a function of x.

Solution

To determine whether the equation x2 + y 2 = 1, which defines the unit circle, is a function, solve the equation for y. x2 + y2 = 1 y2 = 1 - x2 y = { 21 - x2

For values of x between -1 and 1, two values of y result. For example, if x = 0, then y = {1, so two different outputs result from the same input. This means that the equation x2 + y 2 = 1 does not define a function.

Now Work

PROBLEM

33

2 Find the Value of a Function Functions are often denoted by letters such as f, F, g, G, and others. If f is a function, then for each number x in its domain the corresponding image in the range is designated by the symbol f1x2, read as “ f of x” or as “ f at x.” We refer to f1x2 as the value of f at the number x; f1x2 is the number that results when x is given and the function f is applied; f1x2 is the output corresponding to x or the image of x; f1x2 does not mean “ f times x.” For example, the function given in Example 4 3 may be written as y = f1x2 = 2x - 5. Then f a b = -2. 2

67

Functions and Their Graphs

Figure 9 illustrates some other functions. Notice that, in every function, for each x in the domain there is one value in the range. Figure 9 1  f (1)  f (1)

1 1 0

0  f (0)

2

2  f ( 2) x

2

 1–2  F(2)

1

1  F(1)

Range

Domain

0

0  g(0)

0

1

1  g(1)

2

2  g(2)

3  G(0)  G(2)  G(3)

3

2  g(4)

4 x

g(x)  x

Domain

Range

x

G(x)  3

Domain

(c) g(x)  x

Range

(d) G (x)  3

Sometimes it is helpful to think of a function f as a machine that receives as input a number from the domain, manipulates it, and outputs a value. See Figure 10. The restrictions on this input/output machine are as follows:

Figure 10 Input x x

Range

1 (b) F (x)  – x

(a) f (x)  x 2

2

 F(4)

1 F(x )  – x

x

f(x )  x 2

Domain

1– 4

4

1. It only accepts numbers from the domain of the function. 2. For each input, there is exactly one output (which may be repeated for different inputs).

f

Output y  f(x)

E X A MP L E 6

For a function y = f1x2, the variable x is called the independent variable, because it can be assigned any of the permissible numbers from the domain. The variable y is called the dependent variable, because its value depends on x. Any symbols can be used to represent the independent and dependent variables. For example, if f is the cube function, then f can be given by f1x2 = x3 or f1t2 = t 3 or f 1z2 = z 3. All three functions are the same. Each tells us to cube the independent variable to get the output. In practice, the symbols used for the independent and dependent variables are based on common usage, such as using C for cost in business. The independent variable is also called the argument of the function. Thinking of the independent variable as an argument can sometimes make it easier to find the value of a function. For example, if f is the function defined by f1x2 = x3, then f tells us to cube the argument. Thus, f 122 means to cube 2, f1a2 means to cube the number a, and f 1x + h2 means to cube the quantity x + h.

Finding Values of a Function

For the function f defined by f 1x2 = 2x2 - 3x, evaluate

(a) f132

Solution

(e) -f 1x2

(b) f 1x2 + f132 (f) f 13x2

(c) 3f1x2

(g) f1x + 32

(d) f 1 -x2 f1x + h2 - f1x2 (h) h

(a) Substitute 3 for x in the equation for f , f1x2 = 2x2 - 3x, to get f132 = 2132 2 - 3132 = 18 - 9 = 9 The image of 3 is 9.

68

h ⬆0

Functions and Their Graphs

(b) f1x2 + f132 = 12x2 - 3x2 + 192 = 2x2 - 3x + 9 (c) Multiply the equation for f by 3. 3f1x2 = 312x2 - 3x2 = 6x2 - 9x (d) Substitute -x for x in the equation for f and simplify. f1 -x2 = 21 -x2 2 - 31 -x2 = 2x2 + 3x

Notice the use of parentheses here.

(e) -f1x2 = - 12x2 - 3x2 = -2x2 + 3x

(f) Substitute 3x for x in the equation for f and simplify. f13x2 = 213x2 2 - 313x2 = 219x2 2 - 9x = 18x2 - 9x

(g) Substitute x + 3 for x in the equation for f and simplify. f1x + 32 = 21x + 32 2 - 31x + 32

= 21x2 + 6x + 92 - 3x - 9 = 2x2 + 12x + 18 - 3x - 9 = 2x2 + 9x + 9 (h)

3 21x + h2 2 - 31x + h2 4 - 3 2x2 - 3x 4 f1x + h2 - f1x2 = h h c f1x + h2 = 2 1x + h2 2 - 3 1x + h2

= = = = =

21x2 + 2xh + h 2 2 - 3x - 3h - 2x2 + 3x Simplify. h 2x2 + 4xh + 2h 2 - 3h - 2x2 Distribute and combine like terms. h 2 4xh + 2h - 3h Combine like terms. h h(4x + 2h - 3) Factor out h. h Divide out the h's. 4x + 2h - 3

Notice in this example that f1x + 32 ⬆ f1x2 + f132 , f1 -x2 ⬆ -f1x2 , and 3f1x2 ⬆ f13x2. The expression in part (h) is called the difference quotient of f, an important expression in calculus.

Now Work

PROBLEMS

39

AND

75

Most calculators have special keys that allow you to find the value of certain commonly used functions. For example, you should be able to find the square function f1x2 = x2, the square root function f1x2 = 1x, the reciprocal function 1 f1x2 = = x -1, and many others (such as ln x and log x). Verify the results of x Example 7, which follows, on your calculator.

E XAM PL E 7

Finding Values of a Function on a Calculator (a) f1x2 = x2 1 (b) F1x2 = x

f11.2342 = 1.2342 = 1.522756 1 F11.2342 =  0.8103727715 1.234

(c) g1x2 = 1x

g11.2342 = 11.234  1.110855526

69

Functions and Their Graphs

COMMENT Graphing calculators can be used to evaluate any function that you wish. Figure 11 shows the result obtained in Example 6(a) on a TI-84 Plus graphing calculator with the function to be evaluated, f(x) = 2x 2 - 3x, in Y1 . Figure 11



Implicit Form of a Function

COMMENT The explicit form of a function is the form required by a graphing calculator. 

In general, when a function f is defined by an equation in x and y, we say that the function f is given implicitly. If it is possible to solve the equation for y in terms of x, then we write y = f1x2 and say that the function is given explicitly. For example, Implicit Form 3x + y = 5 x2 - y = 6 xy = 4

Explicit Form y = f 1x2 = -3x + 5 y = f 1x2 = x 2 - 6 4 y = f 1x2 = x

SUMMARY Important Facts about Functions (a) For each x in the domain of a function f, there is exactly one image f1x2 in the range; however, an element in the range can result from more than one x in the domain. (b) f is the symbol that we use to denote the function. It is symbolic of the equation (rule) that we use to get from an x in the domain to f1x2 in the range. (c) If y = f 1x2, then x is called the independent variable or argument of f, and y is called the dependent variable or the value of f at x.

3 Find the Domain of a Function Defined by an Equation Often the domain of a function f is not specified; instead, only the equation defining the function is given. In such cases, we agree that the domain of f is the largest set of real numbers for which the value f1x2 is a real number. The domain of a function f is the same as the domain of the variable x in the expression f1x2.

E X A MP L E 8

Finding the Domain of a Function Find the domain of each of the following functions:

Solution

70

(a) f 1x2 = x2 + 5x

(b) g1x2 =

3x x -4

(c) h 1t2 = 24 - 3t

(d) F1x2 =

23x + 12 x-5

2

(a) The function tells us to square a number and then add five times the number. Since these operations can be performed on any real number, we conclude that the domain of f is the set of all real numbers. (b) The function g tells us to divide 3x by x2 - 4. Since division by 0 is not defined, the denominator x2 - 4 can never be 0, so x can never equal -2 or 2. The domain of the function g is 5 x x ⬆ -2, x ⬆ 2 6 .

Functions and Their Graphs

In Words The domain of g found in Example 8(b) is {x | x ⬆ - 2, x ⬆ 2} . This notation is read, “The domain of the function g is the set of all real numbers x such that x does not equal - 2 and x does not equal 2.”

(c) The function h tells us to take the square root of 4 - 3t. But only nonnegative numbers have real square roots, so the expression under the square root (the radicand) must be nonnegative (greater than or equal to zero). This requires that 4 - 3t Ú 0 -3t Ú -4 4 t… 3 The domain of h is e t ` t …

4 4 f or the interval a - , d . 3 3

(d) The function F tells us to take the square root of 3x + 12 and divide this result by x - 5. This requires that 3x + 12 Ú 0, so x Ú -4 , and that x - 5 ⬆ 0 , so x ⬆ 5. Combining these two restrictions, the domain of F is {x  x Ú -4, x ⬆ 5}. For the functions that we will encounter in this text, the following steps may prove helpful for finding the domain of a function that is defined by an equation and whose domain is a subset of the real numbers.

Finding the Domain of a Function Defined by an Equation 1. Start with the domain as the set of real numbers. 2. If the equation has a denominator, exclude any numbers that give a zero denominator. 3. If the equation has a radical of even index, exclude any numbers that cause the expression inside the radical to be negative.

Now Work

PROBLEM

51

If x is in the domain of a function f, we shall say that f is defined at x, or f 1 x2 exists. If x is not in the domain of f, we say that f is not defined at x, or f 1 x2 does x not exist. For example, if f1x2 = 2 , then f102 exists, but f112 and f1 -12 do x -1 not exist. (Do you see why?) We will say more about finding the range when we look at the graph of a function in the next section. When a function is defined by an equation, it can be difficult to find the range. Therefore, we shall usually be content to find just the domain of a function when the function is defined by an equation. We shall express the domain of a function using inequalities, interval notation, set notation, or words, whichever is most convenient. When we use functions in applications, the domain may be restricted by physical or geometric considerations. For example, the domain of the function f defined by f 1x2 = x2 is the set of all real numbers. However, if f is used to obtain the area of a square when the length x of a side is known, then we must restrict the domain of f to the positive real numbers, since the length of a side can never be 0 or negative.

E XAMPL E 9

Finding the Domain in an Application Express the area of a circle as a function of its radius. Find the domain.

Solution Figure 12 A

r

See Figure 12. The formula for the area A of a circle of radius r is A = pr 2. If we use r to represent the independent variable and A to represent the dependent variable, the function expressing this relationship is A(r) = pr 2 In this setting, the domain is 5 r  r 7 0 6. (Do you see why?)

71

Functions and Their Graphs

Observe in the solution to Example 9 that the symbol A is used in two ways: It is used to name the function, and it is used to symbolize the dependent variable. This double use is common in applications and should not cause any difficulty.

Now Work

PROBLEM

89

4 Form the Sum, Difference, Product, and Quotient of Two Functions Next we introduce some operations on functions. We shall see that functions, like numbers, can be added, subtracted, multiplied, and divided. For example, if f1x2 = x2 + 9 and g1x2 = 3x + 5, then f1x2 + g1x2 = 1x2 + 92 + 13x + 52 = x2 + 3x + 14

The new function y = x2 + 3x + 14 is called the sum function f + g. Similarly, f 1x2 # g1x2 = 1x2 + 92 13x + 52 = 3x3 + 5x2 + 27x + 45

The new function y = 3x3 + 5x2 + 27x + 45 is called the product function f # g. The general definitions are given next.

DEFINITION Remember, the symbol x stands for intersection. It means you should find the elements that are common to two sets.

If f and g are functions: The sum f  g is the function defined by 1 f + g2 1x2 = f 1x2 + g1x2

In Words

DEFINITION

The domain of f + g consists of the numbers x that are in the domains of both f and g . That is, domain of f + g = domain of f  domain of g.

The difference f  g is the function defined by 1f - g2 1x2 = f 1x2 - g1x2 The domain of f - g consists of the numbers x that are in the domains of both f and g . That is, domain of f - g = domain of f  domain of g.

DEFINITION

The product f ~ g is the function defined by 1f # g2 1x2 = f1x2 # g1x2 The domain of f # g consists of the numbers x that are in the domains of both f and g . That is, domain of f # g = domain of f  domain of g.

DEFINITION

The quotient

f is the function defined by g f 1x2 f a b 1x2 = g g1x2

72

g1x2 ⬆ 0

Functions and Their Graphs

f consists of the numbers x for which g1x2 ⬆ 0 and that are in g the domains of both f and g . That is, The domain of

domain of

E X AM PL E 1 0

f = 5 x 0 g(x) ⬆ 0 6  domain of f  domain of g g

Operations on Functions Let f and g be two functions defined as f1x2 =

1 x+2

and g1x2 =

x x-1

Find the following, and determine the domain in each case.

Solution

(a) 1f + g2 1x2

(b) 1f - g2 1x2

(c) 1f # g2 1x2

f (d) a b (x) g

The domain of f is 5 x x ⬆ -2 6 and the domain of g is 5 x x ⬆ 1 6 . 1 x (a) 1f + g2 1x2 = f1x2 + g1x2 = + x+2 x-1 =

x1x + 22 x-1 x2 + 3x - 1 + = 1x + 22 1x - 12 1x + 22 1x - 12 1x + 22 1x - 12

The domain of f + g consists of those numbers x that are in the domains of both f and g . Therefore, the domain of f + g is 5 x x ⬆ -2, x ⬆ 1 6.

(b) 1f - g2 1x2 = f 1x2 - g1x2 = =

1 x x+2 x-1

x1x + 22 - 1x2 + x + 12 x-1 = 1x + 22 1x - 12 1x + 22 1x - 12 1x + 22 1x - 12

The domain of f - g consists of those numbers x that are in the domains of both f and g . Therefore, the domain of f - g is 5 x x ⬆ -2, x ⬆ 1 6.

(c) 1f # g2 1x2 = f1x2 # g1x2 =

1 # x x = x+2 x-1 1x + 22 1x - 12

The domain of f # g consists of those numbers x that are in the domains of both f and g . Therefore, the domain of f # g is 5 x x ⬆ -2, x ⬆ 1 6.

1 f 1x2 f 1 #x - 1 x-1 x+2 = = (d) a b 1x2 = = x g x g1x2 x+2 x1x + 22 x-1

f consists of the numbers x for which g(x) ⬆ 0 and that are in g the domains of both f and g . Since g(x) = 0 when x = 0, we exclude 0 as well f as -2 and 1 from the domain. The domain of is 5 x x ⬆ -2, x ⬆ 0, x ⬆ 1 6. g The domain of

Now Work

PROBLEM

63

In calculus, it is sometimes helpful to view a complicated function as the sum, difference, product, or quotient of simpler functions. For example, F1x2 = x2 + 1x is the sum of f1x2 = x2 and g(x) = 1x . x2 - 1 H1x2 = 2 is the quotient of f 1x2 = x2 - 1 and g1x2 = x2 + 1. x +1

73

Functions and Their Graphs

SUMMARY Function

A relation between two sets of real numbers so that each number x in the first set, the domain, has corresponding to it exactly one number y in the second set. A set of ordered pairs 1x, y2 or 1x, f1x22 in which no first element is paired with two different second elements. The range is the set of y values of the function that are the images of the x values in the domain. A function f may be defined implicitly by an equation involving x and y or explicitly by writing y = f 1x2.

Unspecified domain

If a function f is defined by an equation and no domain is specified, then the domain will be taken to be the largest set of real numbers for which the equation defines a real number. y = f 1x2 f is a symbol for the function. x is the independent variable or argument. y is the dependent variable. f 1x2 is the value of the function at x, or the image of x.

Function notation

1 Assess Your Understanding ‘Are You Prepared?’

Answers are given at the end of these exercises.

1. The inequality - 1 6 x 6 3 can be written in interval notation as___________. 1 2. If x = - 2, the value of the expression 3x 2 - 5x + is x _______.

x - 3 3. The domain of the variable in the expression is x + 4 _____________. 4. Solve the inequality: 3 - 2x 7 5. Graph the solution set.

Concepts and Vocabulary 5. If f is a function defined by the equation y = f 1x2, then x is called the variable and y is the ____________ variable. 6. The set of all images of the elements in the domain of a function is called the _________. 7. If the domain of f is all real numbers in the interval 3 0, 7 4 and the domain of g is all real numbers in the interval 3 - 2, 5 4 , the domain of f + g is all real numbers in the interval _________. f consists of numbers x for which 8. The domain of g g(x) 0 that are in the domains of both and ____.

Skill Building

9. If f 1x2 = x + 1 and g(x) = x 3, then____________ = x 3 - (x + 1).

10. True or False

Every relation is a function.

11. True or False The domain of 1f # g2 1x2 consists of the numbers x that are in the domains of both f and g.

12. True or False The independent variable is sometimes referred to as the argument of the function. 13. True or False If no domain is specified for a function f, then the domain of f is taken to be the set of real numbers. x2 - 4 14. True or False The domain of the function f 1x2 = is x 5 x x ⬆ {2 6 .

In Problems 15–26, determine whether each relation represents a function. For each function, state the domain and range. 15.

Person Elvis

Birthday

16.

Father

Jan. 8

Bob

Kaleigh

Mar. 15

John

Linda

Marissa

Sept. 17

Chuck

Marcia

Colleen

74

Daughter Beth Diane

Functions and Their Graphs

17. Hours Worked

18.

Salary

20 Hours

\$200 \$300

30 Hours

\$350

40 Hours

\$425

19. 5 (2, 6), ( - 3, 6), (4, 9), (2, 10) 6

Level of Education

Average Income

\$18,120 \$23,251

20. 5 ( - 2, 5), ( - 1, 3), (3, 7), (4, 12) 6

22. 5 (0, - 2), (1, 3), (2, 3), (3, 7) 6 25. 5 ( - 2, 4), ( - 1, 1), (0, 0), (1, 1) 6

\$36,055 \$45,810 \$67,165

21. 5 (1, 3), (2, 3), (3, 3), (4, 3) 6

23. 5 ( - 2, 4), ( - 2, 6), (0, 3), (3, 7) 6 26. 5 ( - 2, 16), ( - 1, 4), (0, 3), (1, 4) 6

24. 5 ( - 4, 4), ( - 3, 3), ( - 2, 2), ( - 1, 1), ( - 4, 0) 6

In Problems 27–38, determine whether the equation defines y as a function of x.

1 x

27. y = x 2

28. y = x 3

29. y =

31. y 2 = 4 - x 2

32. y = { 21 - 2x

33. x = y 2

30. y = 0 x 0

37. 2x 2 + 3y 2 = 1

38. x 2 - 4y 2 = 1

35. y = 2x 2 - 3x + 4

36. y =

3x - 1 x + 2

34. x + y 2 = 1

In Problems 39– 46, find the following for each function: (a) f 102

(b) f 112

39. f 1x2 = 3x 2 + 2x - 4

(c) f 1 - 12

(d) f 1 - x2

40. f 1x2 = - 2x 2 + x - 1

(e) - f 1x2

(f) f 1x + 12

41. f 1x2 =

x x + 1

2x + 1 3x - 5

44. f 1x2 = 2x 2 + x

45. f 1x2 =

47. f 1x2 = - 5x + 4

48. f 1x2 = x 2 + 2

55. h1x2 = 23x - 12

56. G1x2 = 21 - x

x x + 1 x - 2 53. F 1x2 = 3 x + x 4 57. f 1x2 = 2x - 9

51. g 1x2 =

x x 2 - 16

58. f 1x2 =

x

61. P 1t2 =

52. h1x2 =

59. p1x2 =

2x - 4

2t - 4 3t - 21

2x x2 - 4

2 7x - 1

49. f 1x2 =

(h) f 1x + h2

42. f 1x2 =

2

43. f 1x2 = 0 x 0 + 4

In Problems 47– 62, find the domain of each function.

(g) f 12x2

x2 - 1 x + 4

46. f 1x2 = 1 -

1 1x + 22 2

x2 x + 1 x + 4 54. G1x2 = 3 x - 4x 50. f 1x2 =

2

2

60. q1x2 = 2- x - 2 62. h1z2 =

2z + 3 z - 2

In Problems 63–72, for the given functions f and g, find the following. For parts (a)–(d), also find the domain. (a) 1f + g2 1x2 (e) 1f + g2 132

63. f 1x2 = 3x + 4; 65. f 1x2 = x - 1; 67. f 1x2 = 1x;

(b) 1f - g2 1x2 (f) 1f - g2 142

g 1x2 = 2x - 3

g 1x2 = 2x

73. Given f 1x2 = 3x + 1 and 1f + g2 1x2 = 6 function g.

64. f 1x2 = 2x + 1; 2

68. f 1x2 = 0 x 0 ;

72. f 1x2 = 2x + 1; 74. Given f 1x2 = function g.

g 1x2 = 3x - 2

g(x) = 4x 3 + 1

g 1x2 = x

70. f 1x2 = 2x - 1; 1 x, find the 2

f (h) a b 112 g

(g) ( f # g)(2)

66. f 1x2 = 2x + 3;

2

g 1x2 = 3x - 5 1 1 69. f 1x2 = 1 + ; g 1x2 = x x 2x + 3 4x ; g 1x2 = 71. f 1x2 = 3x - 2 3x - 2

f (d) a b 1x2 g

(c) 1f # g2 1x2

g 1x2 = 24 - x g 1x2 =

2 x

f x + 1 1 , find the and a b 1x2 = 2 x g x - x

75

Functions and Their Graphs

In Problems 75– 82, find the difference quotient of f ; that is, find 75. f 1x2 = 4x + 3 1 79. f 1x2 = 2 x

76. f 1x2 = - 3x + 1 1 80. f 1x2 = x + 3

f 1x + h2 - f 1x2 h

, h ⬆ 0, for each function. Be sure to simplify.

77. f 1x2 = x 2 - x + 4 81. f 1x2 = 2x

[Hint: Rationalize the numerator.]

78. f 1x2 = 3x 2 - 2x + 6 82. f 1x2 = 2x + 1

Applications and Extensions 83. If f 1x2 = 2x 3 + Ax 2 + 4x - 5 and f 122 = 5, what is the value of A?

84. If f 1x2 = 3x 2 - Bx + 4 and f 1 - 12 = 12, what is the value of B? 3x + 8 85. If f 1x2 = and f 102 = 2, what is the value of A? 2x - A 1 2x - B 86. If f 1x2 = and f 122 = , what is the value of B? 3x + 4 2

2x - A 87. If f 1x2 = and f 142 = 0, what is the value of A? x - 3 Where is f not defined? x - B , f 122 = 0 and f 112 is undefined, what are x - A the values of A and B?

88. If f 1x2 =

(b) When is the height of the rock 15 meters? When is it 10 meters? When is it 5 meters? (c) When does the rock strike the ground? 96. Effect of Gravity on Jupiter If a rock falls from a height of 20 meters on the planet Jupiter, its height H (in meters) after x seconds is approximately H(x) = 20 - 13x 2 (a) What is the height of the rock when x = 1 second? x = 1.1 seconds? x = 1.2 seconds? (b) When is the height of the rock 15 meters? When is it 10 meters? When is it 5 meters? (c) When does the rock strike the ground?

89. Geometry Express the area A of a rectangle as a function of the length x if the length of the rectangle is twice its width.

JPL/CalTech/NASA

90. Geometry Express the area A of an isosceles right triangle as a function of the length x of one of the two equal sides. 91. Constructing Functions Express the gross salary G of a person who earns \$10 per hour as a function of the number x of hours worked. 92. Constructing Functions Tiffany, a commissioned sales person, earns \$100 base pay plus \$10 per item sold. Express her gross salary G as a function of the number x of items sold. 93. Population as a Function of Age The function P(a) = 0.015a 2 - 4.962a + 290.580 represents the population P (in millions) of Americans that are a years of age or older. (a) Identify the dependent and independent variables. (b) Evaluate P( 20). Provide a verbal explanation of the meaning of P( 20). (c) Evaluate P (0). Provide a verbal explanation of the meaning of P( 0). 94. Number of Rooms The function N(r) = - 1.44r 2 + 14.52r - 14.96 represents the number N of housing units (in millions) that have r rooms, where r is an integer and 2 … r … 9. (a) Identify the dependent and independent variables. (b) Evaluate N(3). Provide a verbal explanation of the meaning of N(3). 95. Effect of Gravity on Earth If a rock falls from a height of 20 meters on Earth, the height H (in meters) after x seconds is approximately H(x) = 20 - 4.9x 2 (a) What is the height of the rock when x = 1 second? x = 1.1 seconds? x = 1.2 seconds? x = 1.3 seconds?

76

97. Cost of Trans-Atlantic Travel A Boeing 747 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per passenger is given by C(x) = 100 +

36,000 x + 10 x

where x is the ground speed (airspeed { wind). (a) What is the cost per passenger for quiescent (no wind) conditions? (b) What is the cost per passenger with a head wind of 50 miles per hour? (c) What is the cost per passenger with a tail wind of 100 miles per hour? (d) What is the cost per passenger with a head wind of 100 miles per hour? 98. Cross-sectional Area The cross-sectional area of a beam cut from a log with radius 1 foot is given by the function A(x) = 4x 21 - x 2 , where x represents the length, in feet, of half the base of the beam. See the figure. Determine the cross-sectional area of the beam if the length of half the base of the beam is as follows: (a) One-third of a foot (b) One-half of a foot (c) Two-thirds of a foot

Functions and Their Graphs

individual’s tax bill in year x. Determine a function N that represents the individual’s net income (income after taxes) in year x.

A(x )  4x 1  x 2 1

x

99. Economics The participation rate is the number of people in the labor force divided by the civilian population (excludes military). Let L(x) represent the size of the labor force in year x and P(x) represent the civilian population in year x. Determine a function that represents the participation rate R as a function of x. 100. Crimes Suppose that V(x) represents the number of violent crimes committed in year x and P(x) represents the number of property crimes committed in year x. Determine a function T that represents the combined total of violent crimes and property crimes in year x. 101. Health Care Suppose that P(x) represents the percentage of income spent on health care in year x and I(x) represents income in year x. Determine a function H that represents total health care expenditures in year x. 102. Income Tax Suppose that I(x) represents the income of an individual in year x before taxes and T(x) represents the

103. Profit Function Suppose that the revenue R, in dollars, from selling x cell phones, in hundreds, is R(x) = - 1.2x 2 + 220x. The cost C, in dollars, of selling x cell phones is C(x) = 0.05x 3 - 2x 2 + 65x + 500. (a) Find the profit function, P(x) = R(x) - C(x). (b) Find the profit if x = 15 hundred cell phones are sold. (c) Interpret P(15). 104. Profit Function Suppose that the revenue R, in dollars, from selling x clocks is R(x) = 30x. The cost C, in dollars, of selling x clocks is C(x) = 0.1x 2 + 7x + 400. (a) Find the profit function, P(x) = R(x) - C(x). (b) Find the profit if x = 30 clocks are sold. (c) Interpret P(30). 105. Some functions f have the property that f 1a + b2 = f 1a2 + f 1b2 for all real numbers a and b. Which of the following functions have this property? (a) h1x2 = 2x (b) g 1x2 = x 2 1 (c) F 1x2 = 5x - 2 (d) G1x2 = x

Explaining Concepts: Discussion and Writing x2 - 1 106. Are the functions f 1x2 = x - 1 and g 1x2 = the x + 1 same? Explain.

108. Find a function H that multiplies a number x by 3, then subtracts the cube of x and divides the result by your age.

107. Investigate when, historically, the use of the function notation y = f 1x2 first appeared.

‘Are You Prepared?’ Answers 1. ( - 1, 3)

3. 5 x x ⬆ - 4 6

2. 21.5

4. 5 x x 6 - 1 6

1

0

1

2 The Graph of a Function PREPARING FOR THIS SECTION Before getting started, review the following: • Graphs of Equations

• Intercepts

Now Work the ‘Are You Prepared?’ problems.

OBJECTIVES 1 Identify the Graph of a Function 2 Obtain Information from or about the Graph of a Function

In applications, a graph often demonstrates more clearly the relationship between two variables than, say, an equation or table would. For example, Table 1 on the next page shows the average price of gasoline at a particular gas station in Texas (for the years 1981–2010 adjusted for inflation, based on 2008 dollars). If we plot these data and then connect the points, we obtain Figure 13 (also on the next page). 77

Functions and Their Graphs

Table 1 Year

Price

Year

Price

Year

Price

1981

3.26

1991

1.90

2001

1.40

1982

3.15

1992

1.82

2002

1.86

1983

2.51

1993

1.70

2003

1.79

1984

2.51

1994

1.85

2004

2.13

1985

2.46

1995

1.68

2005

2.60

1986

1.63

1996

1.87

2006

2.62

1987

1.90

1997

1.65

2007

3.29

1988

1.77

1998

1.50

2008

2.10

1989

1.83

1999

1.73

2009

2.45

1990

2.25

2000

1.85

2010

3.18

Source: http://www.randomuseless.info/gasprice/gasprice.html

2010

2008

2006

2004

2002

2000

1998

1996

1994

1992

1990

1988

1986

1984

1982

4.00 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00

1980

Price (dollars per gallon)

Figure 13 Average retail price of gasoline (2008 dollars)

Source: http://www.randomuseless.info/gasprice/gasprice.html

We can see from the graph that the price of gasoline (adjusted for inflation) fell from 1981 to 1986 and rose rapidly from 2003 to 2007. The graph also shows that the lowest price occurred in 2001. To learn information such as this from an equation requires that some calculations be made. Look again at Figure 13. The graph shows that for each date on the horizontal axis there is only one price on the vertical axis. The graph represents a function, although the exact rule for getting from date to price is not given. When a function is defined by an equation in x and y, the graph of the function is the graph of the equation, that is, the set of points (x, y) in the xy-plane that satisfy the equation.

1 Identify the Graph of a Function In Words If any vertical line intersects a graph at more than one point, the graph is not the graph of a function.

THEOREM

Not every collection of points in the xy-plane represents the graph of a function. Remember, for a function, each number x in the domain has exactly one image y in the range. This means that the graph of a function cannot contain two points with the same x-coordinate and different y-coordinates. Therefore, the graph of a function must satisfy the following vertical-line test.

Vertical-line Test A set of points in the xy-plane is the graph of a function if and only if every vertical line intersects the graph in at most one point.

E X A MP L E 1

Identifying the Graph of a Function Which of the graphs in Figure 14 are graphs of functions?

78

Functions and Their Graphs

Figure 14

y 6

y 4

y

y 3

1 (1, 1)

4 3

4

3x (a) y  x 2

Solution

4x

(1, 1)

6 x

1 x 1

3

(b) y  x 3

1

(c) x  y 2

(d) x 2  y 2  1

The graphs in Figures 14(a) and 14(b) are graphs of functions, because every vertical line intersects each graph in at most one point. The graphs in Figures 14(c) and 14(d) are not graphs of functions, because there is a vertical line that intersects each graph in more than one point. Notice in Figure 14(c) that the input 1 corresponds to two outputs, -1 and 1. This is why the graph does not represent a function.

Now Work

PROBLEM

15

2 Obtain Information from or about the Graph of a Function If (x, y) is a point on the graph of a function f, then y is the value of f at x; that is, y = f 1x2 . Also if y = f1x2, then (x, y) is a point on the graph of f. For example, if ( -2, 7) is on the graph of f, then f 1 -22 = 7, and if f152 = 8, then the point (5, 8) is on the graph of y = f1x2 . The next example illustrates how to obtain information about a function if its graph is given.

Obtaining Information from the Graph of a Function

E XAMPL E 2

Let f be the function whose graph is given in Figure 15. (The graph of f might represent the distance y that the bob of a pendulum is from its at-rest position at time x. Negative values of y mean that the pendulum is to the left of the at-rest position, and positive values of y mean that the pendulum is to the right of the at-rest position.)

Figure 15 y 4 2

2 4

(

, –– 2

(4, 4)

(2, 4)

(0, 4)

(5––2, 0)

0)

(7––2, 0)

(3––2, 0) (, 4)

x

(3, 4)

(b) (c) (d) (e) (f) (g)

Solution

3p b , and f13p2? 2 What is the domain of f ? What is the range of f ? List the intercepts. (Recall that these are the points, if any, where the graph crosses or touches the coordinate axes.) How many times does the line y = 2 intersect the graph? For what values of x does f1x2 = -4? For what values of x is f1x2 7 0?

(a) What are f102, f a

(a) Since (0, 4) is on the graph of f, the y-coordinate 4 is the value of f at the 3p x-coordinate 0; that is, f102 = 4. In a similar way, we find that when x = , 2 3p then y = 0, so f a b = 0. When x = 3p, then y = -4, so f13p2 = -4. 2 (b) To determine the domain of f, we notice that the points on the graph of f have x-coordinates between 0 and 4p, inclusive; and for each number x between 0 and 4p, there is a point 1x, f1x22 on the graph. The domain of f is 5 x 0 … x … 4p 6 or the interval 3 0, 4p 4 . (c) The points on the graph all have y-coordinates between -4 and 4, inclusive; and for each such number y, there is at least one corresponding number x in the domain. The range of f is 5 y -4 … y … 4 6 or the interval 3 -4, 4 4 .

79

Functions and Their Graphs

(d) The intercepts are the points p 3p 5p (0, 4), a , 0b , a , 0b , a , 0b , and 2 2 2

a

7p , 0b 2

(e) If we draw the horizontal line y = 2 on the graph in Figure 15, we find that it intersects the graph four times. (f) Since (p, -4) and (3p, -4) are the only points on the graph for which y = f1x2 = -4, we have f1x2 = -4 when x = p and x = 3p. (g) To determine where f1x2 7 0, look at Figure 15 and determine the x-values from 0 to 4p for which the y-coordinate is positive. This occurs on p 3p 5p 7p b ´ a , b ´ a , 4p d . Using inequality notation, f 1x2 7 0 for 2 2 2 2 p 3p 5p 7p 0 … x 6 or 6 x6 or 6 x … 4p. 2 2 2 2 c 0,

When the graph of a function is given, its domain may be viewed as the shadow created by the graph on the x-axis by vertical beams of light. Its range can be viewed as the shadow created by the graph on the y-axis by horizontal beams of light. Try this technique with the graph given in Figure 15.

Now Work

E X A M PL E 3

PROBLEMS

9

AND

13

Obtaining Information about the Graph of a Function Consider the function:

f1x2 =

x+1 x+2

(a) Find the domain of f. 1 (b) Is the point a 1, b on the graph of f ? 2 (c) If x = 2, what is f1x2 ? What point is on the graph of f ? (d) If f 1x2 = 2, what is x? What point is on the graph of f ? (e) What are the x-intercepts of the graph of f (if any)? What point(s) are on the graph of f ?

Solution

(a) The domain of f is {x  x ⬆ -2} , since x = -2 results in division by 0. (b) When x = 1, x x 1 f112 = 1

f1x2 =

+ + + +

1 2 1 2 = 2 3

2 1 The point a 1, b is on the graph of f; the point a 1, b is not. 3 2

(c) If x = 2,

f1x2 =

x+1 x+2

f122 =

2+1 3 = 2+2 4

3 The point ¢ 2, ≤ is on the graph of f. 4 80

Functions and Their Graphs

(d) If f1x2 = 2, then f 1x2 x+1 x+2 x+1 x+1 x

=2 =2 = 2(x + 2) = 2x + 4 = -3

Multiply both sides by x + 2. Remove parentheses. Solve for x.

If f1x2 = 2, then x = -3. The point ( -3, 2) is on the graph of f. (e) The x-intercepts of the graph of f are the real solutions of the equation f1x2 = 0 that are in the domain of f. The only real solution of the equation x+1 f1x2 = = 0, is x = -1 , so -1 is the only x-intercept. Since f1 -12 = 0, x+2 the point ( -1 , 0) is on the graph of f.

Now Work

E XAM PL E 4

PROBLEM

25

Average Cost Function The average cost C of manufacturing x computers per day is given by the function C(x) = 0.56x 2 - 34.39x + 1212.57 +

20,000 x

Determine the average cost of manufacturing: (a) 30 computers in a day (b) 40 computers in a day (c) 50 computers in a day (d) Graph the function C = C(x), 0 6 x … 80. (e) Create a TABLE with TblStart = 1 and Tbl = 1. Which value of x minimizes the average cost?

Solution

(a) The average cost of manufacturing x = 30 computers is C(30) = 0.56(30)2 - 34.39(30) + 1212.57 +

20,000 = \$1351.54 30

(b) The average cost of manufacturing x = 40 computers is C(40) = 0.56(40)2 - 34.39(40) + 1212.57 +

20,000 = \$1232.97 40

(c) The average cost of manufacturing x = 50 computers is C(50) = 0.56(50)2 - 34.39(50) + 1212.57 +

20,000 = \$1293.07 50

(d) See Figure 16 for the graph of C = C(x). (e) With the function C = C(x) in Y1 , we create Table 2. We scroll down until we find a value of x for which Y1 is smallest. Table 3 shows that manufacturing x = 41 computers minimizes the average cost at \$1231.74 per computer. Figure 16

Table 2

4000

0

Table 3

80 0

Now Work

PROBLEM

31

81

Functions and Their Graphs

SUMMARY Graph of a Function Vertical Line Test

The collection of points (x, y) that satisfies the equation y = f1x2. A collection of points is the graph of a function provided that every vertical line intersects the graph in at most one point.

2 Assess Your Understanding ‘Are You Prepared?’

Answers are given at the end of these exercises. x 2 + 4y 2 = 16

are

2. True or False The point ( - 2, - 6) is on the graph of the equation x = 2y - 2.

3. A set of points in the xy-plane is the graph of a function if and only if every __________ line intersects the graph in at most one point.

6. True or False A function can have more than one y-intercept.

1. The intercepts of the equation ______________________________.

Concepts and Vocabulary

4. If the point (5, - 3) is a point on the graph of f, then f( ) = ________. 5. Find a so that the point ( - 1, 2) is on the graph of f 1x2 = ax 2 + 4.

7. True or False The graph of a function y = f 1x2 always crosses the y-axis. 8. True or False The y-intercept of the graph of the function y = f 1x2, whose domain is all real numbers, is f 102.

Skill Building 9. Use the given graph of the function f to answer parts (a) – (n). y (0, 3) 4

y

(2, 4) (4, 3)

5

–5

(–6, –3)

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

82

4 (–4, 2)

(10, 0) (11, 1)

(–3, 0)

(–5, –2)

10. Use the given graph of the function f to answer parts (a) – (n).

(6, 0) –3

Find f 102 and f 1 - 62. Find f 162 and f 1112. Is f 132 positive or negative? Is f 1 - 42 positive or negative? For what values of x is f 1x2 = 0? For what values of x is f 1x2 7 0? What is the domain of f ? What is the range of f ? What are the x-intercepts? What is the y-intercept?

(–2, 1) 2 (4, 0)

11 x

–4

–2 (0, 0) –2

(8, – 2)

1 (k) How often does the line y = intersect the 2 graph? (l) How often does the line x = 5 intersect the graph? (m) For what values of x does f 1x2 = 3? (n) For what values of x does f 1x2 = - 2?

(5, 3)

(a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n)

2

4

(6, 0) 6

x

(2, –2)

Find f 102 and f 162. Find f 122 and f 1 - 22. Is f 132 positive or negative? Is f 1 - 12 positive or negative? For what values of x is f 1x2 = 0? For what values of x is f 1x2 6 0? What is the domain of f ? What is the range of f ? What are the x-intercepts? What is the y-intercept? How often does the line y = - 1 intersect the graph? How often does the line x = 1 intersect the graph? For what value of x does f 1x2 = 3? For what value of x does f 1x2 = - 2?

Functions and Their Graphs

In Problems 11–22, determine whether the graph is that of a function by using the vertical-line test. If it is, use the graph to find: (a) The domain and range (b) The intercepts, if any (c) Any symmetry with respect to the x-axis, the y-axis, or the origin 11.

12.

y 3

3x

3

16.

3x

3

y (1, 2) 3

3x

3

(1, 2)

3 3

22.

y

3x

3 3

24. f 1x2 = - 3x 2 + 5x (a) Is the point ( - 1, 2) on the graph of f ? (b) If x = - 2, what is f 1x2? What point is on the graph of f ? (c) If f 1x2 = - 2, what is x? What point(s) are on the graph of f ? (d) What is the domain of f ? (e) List the x-intercepts, if any, of the graph of f. (f) List the y-intercept, if there is one, of the graph of f.

(d) (e) (f)

4 y

(1–2 , 5)

4

6

23. f 1x2 = 2x 2 - x - 1 (a) Is the point ( - 1, 2) on the graph of f ? (b) If x = - 2, what is f 1x2? What point is on the graph of f ? (c) If f 1x2 = - 1, what is x? What point(s) are on the graph of f ? (d) What is the domain of f ? (e) List the x-intercepts, if any, of the graph of f. (f) List the y-intercept, if there is one, of the graph of f.

(a) (b) (c)

(4, 3)

4x

4

9

x + 2 x - 6 Is the point (3, 14) on the graph of f ? If x = 4, what is f 1x2? What point is on the graph of f ? If f 1x2 = 2, what is x? What point(s) are on the graph of f ? What is the domain of f ? List the x-intercepts, if any, of the graph of f. List the y-intercept, if there is one, of the graph of f.

25. f 1x2 =

y 4

(3, 2)

3x

x 3

18.

3

21.

x



–– 2

1

3

y 3



  –– 2



x



–– 2

y 3

3

20.

y 1

 –– 2 1

17.

y 3

3

19.



3

y 3

14.

y 1

3x

3

3

15.

13.

y 3

3 3 x

1 3

26. f 1x2 =

(2, 3)

3

3 x

x2 + 2 x + 4

3 (a) Is the point a1, b on the graph of f ? 5 (b) If x = 0, what is f 1x2? What point is on the graph of f ? 1 (c) If f 1x2 = , what is x? What point(s) are on the graph 2 of f ? (d) What is the domain of f ? (e) List the x-intercepts, if any, of the graph of f. (f) List the y-intercept, if there is one, of the graph of f. 2x 2 x + 1 Is the point ( - 1, 1) on the graph of f ? If x = 2, what is f 1x2? What point is on the graph of f ? If f 1x2 = 1, what is x? What point(s) are on the graph of f ? What is the domain of f ? List the x-intercepts, if any, of the graph of f. List the y-intercept, if there is one, of the graph of f.

27. f 1x2 = (a) (b) (c)

(d) (e) (f)

28. f 1x2 =

4

2x x - 2

2 1 (a) Is the point a , - b on the graph of f ? 2 3 (b) If x = 4, what is f 1x2? What point is on the graph of f ? (c) If f 1x2 = 1, what is x? What point(s) are on the graph of f ? (d) What is the domain of f ? (e) List the x-intercepts, if any, of the graph of f. (f) List the y-intercept, if there is one, of the graph of f.

83

Functions and Their Graphs

Applications and Extensions 29. Free-throw Shots According to physicist Peter Brancazio, the key to a successful foul shot in basketball lies in the arc of the shot. Brancazio determined the optimal angle of the arc from the free-throw line to be 45 degrees. The arc also depends on the velocity with which the ball is shot. If a player shoots a foul shot, releasing the ball at a 45-degree angle from a position 6 feet above the floor, then the path of the ball can be modeled by the function h(x) = -

where x is the horizontal distance that the golf ball has traveled.

44x 2 + x + 6 v2

(a) Determine the height of the ball after it has traveled 8 feet in front of the foul line. (b) Determine h(12).What does this value represent? (c) Find additional points and graph the path of the basketball. (d) The center of the hoop is 10 feet above the floor and 15 feet in front of the foul line. Will the ball go through the hoop? Why or why not? If not, with what initial velocity must the ball be shot in order for the ball to go through the hoop? Source: The Physics of Foul Shots, Discover, Vol. 21, No. 10, October 2000 30. Granny Shots The last player in the NBA to use an underhand foul shot (a “granny” shot) was Hall of Fame forward Rick Barry who retired in 1980. Barry believes that current NBA players could increase their free-throw percentage if they were to use an underhand shot. Since underhand shots are released from a lower position, the angle of the shot must be increased. If a player shoots an underhand foul shot, releasing the ball at a 70-degree angle from a position 3.5 feet above the floor, then the path of the ball can 136x 2 be modeled by the function h(x) = - 2 + 2.7x + 3.5 , v where h is the height of the ball above the floor, x is the forward distance of the ball in front of the foul line, and v is the initial velocity with which the ball is shot in feet per second. (a) The center of the hoop is 10 feet above the floor and 15 feet in front of the foul line. Determine the initial velocity with which the ball must be shot in order for the ball to go through the hoop. (b) Write the function for the path of the ball using the velocity found in part (a). (c) Determine h(9). What does this value represent? (d) Find additional points and graph the path of the basketball. Source: The Physics of Foul Shots, Discover, Vol. 21, No. 10, October 2000 31. Motion of a Golf Ball A golf ball is hit with an initial velocity of 130 feet per second at an inclination of 45° to the horizontal. In physics, it is established that the height h of the golf ball is given by the function h(x) =

84

- 32x 2 + x 1302

Exactostock/SuperStock

where h is the height of the ball above the floor, x is the forward distance of the ball in front of the foul line, and v is the initial velocity with which the ball is shot in feet per second. Suppose a player shoots a ball with an initial velocity of 28 feet per second.

(a) Determine the height of the golf ball after it has traveled 100 feet. (b) What is the height after it has traveled 300 feet? (c) What is h(500)? Interpret this value. (d) How far was the golf ball hit? (e) Use a graphing utility to graph the function h = h(x). (f) Use a graphing utility to determine the distance that the ball has traveled when the height of the ball is 90 feet. (g) Create a TABLE with TblStart = 0 and Tbl = 25. To the nearest 25 feet, how far does the ball travel before it reaches a maximum height? What is the maximum height? (h) Adjust the value of Tbl until you determine the distance, to within 1 foot, that the ball travels before it reaches a maximum height. 32. Cross-sectional Area The cross-sectional area of a beam cut from a log with radius 1 foot is given by the function A(x) = 4x 21 - x 2 , where x represents the length, in feet, of half the base of the beam. See the figure. A(x )  4x 1  x 2 1

x

(a) Find the domain of A. (b) Use a graphing utility to graph the function A = A(x). (c) Create a TABLE with TblStart = 0 and Tbl = 0.1. Which value of x in the domain found in part (a) maximizes the cross-sectional area? What should be the length of the base of the beam to maximize the cross-sectional area? 33. Cost of Trans-Atlantic Travel A Boeing 747 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost C (in dollars) per passenger is given by C 1x2 = 100 +

36,000 x + 10 x

Functions and Their Graphs

where x is the ground speed (airspeed { wind). (a) Use a graphing utility to graph the function C = C(x). (b) Create a TABLE with TblStart = 0 and Tbl = 50. (c) To the nearest 50 miles per hour, what ground speed minimizes the cost per passenger?

W(h) = ma

2 4000 b 4000 + h

(a) If Amy weighs 120 pounds at sea level, how much will she weigh on Pike’s Peak, which is 14,110 feet above sea level? (b) Use a graphing utility to graph the function W = W(h). Use m = 120 pounds. (c) Create a TABLE with TblStart = 0 and Tbl = 0.5 to see how the weight W varies as h changes from 0 to 5 miles. (d) At what height will Amy weigh 119.95 pounds? (e) Does your answer to part (d) seem reasonable? Explain. 35. The graph of two functions, f and g, is illustrated. Use the graph to answer parts (a) – (f). y

yg(x) (4, 1)

2 4

(a) ( f + g)(2) (c) ( f - g)(6) (e) ( f # g)(2)

100,000

(0, 5000)

(50, 51000) (10, 19000) 10

20

30

40 50 60 70 80 Number of computers

90 100 q

37. Reading and Interpreting Graphs Let C be the function whose graph is given below. This graph represents the cost C of using m anytime cell phone minutes in a month for a five-person family plan. C 2000

(6, 1) (6, 0)

(14400, 1822)

x yf(x)

4

(3, 2)

150,000

(30, 32000)

1500

(5, 2)

(4, 3)

(b) ( f + g)(4) (d) 1g - f 2 162 f (f) a b(4) g

36. Reading and Interpreting Graphs Let C be the function whose graph is given in the next column. This graph represents the cost C of manufacturing q computers in a day. (a) Determine C(0). Interpret this value. (b) Determine C(10). Interpret this value. (c) Determine C(50). Interpret this value. (d) What is the domain of C? What does this domain imply in terms of daily production? (e) Describe the shape of the graph. (f) The point (30, 32000) is called an inflection point. Describe the behavior of the graph around the inflection point.

Cost (dollars)

2

200,000

50,000

(2, 2) 2 (2, 1)

(100, 280000)

250,000

Cost (dollars per day)

34. Effect of Elevation on Weight If an object weighs m pounds at sea level, then its weight W (in pounds) at a height of h miles above sea level is given approximately by

C

1000

500 (2000, 210) (0, 80)

(1000, 80) 5000

10000 Number of minutes

15000 m

(a) (b) (c) (d)

Determine C(0). Interpret this value. Determine C(1000). Interpret this value. Determine C(2000). Interpret this value. What is the domain of C? What does this domain imply in terms of the number of anytime minutes? (e) Describe the shape of the graph.

Explaining Concepts: Discussion and Writing 38. Describe how you would proceed to find the domain and range of a function if you were given its graph. How would your strategy change if you were given the equation defining the function instead of its graph? 39. How many x-intercepts can the graph of a function have? How many y-intercepts can the graph of a function have? 40. Is a graph that consists of a single point the graph of a function? Can you write the equation of such a function? 41. Match each of the following functions with the graph on the next page that best describes the situation. (a) The cost of building a house as a function of its square footage (b) The height of an egg dropped from a 300-foot building as a function of time (c) The height of a human as a function of time (d) The demand for Big Macs as a function of price (e) The height of a child on a swing as a function of time?

85

Functions and Their Graphs y

y

y

y

x

x

x

x

x

(III)

(II)

(I)

y

(V)

(IV)

42. Match each of the following functions with the graph that best describes the situation. (a) The temperature of a bowl of soup as a function of time (b) The number of hours of daylight per day over a 2-year period (c) The population of Texas as a function of time (d) The distance traveled by a car going at a constant velocity as a function of time (e) The height of a golf ball hit with a 7-iron as a function of time y

y

y

x

y

x

x

(II)

(I)

44. Consider the following scenario: Jayne enjoys riding her bicycle through the woods. At the forest preserve, she gets on her bicycle and rides up a 2000-foot incline in 10 minutes. She then travels down the incline in 3 minutes. The next 5000 feet is level terrain and she covers the distance in 20 minutes. She rests for 15 minutes. Jayne then travels 10,000 feet in 30 minutes. Draw a graph of Jayne’s distance traveled (in feet) as a function of time. 45. The following sketch represents the distance d (in miles) that Kevin was from home as a function of time t (in hours). Answer the questions based on the graph. In parts (a) – (g), how many hours elapsed and how far was Kevin from home during this time? d (t ) (2.5, 3)

(2.8, 0)

(3.9, 2.8)

(3, 0)

(4.2, 2.8)

(5.3, 0)

( - 4, 0), (4, 0), (0, - 2), (0, 2)

t

x (V)

(d) (e) (f) (g) (h) (i)

From t = 2.8 to t = 3 From t = 3 to t = 3.9 From t = 3.9 to t = 4.2 From t = 4.2 to t = 5.3 What is the farthest distance that Kevin was from home? How many times did Kevin return home?

46. The following sketch represents the speed v (in miles per hour) of Michael’s car as a function of time t (in minutes). v (t ) (7, 50)

(2, 30)

(a) (b) (c) (d) (e) (f)

(8, 38)

(4, 30)

(4.2, 0)

(7.4, 50)

(7.6, 38)

(6, 0)

(9.1, 0)

t

Over what interval of time was Michael traveling fastest? Over what interval(s) of time was Michael’s speed zero? What was Michael’s speed between 0 and 2 minutes? What was Michael’s speed between 4.2 and 6 minutes? What was Michael’s speed between 7 and 7.4 minutes? When was Michael’s speed constant?

47. Draw the graph of a function whose domain is 5 x - 3 … x … 8, x ⬆ 5 6 and whose range is 5 y - 1 … y … 2, y ⬆ 0 6. What point(s) in the rectangle - 3 … x … 8, - 1 … y … 2 cannot be on the graph? Compare your graph with those of other students. What differences do you see? 48. Is there a function whose graph is symmetric with respect to the x-axis? Explain.

(a) From t = 0 to t = 2 (b) From t = 2 to t = 2.5 (c) From t = 2.5 to t = 2.8

1.

x (IV)

(III)

43. Consider the following scenario: Barbara decides to take a walk. She leaves home, walks 2 blocks in 5 minutes at a constant speed, and realizes that she forgot to lock the door. So Barbara runs home in 1 minute. While at her doorstep, it takes her 1 minute to find her keys and lock the door. Barbara walks 5 blocks in 15 minutes and then decides to jog home. It takes her 7 minutes to get home. Draw a graph of Barbara’s distance from home (in blocks) as a function of time.

(2, 3)

y

2. False

Functions and Their Graphs

3 Properties of Functions PREPARING FOR THIS SECTION Before getting started, review the following: • Point–Slope Form of a Line

• Intervals • Intercepts • Slope of a Line

• Symmetry

Now Work the ‘Are You Prepared?’ problems.

OBJECTIVES 1 Determine Even and Odd Functions from a Graph 2 Identify Even and Odd Functions from the Equation 3 Use a Graph to Determine Where a Function is Increasing, Decreasing, or Constant 4 Use a Graph to Locate Local Maxima and Local Minima 5 Use a Graph to Locate the Absolute Maximum and the Absolute Minimum 6 Use a Graphing Utility to Approximate Local Maxima and Local Minima and to Determine Where a Function is Increasing or Decreasing 7 Find the Average Rate of Change of a Function

To obtain the graph of a function y = f 1x2 , it is often helpful to know certain properties that the function has and the impact of these properties on the way that the graph will look.

1 Determine Even and Odd Functions from a Graph The words even and odd, when applied to a function f, describe the symmetry that exists for the graph of the function. A function f is even, if and only if, whenever the point (x, y) is on the graph of f then the point ( -x, y) is also on the graph. Using function notation, we define an even function as follows:

DEFINITION

A function f is even if, for every number x in its domain, the number -x is also in the domain and f1 -x2 = f 1x2

A function f is odd, if and only if, whenever the point (x, y) is on the graph of f then the point ( -x, -y) is also on the graph. Using function notation, we define an odd function as follows:

DEFINITION

A function f is odd if, for every number x in its domain, the number -x is also in the domain and f1 -x2 = -f 1x2

Refer to the tests for symmetry. The following results are then evident.

THEOREM

A function is even if and only if its graph is symmetric with respect to the y-axis. A function is odd if and only if its graph is symmetric with respect to the origin.

87

Functions and Their Graphs

E X A MP L E 1

Determining Even and Odd Functions from the Graph Determine whether each graph given in Figure 17 is the graph of an even function, an odd function, or a function that is neither even nor odd.

Figure 17

y

y

x

x

(b)

(a)

Solution

y

x

(c)

(a) The graph in Figure 17(a) is that of an even function, because the graph is symmetric with respect to the y-axis. (b) The function whose graph is given in Figure 17(b) is neither even nor odd, because the graph is neither symmetric with respect to the y-axis nor symmetric with respect to the origin. (c) The function whose graph is given in Figure 17(c) is odd, because the graph is symmetric with respect to the origin.

Now Work

PROBLEMS

21(a), (b),

AND

(d)

2 Identify Even and Odd Functions from the Equation A graphing utility can be used to conjecture whether a function is even, odd, or neither. Remember that, when the graph of an even function contains the point (x, y), it must also contain the point 1 -x, y2. Therefore, if the graph shows evidence of symmetry with respect to the y-axis, we would conjecture that the function is even. In addition, if the graph shows evidence of symmetry with respect to the origin, we would conjecture that the function is odd.

E X A MP L E 2

Identifying Even and Odd Functions Use a graphing utility to conjecture whether each of the following functions is even, odd, or neither. Then algebraically determine whether the graph is symmetric with respect to the y-axis or with respect to the origin. (a) f 1x2 = x2 - 5

Figure 18 10

4

Solution

4

5

Figure 19 10

3

3

10

88

(b) g1x2 = x3 - 1

(c) h 1x2 = 5x3 - x

(a) Graph the function. See Figure 18. It appears that the graph is symmetric with respect to the y-axis. We conjecture that the function is even. To algebraically verify the conjecture, replace x by -x in f1x2 = x2 - 5. Then f1 -x2 = 1 -x2 2 - 5 = x2 - 5 = f 1x2

Since f1 -x2 = f1x2, we conclude that f is an even function and that the graph is symmetric with respect to the y-axis. (b) Graph the function. See Figure 19. It appears that there is no symmetry. We conjecture that the function is neither even nor odd. To algebraically verify that the function is not even, find g1 -x2 and compare the result with g1x2. g1 -x2 = 1 -x2 3 - 1 = -x3 - 1; g1x2 = x3 - 1

Since g1 -x2 ⬆ g1x2, the function is not even.

Functions and Their Graphs

To algebraically verify that the function is not odd, find -g(x) and compare the result with g( -x). -g(x) = -(x 3 - 1) = -x3 + 1; g( -x) = -x3 - 1 Since g( -x) ⬆ -g(x), the function is not odd. The graph is not symmetric with respect to the y-axis nor is it symmetric with respect to the origin. (c) Graph the function. See Figure 20. It appears that there is symmetry with respect to the origin. We conjecture that the function is odd. To algebraically verify the conjecture, replace x by -x in h(x) = 5x3 - x. Then

Figure 20 3

2

2

h( -x) = 5( -x)3 - ( -x) = -5x3 + x = -(5x3 - x) = -h(x) Since h( -x) = -h(x), h is an odd function and the graph of h is symmetric with respect to the origin.

3

Now Work

PROBLEM

33

3 Use a Graph to Determine Where a Function Is Increasing, Decreasing, or Constant Consider the graph given in Figure 21. If you look from left to right along the graph of the function, you will notice that parts of the graph are going up, parts are going down, and parts are horizontal. In such cases, the function is described as increasing, decreasing, or constant, respectively. Figure 21

y 5 (0, 4)

(6, 0)

(2, 0)

6 x

4 (4, 2)

E XAM PL E 3

(3, 4) y = f (x) (6, 1)

2

Determining Where a Function Is Increasing, Decreasing, or Constant from Its Graph Where is the function in Figure 21 increasing? Where is it decreasing? Where is it constant?

Solution WARNING We describe the behavior of a graph in terms of its x-values. Do not say the graph in Figure 21 is increasing from the point 1 - 4, -22 to the point 10, 42 . Rather, say it is increasing on the interval ( -4, 0) . 

DEFINITIONS

To answer the question of where a function is increasing, where it is decreasing, and where it is constant, we use strict inequalities involving the independent variable x, or we use open intervals* of x-coordinates. The function whose graph is given in Figure 21 is increasing on the open interval ( -4, 0) or for -4 6 x 6 0. The function is decreasing on the open intervals ( -6, -4) and (3, 6) or for -6 6 x 6 -4 and 3 6 x 6 6. The function is constant on the open interval (0, 3) or for 0 6 x 6 3. More precise definitions follow: A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 6 x2 , we have f1x1 2 6 f 1x2 2. A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 6 x2 , we have f1x1 2 7 f 1x2 2.

* The open interval (a, b) consists of all real numbers x for which a 6 x 6 b.

89

Functions and Their Graphs

A function f is constant on an open interval I if, for all choices of x in I, the values of f1x2 are equal.

In Words If a function is decreasing, then, as the values of x get bigger, the values of the function get smaller. If a function is increasing, then, as the values of x get bigger, the values of the function also get bigger. If a function is constant, then, as the values of x get bigger, the values of the function remain unchanged.

Figure 22 illustrates the definitions. Graphs are read like a book—from left to right. So, the graph of an increasing function goes up from left to right, the graph of a decreasing function goes down from left to right, and the graph of a constant function remains at a fixed height. Figure 22 y

y

y

f (x 1) x1

f (x 2) x2

f (x 1) x

f (x 1)

f (x 2)

x1

x2

x

x1

(b) For x 1 < x 2 in l, f (x 1) > f (x 2); f is decreasing on I

(a) For x 1 < x 2 in l, f (x 1) < f (x 2); f is increasing on I

Now Work

PROBLEMS

x2

x

l

l

l

f(x 2)

(c) For all x in I, the values of f are equal; f is constant on I

11, 13, 15,AND 21(c)

4 Use a Graph to Locate Local Maxima and Local Minima Suppose f is a function defined on an open interval containing c. If the value of f at c is greater than or equal to the values of f on I, then f has a local maximum at c *. See Figure 23(a). If the value of f at c is less than or equal to the values of f on I, then f has a local minimum at c. See Figure 23(b). f has a local maximum at c.

Figure 23

f has a local minimum at c.

y

f(c)

y

(c, f(c))

c l

f(c)

x

(a)

DEFINITIONS

(c, f(c))

c l

x

(b)

A function f has a local maximum at c if there is an open interval I containing c so that for all x in I, f 1x2 … f1c2. We call f1c2 a local maximum value of f.

A function f has a local minimum at c if there is an open interval I containing c so that, for all x in I, f 1x2 Ú f1c2. We call f1c2 a local minimum value of f.

If f has a local maximum at c, then the value of f at c is greater than or equal to the values of f near c. If f has a local minimum at c, then the value of f at c is less than or equal to the values of f near c. The word local is used to suggest that it is only near c, that is, in some open interval containing c, that the value f 1c2 has these properties. * Some texts use the term relative instead of local.

90

Functions and Their Graphs

Finding Local Maxima and Local Minima from the Graph of a Function and Determining Where the Function Is Increasing, Decreasing, or Constant

E XAM PL E 4 Figure 24

y

y  f(x)

Figure 24 shows the graph of a function f.

(1, 2)

2 (–1, 1) –2

3

x

Solution

WARNING The y-value is the local maximum value or local minimum value and it occurs at some x-value. For example, in Figure 24, we say f has a local maximum at 1 and the local maximum value is 2. 

(a) At what value(s) of x, if any, does f have a local maximum? List the local maximum values. (b) At what value(s) of x, if any, does f have a local minimum? List the local minimum values. (c) Find the intervals on which f is increasing. Find the intervals on which f is decreasing. The domain of f is the set of real numbers. (a) f has a local maximum at 1, since for all x close to 1, we have f 1x2 … f112. The local maximum value is f112 = 2. (b) f has local minima at -1 and at 3. The local minimum values are f1 -12 = 1 and f 132 = 0. (c) The function whose graph is given in Figure 24 is increasing for all values of x between -1 and 1 and for all values of x greater than 3. That is, the function is increasing on the intervals 1 -1, 12 and 13,  2 or for -1 6 x 6 1 and x 7 3. The function is decreasing for all values of x less than -1 and for all values of x between 1 and 3. That is, the function is decreasing on the intervals 1 - , -12 and 11, 32 or for x 6 -1 and 1 6 x 6 3.

Now Work

PROBLEMS

17

AND

19

5 Use a Graph to Locate the Absolute Maximum and the Absolute Minimum Look at the graph of the function f given in Figure 25. The domain of f is the closed interval 3 a, b 4 . Also, the largest value of f is f 1u2 and the smallest value of f is f1v2 . These are called, respectively, the absolute maximum and the absolute minimum of f on 3 a, b 4 .

Figure 25 y (u, f(u)) y  f(x) (b, f (b))

(a, f(a))

(y, f (y)) a

u

y b

x

DEFINITION Let f denote a function defined on some interval I. If there is a number u in I for which f1x2 … f 1u2 for all x in I, then f 1u2 is the absolute maximum of f on I and we say the absolute maximum of f occurs at u.

If there is a number v in I for which f1x2 Ú f 1v2 for all x in I, then f1v2 is the absolute minimum of f on I and we say the absolute minimum of f occurs at v .

domain: [a, b] for all x in [a, b], f(x)  f (u) for all x in [a, b], f(x)  f (y) absolute maximum: f(u) absolute minimum: f(y)

The absolute maximum and absolute minimum of a function f are sometimes called the extreme values of f on I. The absolute maximum or absolute minimum of a function f may not exist. Let’s look at some examples.

E XAM PL E 5

Finding the Absolute Maximum and the Absolute Minimum from the Graph of a Function For each graph of a function y = f1x2 in Figure 26 on the following page, find the absolute maximum and the absolute minimum, if they exist.

Solution

(a) The function f whose graph is given in Figure 26(a) has the closed interval [0, 5] as its domain. The largest value of f is f 132 = 6 , the absolute maximum. The smallest value of f is f 102 = 1, the absolute minimum.

91

Functions and Their Graphs

Figure 26

(3, 6)

6

4

6

6

(5, 5)

4

(4, 4)

2

2

1

2

(3, 1) 3

5

x

1

3

(1, 1) (2, 1) 5

x

1

3

6

4

4

2

2

(1, 4)

(3, 2)

(0, 0) 5

x

1

3

5

x

1

(d)

(c)

(b)

(a)

6

(0, 3)

(1, 2)

(0, 1)

(5, 4)

4

(5, 3)

y

y

y

y

y

3

5

(e)

(b) The function f whose graph is given in Figure 26(b) has the domain 5 x|1 … x … 5, x ⬆ 3 6 . Note that we exclude 3 from the domain because of the “hole” at (3, 1). The largest value of f on its domain is f152 = 3 , the absolute maximum. There is no absolute minimum. Do you see why? As you trace the graph, getting closer to the point (3, 1), there is no single smallest value. [As soon as you claim a smallest value, we can trace closer to (3, 1) and get a smaller value!] (c) The function f whose graph is given in Figure 26(c) has the interval 3 0, 5 4 as its domain. The absolute maximum of f is f152 = 4. The absolute minimum is 1. Notice that the absolute minimum 1 occurs at any number in the interval 3 1, 2 4 . (d) The graph of the function f given in Figure 26(d) has the interval 3 0, ) as its domain. The function has no absolute maximum; the absolute minimum is f 102 = 0. (e) The graph of the function f in Figure 26(e) has the domain 5 x|1 … x 6 5, x ⬆ 3 6 . The function f has no absolute maximum and no absolute minimum. Do you see why? In calculus, there is a theorem with conditions that guarantee a function will have an absolute maximum and an absolute minimum.

THEOREM

Extreme Value Theorem If f is a continuous function* whose domain is a closed interval 3 a, b 4 , then f has an absolute maximum and an absolute minimum on 3 a, b 4 .

Now Work

PROBLEM

45

6 Use a Graphing Utility to Approximate Local Maxima and Local Minima and to Determine Where a Function Is Increasing or Decreasing To locate the exact value at which a function f has a local maximum or a local minimum usually requires calculus. However, a graphing utility may be used to approximate these values by using the MAXIMUM and MINIMUM features.

E X A MP L E 6

Using a Graphing Utility to Approximate Local Maxima and Minima and to Determine Where a Function Is Increasing or Decreasing (a) Use a graphing utility to graph f1x2 = 6x3 - 12x + 5 for -2 6 x 6 2. Approximate where f has a local maximum and where f has a local minimum. (b) Determine where f is increasing and where it is decreasing. *Although it requires calculus for a precise definition, we’ll agree for now that a continuous function is one whose graph has no gaps or holes and can be traced without lifting the pencil from the paper.

92

x

Functions and Their Graphs

Solution

(a) Graphing utilities have a feature that finds the maximum or minimum point of a graph within a given interval. Graph the function f for -2 6 x 6 2. The MAXIMUM and MINIMUM commands require us to first determine the open interval I. The graphing utility will then approximate the maximum or minimum value in the interval. Using MAXIMUM we find that the local maximum is 11.53 and it occurs at x = -0.82, rounded to two decimal places. See Figure 27(a). Using MINIMUM, we find that the local minimum is -1.53 and it occurs at x = 0.82, rounded to two decimal places. See Figure 27(b).

Figure 27

30

30

2

2

2

2 10 (b)

10 (a)

(b) Looking at Figures 27(a) and (b), we see that the graph of f is increasing from x = -2 to x = -0.82 and from x = 0.82 to x = 2, so f is increasing on the intervals ( -2, -0.822 and (0.82, 2) or for -2 6 x 6 -0.82 and 0.82 6 x 6 2. The graph is decreasing from x = -0.82 to x = 0.82, so f is decreasing on the interval ( -0.82, 0.82) or for -0.82 6 x 6 0.82.

Now Work

PROBLEM

53

7 Find the Average Rate of Change of a Function The slope of a line could be interpreted as the average rate of change. To find the average rate of change of a function between any two points on its graph, calculate the slope of the line containing the two points.

DEFINITION

If a and b, a ⬆ b, are in the domain of a function y = f 1x2, the average rate of change of f from a to b is defined as Average rate of change =

y f1b2 - f1a2 = x b -a

a⬆b

(1)

The symbol y in (1) is the “change in y,” and x is the “change in x.” The average rate of change of f is the change in y divided by the change in x.

E XAM PL E 7

Finding the Average Rate of Change Find the average rate of change of f 1x2 = 3x2: (a) From 1 to 3 (b) From 1 to 5 (c) From 1 to 7

Solution

(a) The average rate of change of f1x2 = 3x2 from 1 to 3 is y f132 - f112 27 - 3 24 = = = = 12 x 3-1 3-1 2 (b) The average rate of change of f 1x2 = 3x2 from 1 to 5 is

y f152 - f 112 75 - 3 72 = = = = 18 x 5-1 5-1 4

(c) The average rate of change of f1x2 = 3x2 from 1 to 7 is y f172 - f112 147 - 3 144 = = = = 24 x 7-1 7-1 6 93

Functions and Their Graphs

See Figure 28 for a graph of f1x2 = 3x2. The function f is increasing for x 7 0. The fact that the average rate of change is positive for any x1, x2, x1 ⬆ x2 in the interval (1, 7) indicates that the graph is increasing on 1 6 x 6 7. Further, the average rate of change is consistently getting larger for 1 6 x 6 7, indicating that the graph is increasing at an increasing rate.

Figure 28 y 160

Now Work

(7, 147) 120

The average rate of change of a function has an important geometric interpretation. Look at the graph of y = f1x2 in Figure 29. We have labeled two points on the graph: 1a, f1a22 and 1b, f1b22. The line containing these two points is called the secant line; its slope is

(5, 75)

Average rate of change  18

40 (3, 27)

(1, 3) (0, 0)

61

The Secant Line

Average rate of change  24

80

PROBLEM

2

4

m sec =

Average rate of change  12 6

f1b2 - f1a2 f1a + h2 - f1a2 = b -a h

x

Figure 29

y

y  f (x ) Secant line (b, f (b ))  (a  h, f (a  h)) f (b )  f (a )  f (a  h)  f (a )

(a, f (a )) bah

a

THEOREM

b ah

x

Slope of the Secant Line The average rate of change of a function f from a to b equals the slope of the secant line containing the two points (a, f (a)) and (b, f (b)) on its graph.

E X A MP L E 8

Finding the Equation of a Secant Line Suppose that g(x) = 3x2 - 2x + 3. (a) Find the average rate of change of g from -2 to 1. (b) Find an equation of the secant line containing ( -2, g( -2)) and (1, g(1)). (c) Using a graphing utility, draw the graph of g and the secant line obtained in part (b) on the same screen.

Solution

(a) The average rate of change of g(x) = 3x2 - 2x + 3 from -2 to 1 is g(1) - g( -2) 1 - ( -2) 4 - 19 g(1) = 3(1)2 - 2(1) + 3 = 4 = g(- 2) = 3(- 2)2 - 2(- 2) + 3 = 19 3 15 = -5 = 3

Average rate of change =

(b) The slope of the secant line containing ( -2, g( -2)) = ( -2, 19) and (1, g(1)) = (1, 4) is m sec = -5. We use the point–slope form to find an equation of the secant line. 94

Functions and Their Graphs

Figure 30

y - y1 = m sec(x - x1) y - 19 = -5(x - ( -2)) y - 19 = -5x - 10 y = -5x + 9

24

Point–slope form of the secant line x1 = - 2, y1 = g( - 2) = 19, m sec = - 5 Simplify. Slope–intercept form of the secant line

(c) Figure 30 shows the graph of g along with the secant line y = -5x + 9. 3

3

Now Work

PROBLEM

4

67

3 Assess Your Understanding ‘Are You Prepared?’

Answers are given at the end of these exercises.

1. The interval (2, 5) can be written as the inequality ___________.

4. Write the point–slope form of the line with slope 5 containing the point (3, - 2).

2. The slope of the line containing the points ( - 2, 3) and (3, 8) is ________.

5. The

intercepts

of

the

equation

y = x2 - 9

are

______________________.

3. Test the equation y = 5x 2 - 1 for symmetry with respect to the x-axis, the y-axis, and the origin.

Concepts and Vocabulary 6. A function f is ___________ on an open interval I if, for any choice of x1 and x2 in I, with x1 6 x2 , we have f 1x1 2 6 f 1x2 2.

9. True or False A function f has a local maximum at c if there is an open interval I containing c so that for all x in I, f 1x2 … f 1c2.

function f is one for which f 1 - x2 = f 1x2 for 7. A(n) every x in the domain of f ; a(n) ________ function f is one for which f 1 - x2 = - f 1x2 for every x in the domain of f.

10. True or False Even functions have graphs that are symmetric with respect to the origin.

8. True or False A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 6 x2 , we have f 1x1 2 7 f 1x2 2.

Skill Building

y

In Problems 11–20, use the graph of the function f given.

(2, 10)

10

11. Is f increasing on the interval ( - 8, - 2)?

(2, 6)

12. Is f decreasing on the interval ( - 8, - 4)? 13. Is f increasing on the interval (2, 10)?

(5, 0)

(5, 0)

14. Is f decreasing on the interval (2, 5)? 10

15. List the interval(s) on which f is increasing. 16. List the interval(s) on which f is decreasing.

(0, 0)

5 (8, 4)

10 x

5

6

17. Is there a local maximum value at 2? If yes, what is it? 18. Is there a local maximum value at 5? If yes, what is it? 19. List the number(s) at which f has a local maximum. What are the local maximum values? 20. List the number(s) at which f has a local minimum. What are the local minimum values? In Problems 21–28, the graph of a function is given. Use the graph to find: (a) The intercepts, if any (b) The domain and range (c) The intervals on which it is increasing, decreasing, or constant (d) Whether it is even, odd, or neither 21.

22.

y 4 (4, 2)

(0, 3)

(3, 3)

y

(3, 3)

3

23.

y

24.

y 3

3

(0, 2)

(4, 2)

(0, 1) 4 (2, 0)

(2, 0)

4x

3

(1, 0)

(1, 0)

3 x

3

3 x

3

(1, 0)

3x

95

Functions and Their Graphs

25.



y 2

26.

y 2

(––2 , 1)

 –– 2



x

( ––2 , 1)

 2



(, 1)

2

( ) 1 0, – 2

(3, 1)

( –1, 2)

 2

x

–3

( )

(, 1)

(2.3, 0)

3 x (1, –1) (2, –1)

1 –, 0 3

2

y 3

28.

y 3

( – 3, 2)

(0, 1)



 –– 2

27.

(2, 1)

(2, 2) (3, 0)

(0, 1)

3 x

3 2

(3, 2)

–3

In Problems 29– 32, the graph of a function f is given. Use the graph to find: (a) The numbers, if any, at which f has a local maximum value. What are the local maximum values? (b) The numbers, if any, at which f has a local minimum value. What are the local minimum values? 29.

30.

y 4

32.

(

1

(0, 2) 

4x

(2, 0)

y 2

y

3

(0, 3)

4 (2, 0)

31.

y

3 (1, 0)

3 x

(1, 0)

 –– 2

 –– , 2

1)



 –– 2

π 2

(π, 1)

1

( ––2 , 1)

 π2

x

π x (π, 1)

2

In Problems 33– 44, determine algebraically whether each function is even, odd, or neither. 33. f 1x2 = 4x 3

34. f 1x2 = 2x 4 - x 2

3 37. F 1x2 = 2x

41. g 1x2 =

35. g 1x2 = - 3x 2 - 5

39. f 1x2 = x + 0 x 0

38. G1x2 = 1x

2

x + 3 x2 - 1

42. h1x2 =

36. h1x2 = 3x 3 + 5 3 40. f 1x2 = 2 2x 2 + 1

3

x x - 1

43. h1x2 =

2

-x 3x 2 - 9

2x 0x0

44. F 1x2 =

In Problems 45–52, for each graph of a function y = f 1x2, find the absolute maximum and the absolute minimum, if they exist. 45.

46.

y

(1, 4)

4

(4, 4)

4

(3, 3)

2

47.

y

2

(2, 2) (5, 1) 1

49.

3

1

x

50.

y 4 2 (1, 1)

3

5

51.

(3, 2) 1

3

1

x

3

5

1 1

3

x

1

x

52.

3

1

(1, 3) 2

(1, 1)

(4, 1) 3

x

y

(3, 2) (2, 0)

(0, 0) 1

(0, 1)

2

2 (0, 2)

(1, 3)

2

y

(2, 4)

(2, 4)

4

(4, 3)

1

x

y 4 (1, 3)

(2, 3)

(3, 4)

(1, 1)

(5, 0)

y

(0, 3)

2

(1, 1)

5

4

(0, 2)

48.

y

(0, 2)

(3, 1)

x 1

(2, 2) 2

1 (2, 0) 3

x

2

(1, 3)

In Problems 53– 60, use a graphing utility to graph each function over the indicated interval and approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. Round answers to two decimal places. 53. f 1x2 = x 3 - 3x + 2 55. f 1x2 = x - x 5

3

1 - 2, 22

1 - 2, 22

57. f 1x2 = - 0.2x 3 - 0.6x 2 + 4x - 6

59. f 1x2 = 0.25x + 0.3x - 0.9x + 3 4

3

2

1 - 6, 42

1 - 3, 22

54. f 1x2 = x 3 - 3x 2 + 5 56. f 1x2 = x 4 - x 2

61. Find the average rate of change of f 1x2 = - 2x + 4 (a) From 0 to 2 (b) From 1 to 3 (c) From 1 to 4

96

2

1 - 1, 32

1 - 2, 22

58. f 1x2 = - 0.4x 3 + 0.6x 2 + 3x - 2

1 - 4, 52

60. f 1x2 = - 0.4x 4 - 0.5x 3 + 0.8x 2 - 2

1 - 3, 22

62. Find the average rate of change of f 1x2 = - x 3 + 1 (a) From 0 to 2 (b) From 1 to 3 (c) From - 1 to 1

Functions and Their Graphs

63. Find the average rate of change of g 1x2 = x 3 - 2x + 1 (a) From - 3 to - 2 (b) From - 1 to 1 (c) From 1 to 3

67. g(x) = x 2 - 2 (a) Find the average rate of change from - 2 to 1. (b) Find an equation of the secant line containing ( - 2, g( - 2)) and (1, g(1)).

64. Find the average rate of change of h1x2 = x 2 - 2x + 3 (a) From - 1 to 1 (b) From 0 to 2 (c) From 2 to 5

68. g(x) = x 2 + 1 (a) Find the average rate of change from - 1 to 2. (b) Find an equation of the secant line containing ( - 1, g( - 1)) and (2, g(2)).

65. f 1x2 = 5x - 2 (a) Find the average rate of change from 1 to 3. (b) Find an equation of the secant line containing 11, f 1122 and 13, f 1322.

69. h(x) = x 2 - 2x (a) Find the average rate of change from 2 to 4. (b) Find an equation of the secant line containing (2, h(2)) and (4, h(4)).

66. f 1x2 = - 4x + 1 (a) Find the average rate of change from 2 to 5. (b) Find an equation of the secant line containing 12, f 1222 and 15, f 1522.

70. h(x) = - 2x 2 + x (a) Find the average rate of change from 0 to 3. (b) Find an equation of the secant line containing (0, h(0)) and (3, h(3)).

Mixed Practice

71. g(x) = x 3 - 27x (a) Determine whether g is even, odd, or neither. (b) There is a local minimum value of - 54 at 3. Determine the local maximum value. 72. f 1x2 = - x 3 + 12x (a) Determine whether f is even, odd, or neither. (b) There is a local maximum value of 16 at 2. Determine the local minimum value. 73. F 1x2 = - x 4 + 8x 2 + 8 (a) Determine whether F is even, odd, or neither. (b) There is a local maximum value of 24 at x = 2. Determine a second local maximum value.

(c) Suppose the area under the graph of F between x = 0 and x = 3 that is bounded below by the x-axis is 47.4 square units. Using the result from part (a), determine the area under the graph of F between x = - 3 and x = 0 bounded below by the x-axis. 74. G1x2 = - x 4 + 32x 2 + 144 (a) Determine whether G is even, odd, or neither. (b) There is a local maximum value of 400 at x = 4. Determine a second local maximum value. (c) Suppose the area under the graph of G between x = 0 and x = 6 that is bounded below by the x-axis is 1612.8 square units. Using the result from part (a), determine the area under the graph of G between x = - 6 and x = 0 bounded below by the x-axis.

Applications and Extensions 75. Minimum Average Cost The average cost per hour in dollars, C , of producing x riding lawn mowers can be modeled by the function 2500 C(x) = 0.3x 2 + 21x - 251 + x

The data below represent the U.S. debt for the years 2000– 2010. Since the debt D depends on the year y and each input corresponds to exactly one output, the debt is a function of the year; so D(y) represents the debt for each year y.

(a) Use a graphing utility to graph C = C(x) . (b) Determine the number of riding lawn mowers to produce in order to minimize average cost. (c) What is the minimum average cost?

Year

Debt (Billions of Dollars)

Year

2000

5674

2006

2001

5807

2007

9008

76. Medicine Concentration The concentration C of a medication in the bloodstream t hours after being administered is modeled by the function

2002

6228

2008

10,025

2003

6783

2009

11,910

2004

7379

2010

13,562

2005

7933

C(t) = - 0.002t 4 + 0.039t 3 - 0.285t 2 + 0.766t + 0.085 (a) After how many hours will the concentration be highest? (b) A woman nursing a child must wait until the concentration is below 0.5 before she can feed him. After taking the medication, how long must she wait before feeding her child? 77. National Debt The size of the total debt owed by the United States federal government has been growing over the past few years. In fact, according to the Department of the Treasury, the debt per person living in the United States is approximately \$45,000 (or over \$300,000 per U.S. household).

Debt (Billions of Dollars) 8507

Source: www.treasurydirect.gov (a) Plot the points (2000, 5.7), (2001, 5.8), and so on in a Cartesian plane. (b) Draw a line segment from the point (2000, 5.7) to (2002, 6.2). What does the slope of this line segment represent? (c) Find the average rate of change of the debt from 2000 to 2002. (d) Find the average rate of change of the debt from 2004 to 2006.

97

Functions and Their Graphs

(e) Find the average rate of change of the debt from 2008 to 2010. (f) What is happening to the average rate of change as time passes?

(e) What is happening to the average rate of change as time passes? 79. For the function f 1x2 = x 2, compute each average rate of change: (a) From 0 to 1 (b) From 0 to 0.5 (c) From 0 to 0.1 (d) From 0 to 0.01 (e) From 0 to 0.001 (f) Use a graphing utility to graph each of the secant lines along with f . (g) What do you think is happening to the secant lines? (h) What is happening to the slopes of the secant lines? Is there some number that they are getting closer to? What is that number?

78. E.coli Growth A strain of E.coli Beu 397-recA441 is placed into a nutrient broth at 30° Celsius and allowed to grow. The data shown in the table are collected. The population is measured in grams and the time in hours. Since population P depends on time t and each input corresponds to exactly one output, we can say that population is a function of time; so P(t) represents the population at time t. Time (hours), t

Population (grams), P

0

0.09

2.5

0.18

3.5

0.26

4.5

0.35

6

0.50

80. For the function f 1x2 = x 2, compute each average rate of change: (a) From 1 to 2 (b) From 1 to 1.5 (c) From 1 to 1.1 (d) From 1 to 1.01 (e) From 1 to 1.001 (f) Use a graphing utility to graph each of the secant lines along with f . (g) What do you think is happening to the secant lines? (h) What is happening to the slopes of the secant lines? Is there some number that they are getting closer to? What is that number?

(a) Plot the points (0, 0.09), (2.5, 0.18), and so on in a Cartesian plane. (b) Draw a line segment from the point (0, 0.09) to (2.5, 0.18). What does the slope of this line segment represent? (c) Find the average rate of change of the population from 0 to 2.5 hours. (d) Find the average rate of change of the population from 4.5 to 6 hours.

Problems 81– 88 require the following discussion of a secant line. The slope of the secant line containing the two points (x, f(x)) and 1x + h, f 1x + h22 on the graph of a function y = f 1x2 may be given as m sec =

f 1x + h2 - f 1x2 1x + h2 - x

=

f 1x + h2 - f 1x2 h

h⬆0

In calculus, this expression is called the difference quotient of f. (a) Express the slope of the secant line of each function in terms of x and h. Be sure to simplify your answer. (b) Find m sec for h = 0.5, 0.1, and 0.01 at x = 1. What value does m sec approach as h approaches 0? (c) Find the equation for the secant line at x = 1 with h = 0.01. (d) Use a graphing utility to graph f and the secant line found in part (c) on the same viewing window. 81. f 1x2 = 2x + 5

85. f 1x2 = 2x 2 - 3x + 1

82. f 1x2 = - 3x + 2

86. f 1x2 = - x 2 + 3x - 2

Explaining Concepts: Discussion and Writing

89. Draw the graph of a function that has the following properties: domain: all real numbers; range: all real numbers; intercepts: (0, - 3) and (3, 0); a local maximum value of - 2 is at - 1; a local minimum value of - 6 is at 2. Compare your graph with those of others. Comment on any differences. 90. Redo Problem 89 with the following additional information: increasing on ( -  , - 1), (2,  ); decreasing on ( - 1, 2). Again compare your graph with others and comment on any differences. 91. How many x-intercepts can a function defined on an interval have if it is increasing on that interval? Explain.

98

83. f 1x2 = x 2 + 2x 1 87. f 1x2 = x

84. f 1x2 = 2x 2 + x 1 88. f 1x2 = 2 x

92. Suppose that a friend of yours does not understand the idea of increasing and decreasing functions. Provide an explanation, complete with graphs, that clarifies the idea. 93. Can a function be both even and odd? Explain. 94. Using a graphing utility, graph y = 5 on the interval ( - 3, 3). Use MAXIMUM to find the local maximum values on ( - 3, 3). Comment on the result provided by the calculator. 95. A function f has a positive average rate of change on the interval 3 2, 5 4 . Is f increasing on 3 2, 5 4 ? Explain. 96. Show that a constant function f(x) = b has an average rate of change of 0. Compute the average rate of change of y = 24 - x 2 on the interval 3 - 2, 2 4 . Explain how this can happen.

Functions and Their Graphs

Interactive Exercises Ask your instructor if the applets below are of interest to you. 97. Open the Secant Line Not Min Point applet. (a) Grab point B and move it just below the x-axis. Verify the slope of the secant line reported in the applet. (b) Move point B closer to point A. What happens to the value of the slope of the secant line? (c) Move point B as close as you can to point A. Describe the secant line as point B approaches point A. Find the equation of the secant line where point B is as close as possible to point A.

(b) Move point B closer to point A, so that the x-coordinate of point B remains greater than the x-coordinate of point A. What happens to the value of the slope of the secant line? Specifically, what is the sign of the slope of the secant line? (c) Move point B so that the x-coordinate of point B is less than the x-coordinate of point A. What happens to the value of the slope of the secant line? Specifically, what is the sign of the slope of the secant line? (d) As point B moves from the left of point A to the right of point A, note the slope of the secant line. Conjecture the value of the slope of the line tangent to the function at point A.

98. Open the Secant Line Min Point applet. In the applet, a local minimum is - 7.33 at x = 3 (labeled as point A). (a) Grab point B and move it just below the x-axis. Verify the slope of the secant line reported in the applet.

‘Are You Prepared?’ Answers 1. 2 6 x 6 5

2. 1

3. symmetric with respect to the y-axis

4. y + 2 = 5(x - 3)

5. ( - 3, 0), (3, 0), (0, - 9)

4 Library of Functions; Piecewise-defined Functions PREPARING FOR THIS SECTION Before getting started, review the following: • Intercepts

• Graphs of Key Equations

Now Work the ‘Are You Prepared?’ problems.

OBJECTIVES 1 Graph the Functions Listed in the Library of Functions 2 Graph Piecewise-defined Functions

1 Graph the Functions Listed in the Library of Functions First we introduce a few more functions, beginning with the square root function. We graphed the equation y = 1x. Figure 31 shows a graph of the function f1x2 = 1x. Based on the graph, we have the following properties: Properties of f1 x 2 ⴝ 1x

Figure 31 y 6 (1, 1)

(4, 2)

(9, 3)

(0, 0) –2

5

10 x

E XAM PL E 1

1. The domain and the range are the set of nonnegative real numbers. 2. The x-intercept of the graph of f1x2 = 1x is 0. The y-intercept of the graph of f1x2 = 1x is also 0. 3. The function is neither even nor odd. 4. The function is increasing on the interval 10,  2. 5. The function has an absolute minimum of 0 at x = 0.

Graphing the Cube Root Function 3 (a) Determine whether f1x2 = 1 x is even, odd, or neither. State whether the graph of f is symmetric with respect to the y-axis or symmetric with respect to the origin. 3 (b) Determine the intercepts, if any, of the graph of f1x2 = 1 x. 3 (c) Graph f1x2 = 1x.

99

Functions and Their Graphs

Solution

(a) Because 3

3

f 1 -x2 = 1 -x = - 1 x = -f 1x2

the function is odd. The graph of f is symmetric with respect to the origin. 3 (b) The y-intercept is f102 = 2 0 = 0. The x-intercept is found by solving the equation f1x2 = 0. f1x2 = 0 3 1 x=0

x=0

3 f1x2 = 1 x

Cube both sides of the equation.

The x-intercept is also 0. (c) Use the function to form Table 4 and obtain some points on the graph. Because of the symmetry with respect to the origin, we find only points 1x, y2 for which 3 x Ú 0. Figure 32 shows the graph of f1x2 = 1 x.

Table 4

x

3 y ⴝf1 x2 ⴝ 1x

(x, y)

0

0

(0, 0)

1 8

1 2

1 1 a , b 8 2

1

1

2

1 2  1.26

8

3

2

Figure 32 y 3 3

(1, 1)

( 1–8, 1–2)

(1, 1)

(2, 2 )

( 1–8 , 1–2)

3

3

x

(0, 0)

3

12, 1 22

3

(1, 1)

(2, 2 )

(8, 2)

3

From the results of Example 1 and Figure 32, we have the following properties of the cube root function. 3 Properties of f1 x 2 ⴝ 2x

1. The domain and the range are the set of all real numbers. 3 2. The x-intercept of the graph of f1x2 = 1 x is 0. The y-intercept of the 3 graph of f1x2 = 1x is also 0. 3. The graph is symmetric with respect to the origin. The function is odd. 4. The function is increasing on the interval 1 - ,  2. 5. The function does not have any local minima or any local maxima.

E X A MP L E 2

Solution

Graphing the Absolute Value Function (a) Determine whether f1x2 = 0 x 0 is even, odd, or neither. State whether the graph of f is symmetric with respect to the y-axis or symmetric with respect to the origin. (b) Determine the intercepts, if any, of the graph of f1x2 = 0 x 0 . (c) Graph f1x2 = 0 x 0 . (a) Because

f1 -x2 = 0 -x 0

= 0 x 0 = f 1x2

the function is even. The graph of f is symmetric with respect to the y-axis. 100

Functions and Their Graphs

(b) The y-intercept is f102 = 0 0 0 = 0. The x-intercept is found by solving the equation f1x2 = 0 or 0 x 0 = 0. So the x-intercept is 0. (c) Use the function to form Table 5 and obtain some points on the graph. Because of the symmetry with respect to the y-axis, we need to find only points 1x, y2 for which x Ú 0. Figure 33 shows the graph of f 1x2 = 0 x 0 . Figure 33

Table 5

x

y ⴝ f (x) ⴝ  x 

1 x, y 2

0

0

(0, 0)

1

1

(1, 1)

2

2

(2, 2)

3

3

(3, 3)

y 3

(3, 3) (2, 2) (1, 1)

3 2 1

(3, 3) (2, 2)

2 1

1

(1, 1) 1 2 (0, 0)

3

x

From the results of Example 2 and Figure 33, we have the following properties of the absolute value function. Properties of f(x) ⴝ  x  1. The domain is the set of all real numbers. The range of f is 5 y  y Ú 0 6 . 2. The x-intercept of the graph of f1x2 = 0 x 0 is 0. The y-intercept of the graph of f1x2 = 0 x 0 is also 0. 3. The graph is symmetric with respect to the y-axis. The function is even. 4. The function is decreasing on the interval 1 - , 02. It is increasing on the interval 10,  2. 5. The function has an absolute minimum of 0 at x = 0.

Seeing the Concept Graph y = 0 x 0 on a square screen and compare what you see with Figure 33. Note that some graphing calculators use abs 1x2 for absolute value.

Below is a list of the key functions that we have discussed. In going through this list, pay special attention to the properties of each function, particularly to the shape of each graph. Knowing these graphs along with key points on each graph will lay the foundation for further graphing techniques. Figure 34 Constant Function

Constant Function

y

f 1x2 = b

f(x) = b (0,b) x

See Figure 34. The domain of a constant function is the set of all real numbers; its range is the set consisting of a single number b. Its graph is a horizontal line whose y-intercept is b. The constant function is an even function.

Figure 35 Identity Function

Identity Function

f (x) = x

y 3

f 1x2 = x

(1, 1) –3 (–1, –1)

b is a real number

(0, 0)

3 x

See Figure 35. The domain and the range of the identity function are the set of all real numbers. Its graph is a line whose slope is 1 and whose y-intercept is 0. The line consists of 101

Functions and Their Graphs

all points for which the x-coordinate equals the y-coordinate. The identity function is an odd function that is increasing over its domain. Note that the graph bisects quadrants I and III. Figure 36 Square Function f(x ) = x 2

y (–2, 4)

Square Function f 1x2 = x2

(2, 4)

4

(–1, 1)

(1, 1) 4 x

(0, 0)

–4

Figure 37 Cube Function

Cube Function

y 4

See Figure 36. The domain of the square function f is the set of all real numbers; its range is the set of nonnegative real numbers. The graph of this function is a parabola whose vertex is at 10, 02, which is also the only intercept. The square function is an even function that is decreasing on the interval 1 - , 02 and increasing on the interval 10,  2. f 1x2 = x3

f(x ) = x 3 (1, 1) (0, 0)

4 (1, 1)

x

4

See Figure 37. The domain and the range of the cube function are the set of all real numbers. The intercept of the graph is at 10, 02. The cube function is odd and is increasing on the interval 1 - ,  2.

4

Square Root Function

Figure 38 Square Root Function y

f(x) =

2

x

(1, 1)

5 x

(0, 0)

1

f 1x2 = 1x

(4, 2)

See Figure 38. The domain and the range of the square root function are the set of nonnegative real numbers. The intercept of the graph is at 10, 02. The square root function is neither even nor odd and is increasing on the interval 10,  2.

Figure 39 Cube Root Function y 3

f(x ) =

3

x

3

(

 1–8 , 1–2

(1, 1)

)

Cube Root Function

(2, 2 )

( 1–8 , 1–2)

3 f 1x2 = 1 x

3 x

3 (0, 0) 3

(1, 1)

(2, 2 )

See Figure 39. The domain and the range of the cube root function are the set of all real numbers. The intercept of the graph is at 10, 02. The cube root function is an odd function that is increasing on the interval 1 - ,  2.

3

Figure 40 Reciprocal Function y 2

(1–2 , 2) f (x ) =

(2,  1–2 )

Reciprocal Function

(1, 1)

f1x2 =

2 x

2 (1, 1)

See Figure 40. 2

102

1 –– x

1 x

Functions and Their Graphs

The domain and the range of the reciprocal function are the set of all nonzero real numbers. The graph has no intercepts. The reciprocal function is decreasing on the intervals 1 - , 02 and 10,  2 and is an odd function.

Absolute Value Function

f 1x2 = 0 x 0 Figure 41 Absolute Value Function y

f(x ) = x 

3 (2, 2)

(2, 2)

(1, 1)

(1, 1) 3

3 x

(0, 0)

See Figure 41. The domain of the absolute value function is the set of all real numbers; its range is the set of nonnegative real numbers. The intercept of the graph is at 10, 02. If x Ú 0, then f1x2 = x, and this part of the graph of f is the line y = x; if x 6 0, then f1x2 = -x, and this part of the graph of f is the line y = -x. The absolute value function is an even function; it is decreasing on the interval 1 - , 02 and increasing on the interval 10,  2. The notation int 1x2 stands for the largest integer less than or equal to x. For example, 1 3 int 112 = 1, int12.52 = 2, inta b = 0, inta - b = -1, int1p2 = 3 2 4

This type of correspondence occurs frequently enough in mathematics that we give it a name.

DEFINITION

Greatest Integer Function f 1x2 = int1x2* = greatest integer less than or equal to x

Table 6 x

-1 1 2 -

y ⴝf (x) ⴝint(x) -1 -1

1 4

-1

0

0

1 4 1 2 3 4

0 0 0

(x, y)

We obtain the graph of f1x2 = int1x2 by plotting several points. See Table 6. For values of x, -1 … x 6 0, the value of f1x2 = int1x2 is -1; for values of x, 0 … x 6 1, the value of f is 0. See Figure 42 for the graph.

(- 1, -1) 1 a - , - 1b 2

Figure 42 Greatest Integer Function y 4

1 a - , - 1b 4

(0, 0)

1 a , 0b 4 1 a , 0b 2 3 a , 0b 4

2 2

2

4

x

3

The domain of the greatest integer function is the set of all real numbers; its range is the set of integers. The y-intercept of the graph is 0. The x-intercepts lie in the interval 3 0, 12. The greatest integer function is neither even nor odd. It is constant on every interval of the form 3 k, k + 12, for k an integer. In Figure 42, we use a solid dot to indicate, for example, that at x = 1 the value of f is f112 = 1; we use an open circle to illustrate that the function does not assume the value of 0 at x = 1. Although a precise definition requires the idea of a limit, discussed in calculus, in a rough sense, a function is said to be continuous if its graph has no gaps or holes and can be drawn without lifting a pencil from the paper on which the graph is drawn. We contrast this with a discontinuous function. A function is discontinuous if *Some books use the notation f 1x2 = 3 x 4 instead of int 1x2.

103

Functions and Their Graphs

Figure 43 f 1x2 = int 1x2 . 6

2

6

its graph has gaps or holes so that its graph cannot be drawn without lifting a pencil from the paper. From the graph of the greatest integer function, we can see why it is also called a step function. At x = 0, x = {1, x = { 2, and so on, this function is discontinuous because, at integer values, the graph suddenly “steps” from one value to another without taking on any of the intermediate values. For example, to the immediate left of x = 3, the y-coordinates of the points on the graph are 2, and at x = 3 and to the immediate right of x = 3, the y-coordinates of the points on the graph are 3. So, the graph has gaps in it. COMMENT When graphing a function using a graphing utility, you can choose either the connected mode, in which points plotted on the screen are connected, making the graph appear without any breaks, or the dot mode, in which only the points plotted appear. When graphing the greatest integer function with a graphing utility, it may be necessary to be in the dot mode. This is to prevent the utility from “connecting the dots” when f1x2 changes from one integer value to the next. See Figure 43. 

2 (a) Connected mode 6

The functions discussed so far are basic. Whenever you encounter one of them, you should see a mental picture of its graph. For example, if you encounter the function f1x2 = x2, you should see in your mind’s eye a picture like Figure 36. 2

6

Now Work 2 (b) Dot mode

PROBLEMS

9

16

THROUGH

2 Graph Piecewise-defined Functions Sometimes a function is defined using different equations on different parts of its domain. For example, the absolute value function f1x2 = 0 x 0 is actually defined by two equations: f1x2 = x if x Ú 0 and f1x2 = -x if x 6 0. For convenience, these equations are generally combined into one expression as f1x2 = 0 x 0 = e

x if x Ú 0 -x if x 6 0

When a function is defined by different equations on different parts of its domain, it is called a piecewise-defined function.

E X A MP L E 3

Analyzing a Piecewise-defined Function The function f is defined as -2x + 1 if -3 … x 6 1 f 1x2 = c 2 if x = 1 x2 if x 7 1

(a) Find f1 -22, f 112, and f122. (c) Locate any intercepts. (e) Use the graph to find the range of f.

Solution

(b) Determine the domain of f. (d) Graph f . (f) Is f continuous on its domain?

(a) To find f1 -22, observe that when x = -2 the equation for f is given by f1x2 = -2x + 1. So f1 -22 = -2( -2) + 1 = 5 When x = 1, the equation for f is f1x2 = 2. That is, f 112 = 2

When x = 2, the equation for f is f 1x2 = x2. So f122 = 22 = 4

104

(b) To find the domain of f, look at its definition. Since f is defined for all x greater than or equal to -3 , the domain of f is {x  x Ú -3} , or the interval 3 -3,  2. (c) The y-intercept of the graph of the function is f 102 . Because the equation for f when x = 0 is f 1x2 = -2x + 1 , the y-intercept is f102 = -2102 + 1 = 1 . The

Functions and Their Graphs

x-intercepts of the graph of a function f are the real solutions to the equation f1x2 = 0 . To find the x-intercepts of f, solve f1x2 = 0 for each “piece” of the function and then determine if the values of x, if any, satisfy the condition that defines the piece. f1x2 = 0 -2x + 1 = 0

f1x2 = 0 -3 … x 6 1

-2x = -1 1 x= 2

Figure 44 y 8

4

(1,2)

(2,4)

(0,1)

( 1–2 , 0)

4

x

(1, 1)

x2 = 0 x=0

x7 1

1 The first potential x-intercept, x = , satisfies the condition -3 … x 6 1 , so 2 1 x = is an x-intercept. The second potential x-intercept, x = 0 , does not satisfy 2 1 the condition x 7 1 , so x = 0 is not an x-intercept. The only x-intercept is . The 2 1 intercepts are (0, 1) and a , 0b . 2 (d) To graph f , we graph “each piece.” First we graph the line y = -2x + 1 and keep only the part for which -3 … x 6 1. Then we plot the point 11, 22 because, when x = 1, f1x2 = 2. Finally, we graph the parabola y = x2 and keep only the part for which x 7 1. See Figure 44. (e) From the graph, we conclude that the range of f is 5 y y 7 -1 6 , or the interval 1 -1,  2. (f ) The function f is not continuous because there is a “jump” in the graph at x = 1.

Now Work

E XAM PL E 4

2=0 x=1 No solution

f1x2 = 0

PROBLEM

29

Cost of Electricity In the summer of 2011, Duke Energy supplied electricity to residences in Ohio for a monthly customer charge of \$5.50 plus 6.4471¢ per kilowatt-hour (kWhr) for the first 1000 kWhr supplied in the month and 7.8391¢ per kWhr for all usage over 1000 kWhr in the month. (a) What is the charge for using 300 kWhr in a month? (b) What is the charge for using 1500 kWhr in a month? (c) If C is the monthly charge for x kWhr, develop a model relating the monthly charge and kilowatt-hours used. That is, express C as a function of x. Source: Duke Energy, 2011.

Solution

(a) For 300 kWhr, the charge is \$5.50 plus 6.4471. = \$0.064471 per kWhr. That is, Charge = \$5.50 + \$0.06447113002 = \$24.84 (b) For 1500 kWhr, the charge is \$5.50 plus 6.4471¢ per kWhr for the first 1000 kWhr plus 7.8391¢ per kWhr for the 500 kWhr in excess of 1000. That is, Charge = \$5.50 + \$0.064471110002 + \$0.07839115002 = \$109.17 (c) Let x represent the number of kilowatt-hours used. If 0 … x … 1000, the monthly charge C (in dollars) can be found by multiplying x times \$0.064471 and adding the monthly customer charge of \$5.50. So, if 0 … x … 1000, then C 1x2 = 0.064471x + 5.50.

For x 7 1000, the charge is 0.064471110002 + 5.50 + 0.0783911x - 10002, since x - 1000 equals the usage in excess of 1000 kWhr, which costs \$0.078391 per kWhr. That is, if x 7 1000, then C1x2 = 0.064471110002 + 5.50 + 0.0783911x - 10002 = 69.971 + 0.0783911x - 10002 = 0.078391x - 8.42

105

Functions and Their Graphs

Figure 45

The rule for computing C follows two equations:

Charge (dollars)

C

(1500, 109.17)

C 1x2 = e

90 60 30 5.50

0.064471x + 5.50 if 0 … x … 1000 0.078391x - 8.42 if x 7 1000

The Model

See Figure 45 for the graph.

(1000, 69.97) (300, 24.84) 500 1000 Usage (kWhr)

1500 x

4 Assess Your Understanding ‘Are You Prepared?’

Answers are given at the end of these exercises.

1. Sketch the graph of y = 1x . 1 2. Sketch the graph of y = . x

3. List the intercepts of the equation y = x 3 - 8.

Concepts and Vocabulary 4. The function f 1x2 = x 2 is decreasing on the interval _________.

5. When functions are defined by more than one equation, they are called __________________ functions.

7. True or False The cube root function is odd and is decreasing on the interval 1 -  ,  2.

8. True or False The domain and the range of the reciprocal function are the set of all real numbers.

6. True or False The cube function is odd and is increasing on the interval 1 -  ,  2.

Skill Building

In Problems 9 –16, match each graph to its function. A. Constant function

B. Identity function

C. Square function

D. Cube function

E. Square root function

F. Reciprocal function

G. Absolute value function

H. Cube root function

9.

10.

11.

12.

13.

14.

15.

16.

In Problems 17–24, sketch the graph of each function. Be sure to label three points on the graph. 17. f 1x2 = x

1 21. f 1x2 = x

18. f 1x2 = x 2 2

x 25. If f 1x2 = c 2 2x + 1 find: (a) f 1 - 22

22. f 1x2 = 0 x 0

3 23. f 1x2 = 1 x

if x 6 0 if x = 0 if x 7 0

(b) f 102

- 3x 26. If f 1x2 = c 0 2x 2 + 1

(c) f 122

2x - 4 if - 1 … x … 2 27. If f 1x2 = e 3 x - 2 if 2 6 x … 3 find: (a) f 102 (b) f 112 (c) f 122

106

19. f 1x2 = x 3

(d) f 132

find: (a) f 1 - 22 x3 28. If f 1x2 = e 3x + 2 find: (a) f 1 - 12

20. f 1x2 = 1x 24. f 1x2 = 3

if x 6 - 1 if x = - 1 if x 7 - 1

(b) f 1 - 12 if - 2 … x 6 1 if 1 … x … 4 (b) f 102

(c) f 102

(c) f 112

(d) f 132

Functions and Their Graphs

In Problems 29–40: (a) Find the domain of each function.

(b) Locate any intercepts.

(d) Based on the graph, find the range.

(e) Is f continuous on its domain?

if x ⬆ 0 if x = 0

29. f 1x2 = b

2x 1

32. f 1x2 = b

x + 3 - 2x - 3

1 + x 35. f 1x2 = b 2 x 38. f 1x2 = b

30. f 1x2 = b

if x ⬆ 0 if x = 0

3x 4

x + 3 33. f 1x2 = c 5 -x + 2

if x 6 - 2 if x Ú - 2

1 x 36. f 1x2 = c 3 1 x

if x 6 0 if x Ú 0

2 - x

if - 3 … x 6 1

2x

if x 7 1

(c) Graph each function.

31. f 1x2 = b

if - 2 … x 6 1 if x = 1 if x 7 1

- 2x + 3 3x - 2

2x + 5 34. f 1x2 = c - 3 - 5x

if x 6 0

37. f 1x2 = b

if x Ú 0

39. f 1x2 = 2 int 1x2

0x0 x3

if x 6 1 if x Ú 1 if - 3 … x 6 0 if x = 0 if x 7 0

if - 2 … x 6 0 if x 7 0

40. f 1x2 = int 12x2

In Problems 41–44, the graph of a piecewise-defined function is given. Write a definition for each function. 41.

y 2

42.

y 2

44.

y (0, 2)

2

(2, 2)

(2, 1)

(2, 1)

(1, 1)

y

43.

(1, 1) (1, 1)

(1, 1)

(1, 0) 2

2 x

(0, 0)

(0, 0)

2

2 x

2

(0, 0)

(2, 0) x

2 x

2

(1, 1)

45. If f 1x2 = int 12x2, find (a) f 11.22

(b) f 11.62

(c) f 1 - 1.82

x 46. If f 1x2 = int a b, find 2 (a) f 11.22

(b) f 11.62

(c) f 1 - 1.82

Applications and Extensions 47. Cell Phone Service Sprint PCS offers a monthly cellular phone plan for \$39.99. It includes 450 anytime minutes and charges \$0.45 per minute for additional minutes. The following function is used to compute the monthly cost for a subscriber:

Determine the fee for parking in the short-term parking garage for (a) 2 hours

(b) 7 hours

(c) 15 hours

(d) 8 hours and 24 minutes Source: O’Hare International Airport

39.99 C 1x2 = b 0.45x - 162.51

if 0 … x … 450 if x 7 450

where x is the number of anytime minutes used. Compute the monthly cost of the cellular phone for use of the following number of anytime minutes: (a) 200

(b) 465

(c) 451

Source: Sprint PCS 48. Parking at O’Hare International Airport The short-term (no more than 24 hours) parking fee F (in dollars) for parking x hours at O’Hare International Airport’s main parking garage can be modeled by the function 2 4 F 1x2 = e 10 5 int 1x + 12 + 2 51

if 0 if 1 if 3 if 4 if 9

6 6 6 6 …

x x x x x

… … … 6 …

1 3 4 9 24

49. Cost of Natural Gas In March 2011, Peoples Energy had the following rate schedule for natural gas usage in singlefamily residences: Monthly service charge \$18.95 Per therm service charge 1st 50 therms \$0.33372/therm Over 50 therms \$0.12360/therm Gas charge \$0.5038/therm (a) What is the charge for using 50 therms in a month? (b) What is the charge for using 500 therms in a month? (c) Develop a model that relates the monthly charge C for x therms of gas. (d) Graph the function found in part (c). Source: Peoples Energy, Chicago, Illinois, 2011

107

Functions and Their Graphs

50. Cost of Natural Gas In March 2011, Nicor Gas had the following rate schedule for natural gas usage in single-family residences: Monthly customer charge \$13.55 Distribution charge 1st 20 therms \$0.1473/therm Next 30 therms \$0.0579/therm Over 50 therms \$0.0519/therm Gas supply charge \$0.51/therm

(a) What is the charge for using 40 therms in a month? (b) What is the charge for using 150 therms in a month? (c) Develop a model that gives the monthly charge C for x therms of gas. (d) Graph the function found in part (c). Source: Nicor Gas, Aurora, Illinois, 2011 51. Federal Income Tax Two 2011 Tax Rate Schedules are given in the table below. If x equals taxable income and y equals the tax due, construct a function y = f 1x2 for Schedule X.

52. Federal Income Tax Refer to the revised 2011 tax rate schedules. If x equals taxable income and y equals the tax due, construct a function y = f 1x2 for Schedule Y-1. REVISED 2011 TAX RATE SCHEDULES

Schedule X—Single If Taxable Income Is Over

But Not Over

The Tax Is This Amount

\$0

\$8,500

-

+

Schedule Y-1—Married Filing Jointly or Qualifying Widow(er) Plus This %

Of the Excess Over

If Taxable Income Is Over

10%

\$0

\$0

But Not Over

The Tax Is This Amount

\$17,000

Plus This %

Of The Excess Over

+

10%

\$0

8,500

34,500

\$850.00

+

15%

8,500

17,000

69,000

\$1,700.00

+

15%

17,000

34,500

83,600

4,750.00

+

25%

34,500

69,000

139,350

9,500.00

+

25%

69,000

83,600

174,400

17,025.00

+

28%

83,600

139,350

212,300

27,087.50

+

28%

139,350

174,400

379,150

42,449.00

+

33%

174,400

212,300

379,150

47,513.50

+

33%

212,300

379,150

110,016.50

+

35%

379,150

379,150

102,574.00

+

35%

379,150

Source: Internal Revenue Service 53. Cost of Transporting Goods A trucking company transports goods between Chicago and New York, a distance of 960 miles. The company’s policy is to charge, for each pound, \$0.50 per mile for the first 100 miles, \$0.40 per mile for the next 300 miles, \$0.25 per mile for the next 400 miles, and no charge for the remaining 160 miles. (a) Graph the relationship between the cost of transportation in dollars and mileage over the entire 960-mile route. (b) Find the cost as a function of mileage for hauls between 100 and 400 miles from Chicago. (c) Find the cost as a function of mileage for hauls between 400 and 800 miles from Chicago. 54. Car Rental Costs An economy car rented in Florida from National Car Rental® on a weekly basis costs \$95 per week. Extra days cost \$24 per day until the day rate exceeds the weekly rate, in which case the weekly rate applies. Also, any part of a day used counts as a full day. Find the cost C of renting an economy car as a function of the number x of days used, where 7 … x … 14. Graph this function. 55. Mortgage Fees Fannie Mae charges an adverse market delivery charge on all mortgages, which represents a fee homebuyers seeking a loan must pay. The rate paid depends on the credit score of the borrower, the amount borrowed, and the loan-to-value (LTV) ratio. The LTV ratio is the ratio of amount borrowed to appraised value of the home. For example, a homebuyer who wishes to borrow \$250,000 with a credit score of 730 and an LTV ratio of 80% will pay 0.5% (0.005) of \$250,000 or \$1250. The table shows the adverse

108

delivery charge for various credit scores and an LTV ratio of 80%.

Credit Score

… 659

3.25%

660–679

2.75%

680–699

1.5%

700–719

1%

720–739

0.5%

Ú 740

0.25%

Source: Fannie Mae.

(a) Construct a function C = C(s)where C is the adverse market delivery charge and s is the credit score of an individual who wishes to borrow \$300,000 with an 80% LTV ratio. (b) What is the adverse market delivery charge on a \$300,000 loan with an 80% LTV ratio for a borrower whose credit score is 725? (c) What is the adverse market delivery charge on a \$300,000 loan with an 80% LTV ratio for a borrower whose credit score is 670? 56. Minimum Payments for Credit Cards Holders of credit cards issued by banks, department stores, oil companies, and so on, receive bills each month that state minimum

Functions and Their Graphs

amounts that must be paid by a certain due date. The minimum due depends on the total amount owed. One such credit card company uses the following rules: For a bill of less than \$10, the entire amount is due. For a bill of at least \$10 but less than \$500, the minimum due is \$10. A minimum of \$30 is due on a bill of at least \$500 but less than \$1000, a minimum of \$50 is due on a bill of at least \$1000 but less than \$1500, and a minimum of \$70 is due on bills of \$1500 or more. Find the function f that describes the minimum payment due on a bill of x dollars. Graph f. 57. Wind Chill The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is t

0 … v 6 1.79

W = d 33 -

110.45 + 10 1v - v2 133 - t2

22.04 33 - 1.5958 133 - t2

1.79 … v … 20 v 7 20

where v represents the wind speed (in meters per second) and t represents the air temperature (°C). Compute the wind chill for the following:

(a) An air temperature of 10°C and a wind speed of 1 meter per second 1m/sec2 (b) An air temperature of 10°C and a wind speed of 5 m/sec

(c) An air temperature of 10°C and a wind speed of 15 m/sec (d) An air temperature of 10°C and a wind speed of 25 m/sec (e) Explain the physical meaning of the equation corresponding to 0 … v 6 1.79. (f) Explain the physical meaning of the equation corresponding to v 7 20. 58. Wind Chill Redo Problem 57(a) – (d) for an air temperature of - 10C. 59. First-class Mail In 2011 the U.S. Postal Service charged \$0.88 postage for first-class mail retail flats (such as an 8.5  by 11  envelope) weighing up to 1 ounce, plus \$0.17 for each additional ounce up to 13 ounces. First-class rates do not apply to flats weighing more than 13 ounces. Develop a model that relates C, the first-class postage charged, for a flat weighing x ounces. Graph the function. Source: United States Postal Service

Explaining Concepts: Discussion and Writing In Problems 60 – 67, use a graphing utility. 60. Exploration Graph y = x 2. Then on the same screen graph y = x 2 + 2, followed by y = x 2 + 4, followed by y = x 2 - 2. What pattern do you observe? Can you predict the graph of y = x 2 - 4? Of y = x 2 + 5?

65. Exploration Graph y = x 3. Then on the same screen graph y = 1x - 12 3 + 2. Could you have predicted the result?

61. Exploration Graph y = x 2. Then on the same screen graph y = 1x - 22 2, followed by y = 1x - 42 2, followed by y = 1x + 22 2. What pattern do you observe? Can you predict the graph of y = 1x + 42 2? Of y = 1x - 52 2?

67. Exploration Graph y = x 3, y = x 5, and y = x 7 on the same screen. What do you notice is the same about each graph? What do you notice that is different?

62. Exploration Graph y = 0 x 0 . Then on the same screen 1 graph y = 2 0 x 0 , followed by y = 4 0 x 0 , followed by y = 0 x 0 . 2 What pattern do you observe? Can you predict the graph of 1 y = 0 x 0 ? Of y = 5 0 x 0 ? 4 63. Exploration Graph y = x 2. Then on the same screen graph y = - x 2. What pattern do you observe? Now try y = 0 x 0 and y = - 0 x 0 . What do you conclude?

64. Exploration Graph y = 1x . Then on the same screen graph y = 1 - x . What pattern do you observe? Now try y = 2x + 1 and y = 2 1 - x2 + 1. What do you conclude?

66. Exploration Graph y = x 2, y = x 4, and y = x 6 on the same screen. What do you notice is the same about each graph? What do you notice that is different?

68. Consider the equation y = b

1 0

if x is rational if x is irrational

Is this a function? What is its domain? What is its range? What is its y-intercept, if any? What are its x-intercepts, if any? Is it even, odd, or neither? How would you describe its graph? 69. Define some functions that pass through 10, 02 and 11, 12 and are increasing for x Ú 0. Begin your list with y = 1x , y = x, and y = x 2. Can you propose a general result about such functions?

y

y 2

2.

2 (1, 1)

(4, 2)

(1, 1)

3. 10, - 82, 12, 02

2 x (0, 0)

4

x

(1, 1)

109

Functions and Their Graphs

5 Graphing Techniques: Transformations OBJECTIVES 1 Graph Functions Using Vertical and Horizontal Shifts 2 Graph Functions Using Compressions and Stretches 3 Graph Functions Using Reflections about the x-Axis or y-Axis

At this stage, if you were asked to graph any of the functions defined by y = x, 1 3 y = x2, y = x3, y = 1x, y = 1 x, y = 0 x 0 , or y = , your response should be, “Yes, x I recognize these functions and know the general shapes of their graphs.” (If this is not your answer, review the previous section, Figures 35 through 41.) Sometimes we are asked to graph a function that is “almost” like one that we already know how to graph. In this section, we develop techniques for graphing such functions. Collectively, these techniques are referred to as transformations. We introduce the method of transformations because it is a more efficient method of graphing than point-plotting.

1 Graph Functions Using Vertical and Horizontal Shifts Exploration

On the same screen, graph each of the following functions: Y1 = x 2 Y2 = x 2 + 2 Y3 = x 2 - 2

Figure 46

6

Y2  x 2  2

What do you observe? Now create a table of values for Y1 , Y2 , and Y3 . What do you observe?

Y1  x 2 6

6 2

Y3  x 2  2

Result Figure 46 illustrates the graphs. You should have observed a general pattern. With Y1 = x 2 on the screen, the graph of Y2 = x 2 + 2 is identical to that of Y1 = x 2, except that it is shifted vertically up 2 units. The graph of Y3 = x 2 - 2 is identical to that of Y1 = x 2, except that it is shifted vertically down 2 units. From Table 7(a), we see that the y-coordinates on Y2 = x 2 + 2 are 2 units larger than the y-coordinates on Y1 = x 2 for any given x-coordinate. From Table 7(b), we see that the y-coordinates on Y3 = x 2 - 2 are 2 units smaller than the y-coordinates on Y1 = x 2 for any given x-coordinate. Notice a vertical shift only affects the range of a function, not the domain. For example, the range of Y1 is 3 0,  2 while the range of Y2 is 3 2,  2 . The domain of both functions is all real numbers.

Table 7

(a)

(b)

We are led to the following conclusions: If a positive real number k is added to the outputs of a function y = f1x2 , the graph of the new function y = f1x2 + k is the graph of f shifted vertically up k units. If a positive real number k is subtracted from the outputs of a function y = f1x2 , the graph of the new function y = f1x2 - k is the graph of f shifted vertically down k units.

E X A MP L E 1

Solution 110

Vertical Shift Down Use the graph of f1x2 = x2 to obtain the graph of h 1x2 = x2 - 4. Find the domain and range of h.

Table 8 lists some points on the graphs of Y1 = f1x2 = x2 and Y2 = h 1x2 = f1x2 - 4 = x2 - 4. Notice that each y-coordinate of h is 4 units less than the corresponding y-coordinate of f.

Functions and Their Graphs

Table 8

To obtain the graph of h from the graph of f, subtract 4 from each y-coordinate on the graph of f. The graph of h is identical to that of f, except that it is shifted down 4 units. See Figure 47. Figure 47 y

2 6 Y1  x

(2, 4)

6

6

y  x2

4

Down 4 units

(2, 4) Down 4 units

(2, 0) (0, 0)

(2, 0) 4 x y  x2  4

4

Y2  x 2  4 5

(0, 4)

The domain of h is the set of all real numbers. The range of h is 3 -4,  2 .

Now Work

Exploration

PROBLEM

39

On the same screen, graph each of the following functions: Y1 = x 2 Y2 = (x - 3)2 Y3 = (x + 2)2 What do you observe?

Figure 48 Y1  x 2

6

6

Y3  (x  2)2

6 2

Y2  (x  3)2

Result Figure 48 illustrates the graphs. You should have observed the following pattern. With the graph of Y1 = x 2 on the screen, the graph of Y2 = (x - 3)2 is identical to that of Y1 = x 2, except it is shifted horizontally to the right 3 units. The graph of Y3 = (x + 2)2 is identical to that of Y1 = x 2, except it is shifted horizontally to the left 2 units. From Table 9(a), we see the x-coordinates on Y2 = (x - 3)2 are 3 units larger than they are for Y1 = x 2 for any given y-coordinate. For example, when Y1 = 0 , then x = 0 , and when Y2 = 0 , then x = 3 . Also, when Y1 = 1 , then x = -1 or 1, and when Y2 = 1, then x = 2 or 4. From Table 9(b), we see the x-coordinates on Y3 = (x + 2)2 are 2 units smaller than they are for Y1 = x 2 for any given y-coordinate. For example, when Y1 = 0 , then x = 0 , and when Y3 = 0 , then x = -2 . Also, when Y1 = 4 , then x = -2 or 2, and when Y3 = 4 , then x = -4 or 0.

Table 9

(a)

(b)

We are led to the following conclusions: If the argument x of a function f is replaced by x - h, h 7 0, the graph of the new function y = f1x - h2 is the graph of f shifted horizontally right h units. If the argument x of a function f is replaced by x + h, h 7 0, the graph of the new function y = f1x + h2 is the graph of f shifted horizontally left h units.

Now Work

PROBLEM

43

111

Functions and Their Graphs

NOTE Vertical shifts result when adding or subtracting a real number k after performing the operation suggested by the basic function, while horizontal shifts result when adding or subtracting a real number h to or from x before performing the operation suggested by the basic function. For example, the graph of f(x) = 1x + 3 is obtained by shifting the graph of y = 1x up 3 units, because we evaluate the square root function first and then add 3. The graph of g(x) = 1x + 3 is obtained by shifting the graph of y = 1x left 3 units, because we first add 3 to x before we evaluate the square root function. 

Vertical and horizontal shifts are sometimes combined.

E X A MP L E 2

Solution

Figure 49

Combining Vertical and Horizontal Shifts Graph the function f1x2 = 1x + 32 2 - 5. Find the domain and range of f .

We graph f in steps. First, notice that the rule for f is basically a square function, so we begin with the graph of y = x2 as shown in Figure 49(a). To get the graph of y = 1x + 32 2, shift the graph of y = x2 horizontally 3 units to the left. See Figure 49(b). Finally, to get the graph of y = 1x + 32 2 - 5, shift the graph of y = 1x + 32 2 vertically down 5 units. See Figure 49(c). Note the points plotted on each graph. Using key points can be helpful in keeping track of the transformation that has taken place. The domain of f is the set of all real numbers. The range of f is 3 -5,  2 .

(1, 1)

(1, 1) 5

y 5

y 5

y 5

(2, 1)

(4, 1)

5 x

5 x 5 (3, 0)

(0, 0)

5 x

5 (2, 4)

5 y  x2

(4, –4) Vertex (3, 5)

5 Replace x by x  3; Horizontal shift left 3 units

y  (x  3)2

(a)

(b)

Subtract 5; Vertical shift down 5 units

5 y  (x  3)2  5

(c)

Check: Graph Y1 = f 1x2 = 1x + 32 2 - 5 and compare the graph to Figure 49(c).

In Example 2, if the vertical shift had been done first, followed by the horizontal shift, the final graph would have been the same. (Try it for yourself.)

Now Work

PROBLEMS

45

AND

77

2 Graph Functions Using Compressions and Stretches Exploration

Y1 = 0 x 0

Y2 = 2 0 x 0 1 Y3 = 0 x 0 2 Then create a table of values and compare the y-coordinates for any given x-coordinate.

Figure 50 Y3 

1– |x| 2

Y1  |x| Y2  2|x| 6

6

6 2

112

On the same screen, graph each of the following functions:

Result Figure 50 illustrates the graphs. Now look at Table 10, where Y1 = 0 x 0 and Y2 = 2 0 x 0 . Notice that the values for Y2 are two times the values of Y1 for each x-value. This means that the graph of Y2 = 2 0 x 0 can be obtained from the graph of Y1 = 0 x 0 by multiplying each y-coordinate of Y1 = 0 x 0 by 2. Therefore, the graph of Y2 will be the graph of Y1 vertically stretched by a factor of 2. 1 Look at Table 11 where Y1 = 0 x 0 and Y3 = 0 x 0 . The values of Y3 are half the values of Y1 for each 2 1 x-value. So, the graph of Y3 = 0 x 0 can be obtained from the graph of Y1 = 0 x 0 by multiplying each 2 1 1 y-coordinate by . Therefore, the graph of Y3 will be the graph of Y1 vertically compressed by a factor of . 2 2

Functions and Their Graphs

Table 10

Table 11

Based on the Exploration, we have the following result: When the right side of a function y = f1x2 is multiplied by a positive number a, the graph of the new function y = af1x2 is obtained by multiplying each y-coordinate on the graph of y = f1x2 by a. The new graph is a vertically compressed (if 0 6 a 6 1 ) or a vertically stretched (if a 7 1 ) version of the graph of y = f1x2.

Now Work

PROBLEM

47

What happens if the argument x of a function y = f1x2 is multiplied by a positive number a, creating a new function y = f1ax2? To find the answer, look at the following Exploration.

Exploration

On the same screen, graph each of the following functions:

Y1 = f 1x2 = 1x

Y2 = f 12x2 = 1 2x

1 1 x x= Y3 = f a xb = 2 A2 A2

Create a table of values to explore the relation between the x- and y-coordinates of each function.

Figure 51 3 Y2 

2x

Y1 

x

Y3  0

0

4

–x 2

Result You should have obtained the graphs in Figure 51. Look at Table 12(a). Notice that (1, 1), (4, 2), and (9, 3) are points on the graph of Y1 = 1x . Also, (0.5, 1), (2, 2), and (4.5, 3) are points on the graph of 1 Y2 = 22x . For a given y-coordinate, the x-coordinate on the graph of Y2 is of the x-coordinate on Y1 . 2 We conclude that the graph of Y2 = 22x is obtained by multiplying the x-coordinate of each point on 1 the graph of Y1 = 1x by . The graph of Y2 = 22x is the graph of Y1 = 1x compressed horizontally. 2 Look at Table 12(b). Notice that (1, 1), (4, 2), and (9, 3) are points on the graph of Y1 = 1x . Also x notice that (2, 1), (8, 2), and (18, 3) are points on the graph of Y3 = . For a given y-coordinate, the A2 x-coordinate on the graph of Y3 is 2 times the x-coordinate on Y1 . We conclude that the graph of x Y3 = is obtained by multiplying the x-coordinate of each point on the graph of Y1 = 1x by 2. The A2 x graph of Y3 = is the graph of Y1 = 1x stretched horizontally. A2

Table 12

(a)

(b)

Based on the results of the Exploration, we have the following result: If the argument x of a function y = f1x2 is multiplied by a positive number a, the graph of the new function y = f1ax2 is obtained by multiplying each 1 x-coordinate of y = f1x2 by . A horizontal compression results if a 7 1, and a a horizontal stretch occurs if 0 6 a 6 1. Let’s look at an example. 113

Functions and Their Graphs

E X A MP L E 3

Graphing Using Stretches and Compressions The graph of y = f1x2 is given in Figure 52. Use this graph to find the graphs of (b) y = f 13x2

(a) y = 2f1x2 Figure 52 y 1

( 2 , 1(

( 52 , 1(

y  f(x)

 2

Solution

x

 3 2 5 3 2 2

1

(32 , 1(

(a) The graph of y = 2f1x2 is obtained by multiplying each y-coordinate of y = f1x2 by 2. See Figure 53. (b) The graph of y = f 13x2 is obtained from the graph of y = f 1x2 by multiplying 1 each x-coordinate of y = f1x2 by . See Figure 54. 3 Figure 53

Figure 54 y 3

( 2 , 2(

y

( 52 , 2(

2

2

1

( 6 , 1( ( 56 , 1(

1  2

3 2



2 52 3

x 1 

(

3

(

3 , 2 2

2

PROBLEM

3

x

2 3

( 2 , 1 (

y  2f(x)

Now Work

2



1

y  f(3x)

69 (e)

AND

(g)

3 Graph Functions Using Reflections about the x-Axis or y-Axis Exploration

Reflection about the x-axis: (a) Graph and create a table of Y1 = x 2 and Y2 = -x 2. (b) Graph and create a table of Y1 = 0 x 0 and Y2 = - 0 x 0 .

(c) Graph and create a table of Y1 = x 2 - 4 and Y2 = -(x 2 - 4) = -x 2 + 4. Result See Tables 13(a), (b), and (c) and Figures 55(a), (b), and (c). For each point (x, y) on the graph of Y1 , the point (x, -y) is on the graph of Y2 . Put another way, Y2 is the reflection about the x-axis of Y1 .

Table 13

(a)

114

(b)

(c)

Functions and Their Graphs

Figure 55

Y1  x 2

4

6

Y1  |x |

Y1  |x |

4

6 6

6 6

4

Y2  |x |

Y2  x 2

2 4 Y1  x  4

Y2  |x |

4

6

4 Y  x 2  4 2

(b)

(a)

(c)

The results of the previous Exploration lead to the following result. When the right side of the function y = f1x2 is multiplied by -1, the graph of the new function y = -f1x2 is the reflection about the x-axis of the graph of the function y = f1x2.

Now Work

Exploration

PROBLEM

51

Reflection about the y-axis: (a) Graph Y1 = 1x , followed by Y2 = 1 -x .

(b) Graph Y1 = x + 1, followed by Y2 = -x + 1. (c) Graph Y1 = x 4 + x, followed by Y2 = ( -x)4 + ( -x) = x 4 - x. Result See Tables 14(a), (b), and (c) and Figures 56(a), (b), and (c). For each point (x, y) on the graph of Y1 , the point ( -x, y) is on the graph of Y2 . Put another way, Y2 is the reflection about the y-axis of Y1 .

Table 14

(c)

(b)

(a)

Figure 56 Y1  x 4  x Y2  x

4

6

Y1  x

Y2  x  1 4

6

4 (a)

6

Y1  x  1

2

6

4 (b)

3

Y2  x 4  x

3

2 (c)

The results of the previous Exploration lead to the following result. When the graph of the function y = f1x2 is known, the graph of the new function y = f1 -x2 is the reflection about the y-axis of the graph of the function y = f1x2. 115

Functions and Their Graphs

SUMMARY OF GRAPHING TECHNIQUES To Graph:

Draw the Graph of f and:

Functional Change to f(x)

Raise the graph of f by k units. Lower the graph of f by k units.

Add k to f 1x2. Subtract k from f 1x2.

y = f1x + h2, h 7 0 y = f 1x - h2, h 7 0

Shift the graph of f to the left h units. Shift the graph of f to the right h units.

Replace x by x + h. Replace x by x - h.

y = af1x2, a 7 0

Multiply each y-coordinate of y = f 1x2 by a. Stretch the graph of f vertically if a 7 1. Compress the graph of f vertically if 0 6 a 6 1.

Multiply f 1x2 by a.

y = f1ax2, a 7 0

1 Multiply each x-coordinate of y = f 1x2 by . a Stretch the graph of f horizontally if 0 6 a 6 1. Compress the graph of f horizontally if a 7 1.

Replace x by ax.

Reflect the graph of f about the x-axis.

Multiply f1x2 by -1.

Reflect the graph of f about the y-axis.

Replace x by -x.

Vertical shifts y = f 1x2 + k, k 7 0 y = f1x2 - k, k 7 0 Horizontal shifts

Compressing or stretching

Reflection about the x-axis y = -f1x2 Reflection about the y-axis y = f1 -x2

E X A MP L E 4

Determining the Function Obtained from a Series of Transformations Find the function that is finally graphed after the following three transformations are applied to the graph of y = 0 x 0 . 1. Shift left 2 units. 2. Shift up 3 units. 3. Reflect about the y-axis.

Solution

1. Shift left 2 units: Replace x by x + 2. 2. Shift up 3 units: Add 3. 3. Reflect about the y-axis: Replace x by -x.

Now Work

E X A MP L E 5

PROBLEM

Combining Graphing Procedures Graph the function f1x2 =

Solution

3 + 1 . Find the domain and the range of f . x-2

1 It is helpful to write f as f1x2 = 3a b + 1 . Now use the following steps to x 2 obtain the graph of f: STEP 1: y =

116

27

y = 0x + 20 y = 0x + 20 + 3 y = 0 -x + 2 0 + 3

1 x

3 1 STEP 2: y = 3 # a b = x x

Reciprocal function 1 Multiply by 3; vertical stretch of the graph of y = by a x factor of 3.

Functions and Their Graphs

3 x-2 3 STEP 4: y = +1 x-2 STEP 3: y =

Replace x by x - 2; horizontal shift to the right 2 units. Add 1; vertical shift up 1 unit.

See Figure 57.

Figure 57 y 4

y 4 (1, 1)

(1, 3)

( )

4 x

4

(3, 4)

(3, 3)

3 2, – 2

(2, 12–)

4

y 4

y 4

(4, 3–2 )

4 x

4

(1, 1)

x

(4, 5–2 ) 4 x

4 (1, 2)

(1, 3)

4 1 x

(a) y  ––

Multiply by 3; Vertical stretch (Step 2)

(1, 3) 4

4

4 Replace x by x  2; Horizontal shift right 2 units (Step 3)

3 x

(b) y  ––

3 x –2

(c) y  –––

Add 1; Vertical shift up 1 unit (Step 4)

3 x –2

(d) y  –––  1

1 The domain of y = is {x  x ⬆ 0} and its range is {y  y ⬆ 0} . Because we x shifted right 2 units and up 1 unit to obtain f , the domain of f is {x  x ⬆ 2} and its range is { y  y ⬆ 1}.

HINT Although the order in which transformations are performed can be altered, you may consider using the following order for consistency: 1. Reflections 2. Compressions and stretches 3. Shifts



Other ordering of the steps shown in Example 5 would also result in the graph of f. For example, try this one: STEP 1:

y=

STEP 2:

y=

STEP 3: STEP 4:

E XAM PL E 6

Solution

1 x

1 x-2 3 y= x-2 y=

3 +1 x-2

Reciprocal function Replace x by x - 2; horizontal shift to the right 2 units. Multiply by 3; vertical stretch of the graph 1 of y = by a factor of 3. x -2 Add 1; vertical shift up 1 unit.

Combining Graphing Procedures Graph the function f1x2 = 21 - x + 2 . Find the domain and the range of f .

Because horizontal shifts require the form x - h , begin by rewriting f1x2 as f1x2 = 21 - x + 2 = 2 -(x - 1) + 2. Now use the following steps:

STEP 1: STEP 2: STEP 3: STEP 4:

y = 1x

Square root function

y = 2 -(x - 1) = 21 - x

Replace x by x - 1 ; horizontal shift to the right 1 unit. Add 2; vertical shift up 2 units.

y = 2 -x

y = 21 - x + 2

Replace x by - x ; reflect about the y-axis.

117

Functions and Their Graphs

See Figure 58. Figure 58

(1, 1) 5

y 5

y 5

y 5 (4, 2)

(0, 3)

(3, 2)

(1, 2)

(0, 1) 5 x 5

(0, 0) (a) y 

(4, 2)

y (3, 4) 5

(1, 1) (0, 0)

5 x 5

(b) y  x

x Replace x by x; Reflect about y-axis (Step 2)

(1, 0)

Replace x by x  1; (c) y  Horizontal shift  right 1 unit  (Step 3)

5 x 5

5 x

(x  1) Add 2; (d) y  x  1 Vertical shift up 2 units 1x (Step 4)

1x2

The domain of f is ( - , 1] and its range is [2, ).

Now Work

PROBLEM

61

5 Assess Your Understanding Concepts and Vocabulary 4. True or False The graph of y = - f 1x2 is the reflection about the x-axis of the graph of y = f 1x2.

1. Suppose that the graph of a function f is known. Then the graph of y = f 1x - 22 may be obtained by a(n) shift of the graph of f to the ________ a distance of 2 units.

5. True or False To obtain the graph of f 1x2 = 2x + 2 , shift the graph of y = 1x horizontally to the right 2 units.

2. Suppose that the graph of a function f is known. Then the graph of y = f 1 - x2 may be obtained by a reflection about the ________-axis of the graph of the function y = f 1x2.

6. True or False To obtain the graph of f 1x2 = x 3 + 5, shift the graph of y = x 3 vertically up 5 units.

3. Suppose that the graph of a function g is known. The graph shift of the of y = g(x) + 2 may be obtained by a graph of g ________ a distance of 2 units.

Skill Building In Problems 7–18, match each graph to one of the following functions: A. y = x 2 + 2 E. y = 1x - 22

C. y = 0 x 0 + 2

B. y = - x 2 + 2

I. y = 2x 2

8.

10.

11.

3x

3 x

3

12.

14.

y 5

y 8

y 3

3 x

3 x

3

13.

y 3

3

y 3

y 1 3

3 x

L. y = - 2 0 x 0

9.

y 3

y 3

3

H. y = - 0 x + 2 0

K. y = 2 0 x 0

J. y = - 2x 2

7.

D. y = - 0 x 0 + 2

G. y = 0 x - 2 0

F. y = - 1x + 22 2

2

3 x

3

6 x

6 3

118

3

1

3 x

3

4

Functions and Their Graphs

15.

16.

17.

y 4

3 x

3

4

y 3

y 4

y 3

4 x

4

18.

4 x

4

4

3

3 x

3

3

In Problems 19–26, write the function whose graph is the graph of y = x 3 , but is: 19. Shifted to the right 4 units

20. Shifted to the left 4 units

21. Shifted up 4 units

22. Shifted down 4 units

25. Vertically stretched by a factor of 4

26. Horizontally stretched by a factor of 4

In Problems 27–30, find the function that is finally graphed after each of the following transformations is applied to the graph of y = 1x in the order stated. 27. (1) Shift up 2 units

28. (1) Reflect about the x-axis

(2) Shift right 3 units

(3) Shift down 2 units 30. (1) Shift up 2 units

29. (1) Reflect about the x-axis (2) Shift up 2 units

(3) Shift left 3 units

(3) Shift left 3 units

31. If 13, 62 is a point on the graph of y = f 1x2, which of the following points must be on the graph of y = - f 1x2? (a) 16, 32 (b) 16, - 32 (c) 13, - 62 (d) 1 - 3, 62

32. If 13, 62 is a point on the graph of y = f 1x2, which of the following points must be on the graph of y = f 1 - x2? (a) 16, 32 (b) 16, - 32 (c) 13, - 62 (d) 1 - 3, 62

35. Suppose that the x-intercepts of the graph of y = f 1x2 are - 5 and 3. (a) What are the x-intercepts of the graph of y = f 1x + 22? (b) What are the x-intercepts of the graph of y = f 1x - 22? (c) What are the x-intercepts of the graph of y = 4f 1x2? (d) What are the x-intercepts of the graph of y = f 1 - x2?

36. Suppose that the x-intercepts of the graph of y = f 1x2 are - 8 and 1. (a) What are the x-intercepts of the graph of y = f 1x + 42? (b) What are the x-intercepts of the graph of y = f 1x - 32? (c) What are the x-intercepts of the graph of y = 2f 1x2? (d) What are the x-intercepts of the graph of y = f 1 - x2?

33. If 11, 32 is a point on the graph of y = f 1x2, which of the following points must be on the graph of y = 2f 1x2? 3 (a) a1, b (b) 12, 32 2 1 (c) 11, 62 (d) a , 3b 2

37. Suppose that the function y = f 1x2 is increasing on the interval 1 - 1, 52. (a) Over what interval is the graph of y = f 1x + 22 increasing? (b) Over what interval is the graph of y = f 1x - 52 increasing? (c) What can be said about the graph of y = - f 1x2? (d) What can be said about the graph of y = f 1 - x2?

34. If 14, 22 is a point on the graph of y = f 1x2, which of the following points must be on the graph of y = f 12x2 ? (a) 14, 12 (b) 18, 22 (c) 12, 22 (d) 14, 42

38. Suppose that the function y = f 1x2 is decreasing on the interval 1 - 2, 72. (a) Over what interval is the graph of y = f 1x + 22 decreasing? (b) Over what interval is the graph of y = f 1x - 52 decreasing? (c) What can be said about the graph of y = - f 1x2? (d) What can be said about the graph of y = f 1 - x2?

In Problems 39–68, graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, y = x 2 ) and show all stages. Be sure to show at least three key points. Find the domain and the range of each function.Verify your results using a graphing utility. 39. f 1x2 = x 2 - 1

42. g 1x2 = x 3 - 1

40. f 1x2 = x 2 + 4

43. h1x2 = 2x - 2

41. g 1x2 = x 3 + 1

44. h1x2 = 2x + 1

119

Functions and Their Graphs

45. f 1x2 = 1x - 12 3 + 2 1 48. g 1x2 = 1x 2 3 51. f 1x2 = - 2 x

46. f 1x2 = 1x + 22 3 - 3 1 49. h1x2 = 2x 52. f 1x2 = - 1x

47. g (x) = 4 1x

57. f 1x2 = 2 1x + 12 2 - 3

58. f 1x2 = 3(x - 2)2 + 1

59. g 1x2 = 2 1x - 2 + 1

63. f 1x2 = - 1x + 12 3 - 1

64. f 1x2 = - 4 1x - 1

54. g 1x2 =

1 -x

3 50. h1x2 = 1 2x

3 53. g (x) = 2 -x

55. h1x2 = - x 3 + 2

60. g 1x2 = 3|x + 1| - 3

56. h1x2 =

4 + 2 x 65. g (x) = 2|1 - x|

61. h1x2 = 1 -x - 2

66. g(x) = 4 12 - x

1 + 2 -x

62. h1x2 =

68. h1x2 = int 1 - x2

67. h(x) = 2 int (x - 1)

In Problems 69–72, the graph of a function f is illustrated. Use the graph of f as the first step toward graphing each of the following functions: (a) F1x2 = f 1x2 + 3 1 (e) Q1x2 = f 1x2 2 69. y 4 (0, 2)

(b) G1x2 = f 1x + 22 (f) g 1x2 = f 1 - x2

(c) P 1x2 = - f 1x2

(d) H1x2 = f 1x + 12 - 2

(g) h1x2 = f 12x2 70.

y 4

(2, 2)

2

(2, 2)

(4, 0) 4 (4, 2)

71.

4 2 (4, 2)

1 π –2

2

2 (2, 2)

2

y

x

2

72.

1 π ( –2 , 1)

2

(4, 2)

y

(π–2 , 1) π–

4 x

1

π

x

π –2

(π, 1)

π– 1

2

π

x

(π, 1)

Mixed Practice 73. (a) Using a graphing utility, graph f 1x2 = x 3 - 9x for - 4 6 x 6 4. (b) Find the x-intercepts of the graph of f. (c) Approximate any local maxima and local minima. (d) Determine where f is increasing and where it is decreasing. (e) Without using a graphing utility, repeat parts (b) – (d) for y = f 1x + 22 . (f) Without using a graphing utility, repeat parts (b) – (d) for y = 2f 1x2 . (g) Without using a graphing utility, repeat parts (b) – (d) for y = f 1 - x2 .

74. (a) Using a graphing utility, graph f(x) = x 3 - 4x for - 3 6 x 6 3. (b) Find the x-intercepts of the graph of f. (c) Approximate any local maxima and local minima. (d) Determine where f is increasing and where it is decreasing. (e) Without using a graphing utility, repeat parts (b) – (d) for y = f 1x - 42 . (f) Without using a graphing utility, repeat parts (b) – (d) for y = f 12x2 . (g) Without using a graphing utility, repeat parts (b) – (d) for y = - f 1x2 .

In Problems 75–82, complete the square of each quadratic expression. Then graph each function using the technique of shifting. 75. f 1x2 = x 2 + 2x

79. f 1x2 = 2x - 12x + 19 2

76. f 1x2 = x 2 - 6x

80. f 1x2 = 3x + 6x + 1 2

77. f 1x2 = x 2 - 8x + 1

81. f 1x2 = - 3x - 12x - 17 2

78. f 1x2 = x 2 + 4x + 2

82. f 1x2 = - 2x 2 - 12x - 13

Applications and Extensions 83. The equation y = 1x - c2 2 defines a family of parabolas, one parabola for each value of c. On one set of coordinate

120

axes, graph the members of the family for c = 0, c = 3, and c = - 2.

Functions and Their Graphs

84. Repeat Problem 83 for the family of parabolas y = x 2 + c. 85. Thermostat Control Energy conservation experts estimate that homeowners can save 5% to 10% on winter heating bills by programming their thermostats 5 to 10 degrees lower while sleeping. In the given graph, the temperature T (in degrees Fahrenheit) of a home is given as a function of time t (in hours after midnight) over a 24-hour period.

88. Period of a Pendulum The period T (in seconds) of a simple pendulum is a function of its length l (in feet) defined by the equation l T = 2p Ag where g  32.2 feet per second per second is the acceleration of gravity.

T

76 72 68 64 60

Kg Kua/Dreamstime

Temperature (°F )

80

56 0

t 4 8 12 16 20 24 Time (hours after midnight)

(a) At what temperature is the thermostat set during daytime hours? At what temperature is the thermostat set overnight? (b) The homeowner reprograms the thermostat to y = T 1t2 - 2. Explain how this affects the temperature in the house. Graph this new function. (c) The homeowner reprograms the thermostat to y = T 1t + 12. Explain how this affects the temperature in the house. Graph this new function.

Source: Roger Albright, 547 Ways to Be Fuel Smart, 2000

86. Digital Music Revenues The total projected worldwide digital music revenues R, in millions of dollars, for the years 2005 through 2010 can be estimated by the function R 1x2 = 170.7x 2 + 1373x + 1080

where x is the number of years after 2005. (a) Find R 102, R 132, and R 152 and explain what each value represents. (b) Find r = R 1x - 52. (c) Find r 152, r 182, and r 1102 and explain what each value represents. (d) In the model r, what does x represent? (e) Would there be an advantage in using the model r when estimating the projected revenues for a given year instead of the model R? Source: eMarketer.com, May 2006 87. Temperature Measurements The relationship between the Celsius (°C) and Fahrenheit (°F) scales for measuring temperature is given by the equation F =

9 C + 32 5

The relationship between the Celsius (°C) and Kelvin (K) 9 scales is K = C + 273. Graph the equation F = C + 32 5 using degrees Fahrenheit on the y-axis and degrees Celsius on the x-axis. Use the techniques introduced in this section to obtain the graph showing the relationship between Kelvin and Fahrenheit temperatures.

(a) Use a graphing utility to graph the function T = T 1l2. (b) Now graph the functions T = T 1l + 12, T = T 1l + 22, and T = T 1l + 32. (c) Discuss how adding to the length l changes the period T. (d) Now graph the functions T = T 12l2, T = T 13l2, and T = T 14l2. (e) Discuss how multiplying the length l by factors of 2, 3, and 4 changes the period T. 89. Cigar Company Profits The daily profits of a cigar company from selling x cigars are given by p1x2 = - 0.05x 2 + 100x - 2000 The government wishes to impose a tax on cigars (sometimes called a sin tax) that gives the company the option of either paying a flat tax of \$10,000 per day or a tax of 10% on profits. As chief financial officer (CFO) of the company, you need to decide which tax is the better option for the company. (a) On the same screen, graph Y1 = p1x2 - 10,000 and Y2 = 11 - 0.102p1x2. (b) Based on the graph, which option would you select? Why? (c) Using the terminology learned in this section, describe each graph in terms of the graph of p1x2. (d) Suppose that the government offered the options of a flat tax of \$4800 or a tax of 10% on profits. Which would you select? Why? 90. The graph of a function f is illustrated in the figure. (a) Draw the graph of y = 0 f 1x2 0 . (b) Draw the graph of y = f 1 0 x 0 2. y 2

(1, 1) 3 (2, 1)

(2, 0) 3 x (1, 1) 2

121

Functions and Their Graphs

91. The graph of a function f is illustrated in the figure. (a) Draw the graph of y = 0 f 1x2 0 . (b) Draw the graph of y = f 1 0 x 0 2. y 2

(2, 0)

(1, 1) (2, 0) 3 x

3 (1, 1) 2

92. Suppose 1 1, 3 2 is a point on the graph of y = f 1x2 . (a) What point is on the graph of y = f 1x + 32 - 5? (b) What point is on the graph of y = -2f 1x - 22 + 1? (c) What point is on the graph of y = f 12x + 32?

93. Suppose 1 - 3, 52 is a point on the graph of y = g 1x2 . (a) What point is on the graph of y = g 1x + 12 - 3? (b) What point is on the graph of y = - 3g 1x - 42 + 3? (c) What point is on the graph of y = g 13x + 92?

Explaining Concepts: Discussion and Writing 94. Suppose that the graph of a function f is known. Explain how the graph of y = 4f 1x2 differs from the graph of y = f 14x2.

95. Suppose that the graph of a function f is known. Explain how the graph of y = f 1x2 - 2 differs from the graph of y = f 1x - 22. 96. The area under the curve y = 1x bounded below by the 16 x-axis and on the right by x = 4 is square units. Using the 3

ideas presented in this section, what do you think is the area under the curve of y = 1 - x bounded below by the x-axis and on the left by x = - 4? Justify your answer. 97. Explain how the range of a function y = f 1x2 = x 2 compares to y = g 1x2 = f 1x2 + k. 98. Explain how the domain of y = g(x) = 1x compares to y = g 1x - k2, k Ú 0.

Interactive Exercises: Exploring Transformations Ask your instructor if the applets below are of interest to you. 99. Vertical Shifts Open the vertical shift applet. Use your mouse to grab the slider and change the value of k. Note the role k plays in the graph of g 1x2 = f 1x2 + k, where f 1x2 = x 2.

100. Horizontal Shifts Open the horizontal shift applet. Use your mouse to grab the slider and change the value of h. Note the role h plays in the graph of g 1x2 = f 1x - h2, where f 1x2 = x 2.

101. Vertical Stretches Open the vertical stretch applet. Use your mouse to grab the slider and change the value of a. Note the role a plays in the graph of g 1x2 = af 1x2, where f 1x2 =  x  .

102. Horizontal Stretches Open the horizontal stretch applet. (a) Use your mouse to grab the slider and change the value of a. Note the role a plays in the graph of g 1x2 = f 1ax2 = 2ax, where f 1x2 = 2x. What happens to the points on the graph of g when 0 6 a 6 1? What happens to the points on the graph when a 7 1? (b) To further understand the concept of horizontal compressions, fill in the spreadsheet to the right of the graph as follows: (i) What x-coordinate is required on the graph of g 1x2 = 22x , if the y-coordinate is to be 1?

(ii) What x-coordinate is required on the graph g 1x2 = 22x , if the y-coordinate is to be 2? (iii) What x-coordinate is required on the graph g 1x2 = 22x , if the y-coordinate is to be 3? (iv) What x-coordinate is required on the graph 1 g 1x2 = x , if the y-coordinate is to be 1? A2 (v) What x-coordinate is required on the graph 1 g 1x2 = x , if the y-coordinate is to be 2? A2

of of of

of

(vi) What x-coordinate is required on the graph of 1 g 1x2 = x , if the y-coordinate is to be 3? A2

103. Reflection about the y-axis Open the reflection about the yaxis applet. Move your mouse to grab the slider and change the value of a from 1 to - 1 . 104. Reflection about the x-axis Open the reflection about the xaxis applet. Move your mouse to grab the slider and change the value of a from 1 to - 1 .

6 Mathematical Models: Building Functions OBJECTIVE 1 Build and Analyze Functions

1 Build and Analyze Functions Real-world problems often result in mathematical models that involve functions. These functions need to be constructed or built based on the information given. In building functions, we must be able to translate the verbal description into the 122

Functions and Their Graphs

language of mathematics. We do this by assigning symbols to represent the independent and dependent variables and then by finding the function or rule that relates these variables.

E XAM PL E 1

Finding the Distance from the Origin to a Point on a Graph Let P = 1x, y2 be a point on the graph of y = x2 - 1.

(a) Express the distance d from P to the origin O as a function of x. (b) What is d if x = 0? (c) What is d if x = 1? 22 ? (d) What is d if x = 2 (e) Use a graphing utility to graph the function d = d 1x2, x Ú 0. Rounded to two decimal places, find the value(s) of x at which d has a local minimum. [This gives the point(s) on the graph of y = x2 - 1 closest to the origin.]

Figure 59

Solution

(a) Figure 59 illustrates the graph of y = x2 - 1. The distance d from P to O is

y

d = 21x - 02 2 + 1y - 02 2 = 2x2 + y 2

2

Since P is a point on the graph of y = x2 - 1, substitute x2 - 1 for y. Then

y  x2  1 1 P (x, y) (0, 0) d 1 1 2 x

d 1x2 = 2x2 + 1x2 - 12 2 = 2x4 - x2 + 1

1

The distance d is expressed as a function of x. (b) If x = 0, the distance d is

d 102 = 204 - 02 + 1 = 21 = 1

(c) If x = 1, the distance d is

(d) If x =

d 112 = 214 - 12 + 1 = 1

22 , the distance d is 2

Figure 60

da

2

2

0 0

22 1 1 12 4 22 2 13 b = b - a b +1 = - +1= a 2 2 2 B 2 B4 2

(e) Figure 60 shows the graph of Y1 = 2x4 - x2 + 1. Using the MINIMUM feature on a graphing utility, we find that when x  0.71 the value of d is smallest. The local minimum is d  0.87 rounded to two decimal places. Since d 1x2 is even, by symmetry, it follows that when x  -0.71 the value of d is also a local minimum. Since 1{ 0.712 2 - 1  -0.50 , the points 1 -0.71, -0.502 and 10.71, -0.502 on the graph of y = x2 - 1 are closest to the origin.

Now Work

E XAMPL E 2

PROBLEM

1

Area of a Rectangle A rectangle has one corner in quadrant I on the graph of y = 25 - x2, another at the origin, a third on the positive y-axis, and the fourth on the positive x-axis. See Figure 61 on the next page. (a) Express the area A of the rectangle as a function of x. (b) What is the domain of A? (c) Graph A = A1x2. (d) For what value of x is the area largest? 123

Functions and Their Graphs

Figure 61

Solution

y 30 (x, y)

20

y  25  x 2

10 1

1 2 3 4 5

x

(0,0)

(a) The area A of the rectangle is A = xy, where y = 25 - x 2. Substituting this expression for y, we obtain A1x2 = x125 - x2 2 = 25x - x3. (b) Since 1x, y2 is in quadrant I, we have x 7 0. Also, y = 25 - x2 7 0, which implies that x2 6 25, so -5 6 x 6 5. Combining these restrictions, we have the domain of A as 5 x  0 6 x 6 5 6 , or 10, 52 using interval notation. (c) See Figure 62 for the graph of A = A1x2. (d) Using MAXIMUM, we find that the maximum area is 48.11 square units at x = 2.89 units, each rounded to two decimal places. See Figure 63. Figure 62

Figure 63 50

50

0

0

5

5 0

0

Now Work

PROBLEM

7

Close Call?

E X A MP L E 3

Solution

Suppose two planes flying at the same altitude are headed toward each other. One plane is flying due South at a groundspeed of 400 miles per hour and is 600 miles from the potential intersection point of the planes. The other plane is flying due West with a groundspeed of 250 miles per hour and is 400 miles from the potential intersection point of the planes. See Figure 64. (a) Build a model that expresses the distance d between the planes as a function of time t. (b) Use a graphing utility to graph d = d 1t2. How close do the planes come to each other? At what time are the planes closest? (a) Refer to Figure 64. The distance d between the two planes is the hypotenuse of a right triangle. At any time t the length of the North/South leg of the triangle is 600 - 400t. At any time t, the length of the East/West leg of the triangle is 400 - 250t. Using the Pythagorean Theorem, the square of the distance between the two planes is d 2 = 1600 - 400t2 2 + 1400 - 250t2 2 Therefore, the distance between the two planes as a function of time is given by the model d 1t2 = 21600 - 400t2 2 + 1400 - 250t2 2

(b) Figure 65(a) shows the graph of d = d 1t2 . Using MINIMUM, the minimum distance between the planes is 21.20 miles and the time at which the planes are closest is after 1.53 hours, each rounded to two decimal places. See Figure 65(b). Figure 64

Figure 65

N Plane

600 miles

500 400 mph

d Plane 250 mph 400 miles

124

0 E

2

50 (a)

(b)

Functions and Their Graphs

6 Assess Your Understanding Applications and Extensions 1. Let P = 1x, y2 be a point on the graph of y = x 2 - 8. (a) Express the distance d from P to the origin as a function of x. (b) What is d if x = 0? (c) What is d if x = 1? (d) Use a graphing utility to graph d = d1x2. (e) For what values of x is d smallest? 2. Let P = 1x, y2 be a point on the graph of y = x - 8.

y y  16  x 2

16

(x, y) 8

(0,0)

4

x

2

(a) Express the distance d from P to the point 10, - 12 as a function of x. (b) What is d if x = 0? (c) What is d if x = - 1? (d) Use a graphing utility to graph d = d1x2. (e) For what values of x is d smallest?

3. Let P = 1x, y2 be a point on the graph of y = 1x .

(a) Express the distance d from P to the point 11, 02 as a function of x. (b) Use a graphing utility to graph d = d1x2. (c) For what values of x is d smallest? 1 4. Let P = 1x, y2 be a point on the graph of y = . x (a) Express the distance d from P to the origin as a function of x. (b) Use a graphing utility to graph d = d1x2. (c) For what values of x is d smallest? 5. A right triangle has one vertex on the graph of y = x 3, x 7 0, at 1x, y2, another at the origin, and the third on the positive y-axis at 10, y2, as shown in the figure. Express the area A

8. A rectangle is inscribed in a semicircle of radius 2. See the figure. Let P = 1x, y2 be the point in quadrant I that is a vertex of the rectangle and is on the circle. y y  4  x2

2

yx

x

2

(a) Express the area A of the rectangle as a function of x. (b) Express the perimeter p of the rectangle as a function of x. (c) Graph A = A1x2. For what value of x is A largest? (d) Graph p = p 1x2. For what value of x is p largest?

9. A rectangle is inscribed in a circle of radius 2. See the figure. Let P = 1x, y2 be the point in quadrant I that is a vertex of the rectangle and is on the circle. y

2

of the triangle as a function of x. y

P  (x, y )

P  (x, y)

3

2

2

x

2 (0, y)

(x, y)

(0, 0)

2

2 x y 4

x

6. A right triangle has one vertex on the graph of y = 9 - x 2, x 7 0, at 1x, y2, another at the origin, and the third on the positive x-axis at 1x, 02. Express the area A of the triangle as a function of x.

7. A rectangle has one corner in quadrant I on the graph of y = 16 - x 2, another at the origin, a third on the positive y-axis, and the fourth on the positive x-axis. See the figure. (a) Express the area A of the rectangle as a function of x. (b) What is the domain of A? (c) Graph A = A 1x2. For what value of x is A largest?

(a) Express the area A of the rectangle as a function of x. (b) Express the perimeter p of the rectangle as a function of x. (c) Graph A = A 1x2. For what value of x is A largest? (d) Graph p = p1x2. For what value of x is p largest?

10. A circle of radius r is inscribed in a square. See the figure. r

(a) Express the area A of the square as a function of the radius r of the circle. (b) Express the perimeter p of the square as a function of r.

125

Functions and Their Graphs

11. Geometry A wire 10 meters long is to be cut into two pieces. One piece will be shaped as a square, and the other piece will be shaped as a circle. See the figure. x 4x 10 m

18. Uniform Motion Two cars leave an intersection at the same time. One is headed south at a constant speed of 30 miles per hour, and the other is headed west at a constant speed of 40 miles per hour (see the figure). Build a model that expresses the distance d between the cars as a function of the time t. [Hint: At t = 0, the cars leave the intersection.]

10  4x

N W S

(a) Express the total area A enclosed by the pieces of wire as a function of the length x of a side of the square. (b) What is the domain of A? (c) Graph A = A 1x2. For what value of x is A smallest?

12. Geometry A wire 10 meters long is to be cut into two pieces. One piece will be shaped as an equilateral triangle, and the other piece will be shaped as a circle. (a) Express the total area A enclosed by the pieces of wire as a function of the length x of a side of the equilateral triangle. (b) What is the domain of A? (c) Graph A = A 1x2. For what value of x is A smallest? 13. A wire of length x is bent into the shape of a circle. (a) Express the circumference C of the circle as a function of x. (b) Express the area A of the circle as a function of x. 14. A wire of length x is bent into the shape of a square. (a) Express the perimeter p of the square as a function of x. (b) Express the area A of the square as a function of x. 15. Geometry A semicircle of radius r is inscribed in a rectangle so that the diameter of the semicircle is the length of the rectangle. See the figure. r

(a) Express the area A of the rectangle as a function of the radius r of the semicircle. (b) Express the perimeter p of the rectangle as a function of r. 16. Geometry An equilateral triangle is inscribed in a circle of radius r. See the figure. Express the circumference C of the circle as a function of the length x of a side of the triangle. x2 [Hint: First show that r 2 = . ] 3

E

d

19. Uniform Motion Two cars are approaching an intersection. One is 2 miles south of the intersection and is moving at a constant speed of 30 miles per hour. At the same time, the other car is 3 miles east of the intersection and is moving at a constant speed of 40 miles per hour. (a) Build a model that expresses the distance d between the cars as a function of time t. [Hint: At t = 0, the cars are 2 miles south and 3 miles east of the intersection, respectively.] (b) Use a graphing utility to graph d = d1t2. For what value of t is d smallest? 20. Inscribing a Cylinder in a Sphere Inscribe a right circular cylinder of height h and radius r in a sphere of fixed radius R. See the illustration. Express the volume V of the cylinder as a function of h. [Hint: V = pr 2 h. Note also the right triangle.] r

R

h

Sphere x

x r x

17. Geometry An equilateral triangle is inscribed in a circle of radius r. See the figure in Problem 16. Express the area A within the circle, but outside the triangle, as a function of the length x of a side of the triangle.

126

21. Inscribing a Cylinder in a Cone Inscribe a right circular cylinder of height h and radius r in a cone of fixed radius R and fixed height H. See the illustration. Express the volume V of the cylinder as a function of r. [Hint: V = pr 2 h. Note also the similar triangles.]

Functions and Their Graphs

per hour, build a model that expresses the time T that it takes to go from the island to town as a function of the distance x from P to where the person lands the boat. (b) What is the domain of T  ? (c) How long will it take to travel from the island to town if the person lands the boat 4 miles from P?  (d) How long will it take if the person lands the boat 8 miles from P ?

r

H h

R Cone

22. Installing Cable TV  MetroMedia Cable is asked to provide service to a customer whose house is located 2 miles from the road along which the cable is buried. The nearest connection box for the cable is located 5 miles down the road. See the figure.

24. Filling a Conical Tank  Water is poured into a container in the shape of a right circular cone with radius 4 feet and height 16 feet. See the figure. Express the volume V of the water in the cone as a function of the height h of the water. [Hint: The volume V of a cone of radius r and height h is 1 V = pr 2 h.] 3

House 4

Stream 2 mi

Box 5 mi

x

(a) If the installation cost is \$500 per mile along the road and \$700 per mile off the road, build a model that expresses the total cost C of installation as a function of the distance x (in miles) from the connection box to the point where the cable installation turns off the road. Give the domain. (b) Compute the cost if x = 1 mile. (c) Compute the cost if x = 3 miles. (d) Graph the function C = C 1x2. Use TRACE to see how the cost C varies as x changes from 0 to 5. (e) What value of x results in the least cost? 23. Time Required to Go from an Island to a Town  An island is 2 miles from the nearest point P on a straight shoreline. A town is 12 miles down the shore from P. See the illustration.

d2 Town

P

2 mi

x d1

16

12  x 12 mi

Island

(a) If a person can row a boat at an average speed of 3 miles per hour and the same person can walk 5 miles

r h

25. Constructing an Open Box  An open box with a square base is to be made from a square piece of cardboard 24 inches on a side by cutting out a square from each corner and turning up the sides. See the figure. x

x

x

x 24 in.

x

x x

24 in.

x

(a) Express the volume V of the box as a function of the length x of the side of the square cut from each corner. (b) What is the volume if a 3-inch square is cut out?  (c) What is the volume if a 10-inch square is cut out? (d) Graph V = V 1x2. For what value of x is V largest?

26. Constructing an Open Box  An open box with a square base is required to have a volume of 10 cubic feet. (a) Express the amount A of material used to make such a box as a function of the length x of a side of the square base. (b) How much material is required for a base 1 foot by 1 foot? (c) How much material is required for a base 2 feet by 2 feet? (d) Use a graphing utility to graph A = A 1x2. For what value of x is A smallest?

127

Functions and Their Graphs

CHAPTER REVIEW Library of Functions Constant function f 1x2 = b The graph is a horizontal line with y-intercept b.

Identity function f 1x2 = x The graph is a line with slope 1 and y-intercept 0.

Square function f 1x2 = x 2 The graph is a parabola with intercept at (0, 0). y

y 3

y f (x) = b

( – 2, 4)

(2, 4)

4

(0,b) (1, 1) x

(0, 0)

–3 ( – 1, – 1)

(– 1, 1)

3 x

Square root function f 1x2 = 1x

Cube root function 3 f 1x2 = 1 x

y

y 4

2

(1, 1)

(0, 0)

4 (1, 1)

4

x

y 3

(4, 2)

(1, 1) 1

4 x

(0, 0)

–4

Cube function f 1x2 = x 3

(1, 1)

3

(

 1–8 , 1–2

5 x

(0, 0)

(1, 1)

)

(2, 2 )

( 1–8 , 1–2) 3 x

3 (0, 0) 3

4

(1, 1)

(2, 2 )

3

Reciprocal function 1 f 1x2 = x

Absolute value function f 1x2 = 0 x 0

Greatest integer function f 1x2 = int 1x2

y

y

y 4

3

2

(2, 2)

(2, 2) (1, 1)

(1, 1) 2 x

2

3

(0, 0)

2

(1, 1) 3 x

2

(1, 1)

2

4

x

3 2

Things to Know Function

A relation between two sets so that each element x in the first set, the domain, has corresponding to it exactly one element y in the second set. The range is the set of y values of the function for the x values in the domain. A function can also be characterized as a set of ordered pairs (x, y) in which no first element is paired with two different second elements.

Function notation

y = f 1x2

f is a symbol for the function.

x is the argument, or independent variable. y is the dependent variable. f 1x2 is the value of the function at x, or the image of x.

Difference quotient of f

128

A function f may be defined implicitly by an equation involving x and y or explicitly by writing y = f 1x2. f 1x + h2 - f 1x2 h

h⬆0

Functions and Their Graphs

Domain

If unspecified, the domain of a function f defined by an equation is the largest set of real numbers for which f 1x2 is a real number.

Vertical-line test

A set of points in the plane is the graph of a function if and only if every vertical line intersects the graph in at most one point.

Even function f

f 1 -x2 = f 1x2 for every x in the domain ( -x must also be in the domain).

Odd function f Increasing function Decreasing function Constant function Local maximum

f 1 - x2 = -f 1x2 for every x in the domain ( -x must also be in the domain).

A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 6 x2 , we have f 1x1 2 6 f 1x2 2.

A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 6 x2 , we have f 1x1 2 7 f 1x2 2.

A function f is constant on an open interval I if, for all choices of x in I, the values of f 1x2 are equal.

A function f has a local maximum at c if there is an open interval I containing c so that, for all x in I, f 1x2 … f 1c2.

Local minimum Absolute maximum and Absolute minimum

Average rate of change of a function

A function f has a local minimum at c if there is an open interval I containing c so that, for all x in I, f 1x2 Ú f 1c2.

Let f denote a function defined on some interval I. If there is a number u in I for which f 1x2 … f 1u2 for all x in I, then f 1u2 is the absolute maximum of f on I and we say the absolute maximum of f occurs at u. If there is a number v in I for which f 1x2 Ú f 1v2 , for all x in I, then f 1v2 is the absolute minimum of f on I and we say the absolute minimum of f occurs at v . The average rate of change of f from a to b is y x

=

f 1b2 - f 1a2 b-a

a ⬆b

Objectives Section 1

You should be able to . . . 1 2 3 4

2

1 2

3

Identify the graph of a function Obtain information from or about the graph of a function

1 2–4

27, 28 16(a) – (e), 17(a), 17(e), 17(g)

22, 23, 40(d), 41(b) 24–26

2

Graph the functions listed in the library of functions Graph piecewise-defined functions

1, 2 3, 4

29, 30 37, 38

1

Graph functions using vertical and horizontal shifts

1, 2, 5, 6

2 3

Graph functions using compressions and stretches Graph functions using reflections about the x-axis or y-axis

1

Build and analyze functions

4 5 6 7

6

1, 2 3–5, 15, 39 6–11 12–14

6 7, 8

3

5

Review Exercises

1–5 6, 7 8, 9 10

Determine even and odd functions from a graph Identify even and odd functions from the equation Use a graph to determine where a function is increasing, decreasing, or constant Use a graph to locate local maxima and local minima Use a graph to locate the absolute maximum and the absolute minimum Use a graphing utility to approximate local maxima and local minima and to determine where a function is increasing or decreasing Find the average rate of change of a function

1 2

4

Examples

Determine whether a relation represents a function Find the value of a function Find the domain of a function defined by an equation Form the sum, difference, product, and quotient of two functions

1

1 2

17(f) 18 – 21

3 4

17(b) 17(c)

5

17(d)

3, 5, 6 6 1–3

16(f), 31, 33, 34, 35, 36 16(g), 32, 36 16(h), 32, 34, 36 40, 41

129

Functions and Their Graphs

Review Exercises In Problems 1 and 2, determine whether each relation represents a function. For each function, state the domain and range. 1. 5 ( - 1, 0), (2, 3), (4, 0) 6

2. 5 (4, - 1), (2, 1), (4, 2) 6

In Problems 3–5, find the following for each function: (a) f 122

3. f 1x2 =

(b) f 1 - 22

3x x2 - 1

(c) f 1 - x2

(d) - f 1x2

4. f 1x2 = 2x 2 - 4

In Problems 6–11, find the domain of each function. x 6. f 1x2 = 2 7. f 1x2 = 22 - x x - 9 x 1x + 1 9. f 1x2 = 2 10. f 1x2 = 2 x + 2x - 3 x - 4

(e) f 1x - 22 5. f 1x2 =

8. g(x) = 11. g(x) =

2

x - 4 x2

(f) f 12x2

0x0 x

x 1x + 8

f for each pair of functions. State the domain of each of these functions. g x + 1 1 12. f 1x2 = 2 - x; g(x) = 3x + 1 13. f 1x2 = 3x 2 + x + 1; g(x) = 3x 14. f 1x2 = ; g(x) = x - 1 x f 1x + h2 f 1x2 15. Find the difference quotient of f 1x2 = - 2x 2 + x + 1; that is, find , h ⬆ 0. h

In Problems 12–14, find f + g, f - g, f # g, and

16. Use the graph of the function f shown to find: (a) (b) (c) (d) (e) (f)

Find the domain and the range of f. List the intercepts. Find f 1 - 22. Find the value(s) of x for which f 1x2 = - 3. Solve f 1x2 7 0. Graph y = f 1x - 32. 1 (g) Graph y = f a x b. 2 (h) Graph y = - f 1x2.

17. Use the graph of the function f shown to find: (a) The domain and the range of f . (b) The intervals on which f is increasing, decreasing, or constant. (c) The local minimum values and local maximum values. (d) The absolute maximum and absolute minimum. (e) Whether the graph is symmetric with respect to the x-axis, the y-axis, or the origin. (f) Whether the function is even, odd, or neither. (g) The intercepts, if any. y 4

y

4

(2, 1)

(3, 3)

6 (4,3) (3, 0) 5 (2, 1) (4, 3)

(0, 0)

5

x

(3, 0)

(4, 3)

6 x (2, 1)

4

4

In Problems 18–21, determine (algebraically) whether the given function is even, odd, or neither. 4 + x2 19. g(x) = 20. G(x) = 1 - x + x 3 18. f 1x2 = x 3 - 4x 1 + x4

21.

f 1x2 =

x 1 + x2

In Problems 22 and 23, use a graphing utility to graph each function over the indicated interval. Approximate any local maximum values and local minimum values. Determine where the function is increasing and where it is decreasing. 22. f 1x2 = 2x 3 - 5x + 1

( - 3, 3)

24. Find the average rate of change of f 1x2 = 8x - x. (a) From 1 to 2 (b) From 0 to 1 2

23. f 1x2 = 2x 4 - 5x 3 + 2x + 1 (c) From 2 to 4

In Problems 25 and 26, find the average rate of change from 2 to 3 for each function f. Be sure to simplify. 25. f 1x2 = 2 - 5x

130

26. f 1x2 = 3x - 4x 2

( - 2, 3)

Functions and Their Graphs

In Problems 27 and 28, is the graph shown the graph of a function? 27.

28.

y

y

x

x

In Problems 29 and 30, sketch the graph of each function. Be sure to label at least three points. 29. f 1x2 = 0 x 0

30. f 1x2 = 1x

31. F 1x2 = 0 x 0 - 4

32. g 1x2 = - 2 0 x 0

In Problems 31–36, graph each function using the techniques of shifting, compressing or stretching, and reflections. Identify any intercepts on the graph. State the domain and, based on the graph, find the range. 34. f 1x2 = 21 - x

In Problems 37 and 38, (a) Find the domain of each function. (d) Based on the graph, find the range. 37. f 1x2 = b

3x x + 1

x 38. f 1x2 = c 1 3x

33. h(x) = 2x - 1

35. h(x) = (x - 1)2 + 2

36. g(x) = - 2(x + 2)3 - 8

(b) Locate any intercepts. (e) Is f continuous on its domain?

(c) Graph each function.

39. A function f is defined by

if - 2 6 x … 1 if x 7 1

f 1x2 =

if - 4 … x 6 0 if x = 0 if x 7 0

Ax + 5 6x - 2

If f 112 = 4, find A.

40. Constructing a Closed Box A closed box with a square base is required to have a volume of 10 cubic feet. (a) Build a model that expresses the amount A of material used to make such a box as a function of the length x of a side of the square base. (b) How much material is required for a base 1 foot by 1 foot? (c) How much material is required for a base 2 feet by 2 feet? (d) Graph A = A(x). For what value of x is A smallest?

41. A rectangle has one vertex in quadrant I on the graph of y = 10 - x 2, another at the origin, one on the positive x-axis, and one on the positive y-axis. (a) Express the area A of the rectangle as a function of x. (b) Find the largest area A that can be enclosed by the rectangle.

The Chapter Test Prep Videos are step-by-step test solutions available in , or on this text’s Channel.

CHAPTER TEST

1. Determine whether each relation represents a function. For each function, state the domain and the range. (a) 5 (2, 5), (4, 6), (6, 7), (8, 8) 6 (b) 5 (1, 3), (4, - 2), ( - 3, 5), (1, 7) 6 (c)

(d)

y 6 4

y 6

2 x

4

4

2

2

4

2

2 x 4

2

2

4

2 4

131

Functions and Their Graphs

In Problems 2– 4, find the domain of each function and evaluate each function at x = - 1. x + 2 2. f 1x2 = 24 - 5x 3. g(x) = 0x + 20 x - 4 4. h(x) = 2 x + 5x - 36 5. Using the graph of the function f : y 4 (1, 3) (0, 2) (2, 0)

(2, 0)

x 4

4 2 (5, 3)

(a) (b) (c) (d) (e)

4

(5, 2) (3, 3)

Find the domain and the range of f. List the intercepts. Find f 112. For what value(s) of x does f 1x2 = - 3? Solve f 1x2 6 0.

6. Use a graphing utility to graph the function f 1x2 = - x 4 + 2x 3 + 4x 2 - 2 on the interval ( - 5, 5). Approximate any local maximum values and local minimum values rounded to two decimal places. Determine where the function is increasing and where it is decreasing. 2x + 1 7. Consider the function g(x) = b x - 4 (a) Graph the function. (b) List the intercepts. (c) Find g( - 5). (d) Find g(2).

if x 6 - 1 if x Ú - 1

8. For the function f 1x2 = 3x 2 - 2x + 4, find the average rate of change of f from 3 to 4.

132

9. For the functions f 1x2 = 2x 2 + 1 and g(x) = 3x - 2, find the following and simplify: (a) f - g (b) f # g (c) f 1x + h2 - f 1x2

10. Graph each function using the techniques of shifting, compressing or stretching, and reflections. Start with the graph of the basic function and show all stages. (a) h(x) = - 2(x + 1)3 + 3 (b) g(x) = 0 x + 4 0 + 2

11. The variable interest rate on a student loan changes each July 1 based on the bank prime loan rate. For the years 1992–2007, this rate can be approximated by the model r(x) = - 0.115x 2 + 1.183x + 5.623, where x is the number of years since 1992 and r is the interest rate as a percent. (a) Use a graphing utility to estimate the highest rate during this time period. During which year was the interest rate the highest? (b) Use the model to estimate the rate in 2010. Does this value seem reasonable? Source: U.S. Federal Reserve 12. A community skating rink is in the shape of a rectangle with semicircles attached at the ends. The length of the rectangle is 20 feet less than twice the width. The thickness of the ice is 0.75 inch. (a) Build a model that expresses the ice volume, V, as a function of the width, x. (b) How much ice is in the rink if the width is 90 feet?

Functions and Their Graphs

CHAPTER PROJECTS 4. Suppose you expect to use 500 anytime minutes with unlimited texting and 20 MB of data. What would be the monthly cost of each plan you are considering? Stephen Coburn/Shutterstock

5. Build a model that describes the monthly cost C as a function of the number of anytime minutes used m assuming unlimited texting and 20 MB of data each month for each plan you are considering.

I.

6. Graph each function from Problem 5. 7. Based on your particular usage, which plan is best for you? 8. Now, develop an Excel spreadsheet to analyze the various plans you are considering. Suppose you want a plan that offers 700 anytime minutes with additional minutes costing \$0.40 per minute that costs \$39.99 per month. In addition, you want unlimited texting, which costs an additional \$20 per month, and a data plan that offers up to 25 MB of data each month, with each additional MB costing \$0.20. Because cellular telephone plans’ cost structure is based on piecewise-defined functions, we need “if-then” statements within Excel to analyze the cost of the plan. Use the Excel spreadsheet below as a guide in developing your worksheet. Enter into your spreadsheet a variety of possible minutes and data used to help arrive at a decision regarding which plan is best for you.

Internet-based Project Choosing a Cellular Telephone Plan Collect information from your family, friends, or consumer agencies such as Consumer Reports. Then decide on a cellular telephone provider, choosing the company that you feel offers the best service. Once you have selected a service provider, research the various types of individual plans offered by the company by visiting the provider’s website. 1. Suppose you expect to use 400 anytime minutes without a texting or data plan. What would be the monthly cost of each plan you are considering? 2.

Suppose you expect to use 600 anytime minutes with unlimited texting, but no data plan. What would be the monthly cost of each plan you are considering?

3. Suppose you expect to use 500 anytime minutes with unlimited texting and an unlimited data plan. What would be the monthly cost of each plan you are considering?

9. Write a paragraph supporting the choice in plans that best meets your needs. 10. How are “if/then” loops similar to a piecewise-defined function? Citation: Excel © 2010 Microsoft Corporation. Used with permission from Microsoft.

A 1 2 Monthly Fee \$ 3 Alloted number of anytime minutes 4 5 6 7 8 9 10 11 12 13 14 15 16

Number of anytime minutes used: Cost per additional minute Monthly cost of text messaging: Monthly cost of data plan Alloted data per month (MB) Data used Cost per additional MB of data

\$ \$ \$

\$

B

C

D

39.99 700 700 0.40 20.00 9.99 25 30 0.20

Cost of phone minutes Cost of data

=IF(B4 1 , the series diverges.

57. a1 = 8 , r =

1 4 Since r < 1, the series converges.

59. a1 = 5, r =

S∞ =

a1 5 5 20 = = = 1− r ⎛ 1 ⎞ ⎛ 3 ⎞ 3 ⎜1 − ⎟ ⎜ ⎟ ⎝ 4⎠ ⎝4⎠

1791

Sequences; Induction; the Binomial Theorem

1 , r =3 2 Since r > 1 , the series diverges.

73. 1, 3, 6, 10, ... There is no common difference and there is no common ratio. Therefore the sequence is neither arithmetic nor geometric.

2 3 Since r < 1, the series converges.

⎧⎪ ⎛ 2 ⎞ n 75. ⎨ ⎜ ⎟ ⎪⎩ ⎝ 3 ⎠

61. a1 =

63. a1 = 6, r = −

n +1

⎛2⎞ n +1− n ⎜ ⎟ 2 3⎠ ⎛2⎞ ⎝ r= =⎜ ⎟ = n 3 3 ⎝ ⎠ ⎛2⎞ ⎜ ⎟ ⎝3⎠ The ratio of consecutive terms is constant. Therefore the sequence is geometric.

a 6 6 18 = = S∞ = 1 = 1− r ⎛ ⎛ 2 ⎞⎞ ⎛ 5 ⎞ 5 ⎜1 − ⎜ − 3 ⎟ ⎟ ⎜ 3 ⎟ ⎠⎠ ⎝ ⎠ ⎝ ⎝

65.

k

S∞ =

67.

k −1

∞ ∞ 2 ⎛2⎞ ⎛2⎞ ⎛2⎞ ∑ 3⎜ 3 ⎟ = ∑ 3 ⋅ 3 ⋅ ⎜ 3 ⎟ = ∑ 2 ⎜ 3 ⎟ ⎝ ⎠ k =1 ⎝ ⎠ k =1 k =1 ⎝ ⎠ 2 a1 = 2 , r = 3 Since r < 1, the series converges.

k −1

a1 2 2 = = =6 1− r 1− 2 1 3 3

{ n + 2} d = (n + 1 + 2) − (n + 2) = n + 3 − n − 2 = 1 The difference between consecutive terms is constant. Therefore the sequence is arithmetic. 50

50

50

k =1

k =1

k =1

S50 = ∑ (k + 2) = ∑ k + ∑ 2 =

69.

{

4n 2

50(50 + 1) + 2(50) = 1275 + 100 = 1375 2

}

Examine the terms of the sequence: 4,

16, 36, 64, 100, ... There is no common difference and there is no common ratio. Therefore the sequence is neither arithmetic nor geometric. 2 ⎫ ⎧ 71. ⎨ 3 − n ⎬ 3 ⎭ ⎩ 2 2 ⎞ ⎛ ⎞ ⎛ d = ⎜ 3 − (n + 1) ⎟ − ⎜ 3 − n ⎟ 3 3 ⎠ ⎝ ⎠ ⎝ 2 2 2 2 = 3− n − −3+ n = − 3 3 3 3 The difference between consecutive terms is constant. Therefore the sequence is arithmetic. 50 2 ⎞ 50 2 50 ⎛ S50 = ∑ ⎜ 3 − k ⎟ = ∑ 3 − ∑ k 3 ⎠ k =1 3 k =1 k =1 ⎝ 2 ⎛ 50(50 + 1) ⎞ = 3(50) − ⎜ ⎟ = 150 − 850 = −700 3⎝ 2 ⎠

1792

⎫⎪ ⎬ ⎪⎭

⎛2⎞ 1− ⎜ ⎟ 50 2 3 ⎛ 2⎞ S50 = ∑ ⎜ ⎟ = ⋅ ⎝ ⎠ 2 3 3 ⎠ k =1 ⎝ 1− 3

50

k

= 1.999999997

77. –1, 2, –4, 8, ... 2 −4 8 r= = = = −2 −1 2 −4 The ratio of consecutive terms is constant. Therefore the sequence is geometric. 50 1 − (−2)50 S50 = ∑ −1 ⋅ (−2) k −1 = −1 ⋅ 1 − (−2) k =1 ≈ 3.752999689 × 1014

79.

{3 } n/2

r=

⎛ n +1 ⎞ ⎜ ⎟ 3⎝ 2 ⎠

⎛ n +1 n ⎞ − ⎟ ⎜ 2 2⎠

= 3⎝

⎛n⎞ ⎜ ⎟ 3⎝ 2 ⎠

= 31/ 2

The ratio of consecutive terms is constant. Therefore the sequence is geometric. 50

S50 = ∑ 3 k =1

k/2

=3

1/ 2

1 − ( 31/ 2 )

50

1 − 31/ 2

≈ 2.004706374 × 1012

81. Find the common ratio of the terms and solve the system of equations: x+2 x+3 = r; =r x x+2 x+2 x+3 = → x 2 + 4 x + 4 = x 2 + 3x → x = − 4 x x+2 83. This is a geometric series with a1 = \$18, 000, r = 1.05, n = 5 . Find the 5th

Sequences; Induction; the Binomial Theorem

term: 5 −1

a5 = 18000 (1.05 ) 85. a.

= 18000 (1.05 ) = \$21,879.11 4

Find the 10th term of the geometric sequence: a1 = 2, r = 0.9, n = 10 a10 = 2(0.9)10 −1 = 2(0.9)9 = 0.775 feet

b. Find n when an < 1 : 2(0.9)n −1 < 1

( 0.9 )n −1 < 0.5 (n − 1) log ( 0.9 ) < log ( 0.5 ) log ( 0.5 ) n −1 > log ( 0.9 ) log ( 0.5 ) + 1 ≈ 7.58 n> log ( 0.9 ) On the 8th swing the arc is less than 1 foot. c.

Find the sum of the first 15 swings: ⎛ 1 − ( 0.9 )15 ⎞ ⎛ 1 − (0.9)15 ⎞ ⎟ S15 = 2 ⎜⎜ ⎟⎟ = 2 ⎜ ⎜ 0.1 ⎟ ⎝ 1 − 0.9 ⎠ ⎝ ⎠

(

= 20 1 − ( 0.9 )

15

) = 15.88 feet

d. Find the infinite sum of the geometric series: 2 2 S∞ = = = 20 feet 1 − 0.9 0.1 87. This is a geometric sequence with a1 = 1, r = 2, n = 64 . Find the sum of the geometric series: ⎛ 1 − 264 ⎞ 1 − 264 S64 = 1⎜⎜ = 264 − 1 ⎟⎟ = − − 1 2 1 ⎝ ⎠ = 1.845 × 1019 grains

89. The common ratio, r = 0.90 < 1 . The sum is: 1 1 S= = = 10 . 1 − 0.9 0.10 The multiplier is 10. 91. This is an infinite geometric series with 1.03 a = 4, and r = . 1.09 4 Find the sum: Price = ≈ \$72.67 . 1.03 ⎛ ⎞ − 1 ⎜ ⎟ ⎝ 1.09 ⎠

93. Given: a1 = 1000, r = 0.9 Find n when an < 0.01 : 1000(0.9) n −1 < 0.01

( 0.9 )n −1 < 0.00001 (n − 1) log ( 0.9 ) < log ( 0.00001) log ( 0.00001) n −1 > log ( 0.9 ) log ( 0.00001) + 1 ≈ 110.27 n> log ( 0.9 ) On the 111th day or December 20, 2007, the amount will be less than \$0.01. Find the sum of the geometric series: ⎛ 1 − ( 0.9 )111 ⎞ ⎛ 1− rn ⎞ ⎜ ⎟ = S111 = a1 ⎜⎜ 1000 ⎟⎟ ⎜ 1 − 0.9 ⎟ ⎝ 1− r ⎠ ⎝ ⎠ ⎛ 1 − ( 0.9 )111 ⎞ ⎟ = \$9999.92 = 1000 ⎜ ⎜ ⎟ 0.1 ⎝ ⎠ 95. Find the sum of each sequence: A: Arithmetic series with: a1 = \$1000, d = −1, n = 1000 Find the sum of the arithmetic series: 1000 S1000 = (1000 + 1) = 500(1001) = \$500,500 2 B: This is a geometric sequence with a 1 = 1, r = 2, n = 19 . Find the sum of the geometric series: ⎛ 1 − 219 ⎞ 1 − 219 S19 = 1⎜⎜ = 219 − 1 = \$524, 287 ⎟⎟ = − − 1 2 1 ⎝ ⎠

B results in more money. 97. The amount paid each day forms a geometric sequence with a1 = 0.01 and r = 2 . 1 − r 22 1 − 222 = 0.01 ⋅ = 41,943.03 1− r 1− 2 The total payment would be \$41,943.03 if you worked all 22 days. S22 = a1 ⋅

a22 = a1 ⋅ r 22 −1 = 0.01( 2 ) = 20,971.52 21

The payment on the 22nd day is \$20,971.52. Answers will vary. With this payment plan, the bulk of the payment is at the end so missing even one day can dramatically reduce the overall payment. Notice that with one sick day you would lose the amount paid on the 22nd day

1793

Sequences; Induction; the Binomial Theorem

which is about half the total payment for the 22 days.

II: If 2 + 5 + 8 + " + (3k − 1) =

then 2 + 5 + 8 + " + (3k − 1) + [3(k + 1) − 1]

= [ 2 + 5 + 8 + " + (3k − 1) ] + (3k + 2)

101. Answers will vary. Both increase (or decrease) exponentially, but the domain of a geometric sequence is the set of natural numbers while the domain of an exponential function is the set of all real numbers.

1 3 1 ⋅ k (3k + 1) + (3k + 2) = k 2 + k + 3k + 2 2 2 2 3 2 7 1 2 = k + k + 2 = ⋅ 3k + 7 k + 4 2 2 2 1 = ⋅ (k + 1)(3k + 4) 2 1 = ⋅ (k + 1) ( 3 ( k + 1) + 1) 2 =

(

Section 4 1. I:

n = 1: 2 ⋅1 = 2 and 1(1 + 1) = 2

II: If 2 + 4 + 6 + " + 2k = k (k + 1) , then 2 + 4 + 6 + " + 2k + 2(k + 1) = [ 2 + 4 + 6 + " + 2k ] + 2(k + 1) = k (k + 1) + 2(k + 1) = ( k + 1) ( ( k + 1) + 1)

Conditions I and II are satisfied; the statement is true. 7. I:

1 ⋅1(1 + 5) = 3 2

1 ⋅ k (k + 5) , then 2 3 + 4 + 5 + " + (k + 2) + [(k + 1) + 2]

II: If 3 + 4 + 5 + " + (k + 2) =

= [3 + 4 + 5 + " + (k + 2) ] + (k + 3)

1 ⋅ k (k + 5) + (k + 3) 2 1 5 = k2 + k + k + 3 2 2 1 7 = k2 + k + 3 2 2 1 = ⋅ k 2 + 7k + 6 2 1 = ⋅ (k + 1)(k + 6) 2 1 = ⋅ (k + 1) ( ( k + 1) + 5 ) 2 Conditions I and II are satisfied; the statement is true. =

(

5. I:

1794

n = 1: 21−1 = 1 and 21 − 1 = 1

1 + 2 + 22 + " + 2k −1 + 2k +1−1 = ⎡⎣1 + 2 + 22 + " + 2k −1 ⎤⎦ + 2k

Conditions I and II are satisfied; the statement is true. n = 1: 1 + 2 = 3 and

)

II: If 1 + 2 + 22 + " + 2k −1 = 2k − 1 , then

= (k + 1)(k + 2)

3. I:

1 ⋅ k (3k + 1) , 2

)

n = 1: 3 ⋅1 − 1 = 2 and

1 ⋅1(3 ⋅1 + 1) = 2 2

= 2k − 1 + 2k = 2 ⋅ 2k − 1 = 2k +1 − 1 Conditions I and II are satisfied; the statement is true.

9. I:

(

)

1 n = 1: 41−1 = 1 and ⋅ 41 − 1 = 1 3

(

)

1 II: If 1 + 4 + 42 + " + 4k −1 = ⋅ 4k − 1 , then 3 1 + 4 + 42 + " + 4k −1 + 4k +1−1 = ⎡⎣1 + 4 + 42 + " + 4k −1 ⎤⎦ + 4k

(

)

1 1 1 = ⋅ 4k − 1 + 4k = ⋅ 4k − + 4k 3 3 3 4 k 1 1 = ⋅ 4 − = 4 ⋅ 4k − 1 3 3 3 1 k +1 = ⋅ 4 −1 3 Conditions I and II are satisfied; the statement is true.

(

(

)

)

Sequences; Induction; the Binomial Theorem

11. I:

n = 1:

II: If

1 1 1 1 = and = 1(1 + 1) 2 1+1 2

1 1 1 1 k + + +"+ = , then k (k + 1) k + 1 1⋅ 2 2 ⋅ 3 3 ⋅ 4

⎡ 1 1 1 1 1 1 1 1 1 ⎤ 1 + + +"+ + =⎢ + + +"+ + ⎥ k (k + 1) (k + 1)(k + 1 + 1) ⎣1 ⋅ 2 2 ⋅ 3 3 ⋅ 4 k (k + 1) ⎦ (k + 1)(k + 2) 1⋅ 2 2 ⋅ 3 3 ⋅ 4 k k k +2 1 1 = + = ⋅ + k + 1 (k + 1)(k + 2) k + 1 k + 2 (k + 1)(k + 2)

=

k 2 + 2k + 1 k +1 (k + 1)(k + 1) k + 1 = = = (k + 1)(k + 2) (k + 1)(k + 2) k + 2 ( k + 1) + 1

Conditions I and II are satisfied; the statement is true. 13. I:

n = 1: 12 = 1 and

1 ⋅1(1 + 1)(2 ⋅1 + 1) = 1 6

II: If 12 + 22 + 32 + " + k 2 =

1 ⋅ k (k + 1)(2k + 1) , then 6

1 12 + 22 + 32 + " + k 2 + (k + 1) 2 = ⎡⎣12 + 22 + 32 + " + k 2 ⎤⎦ + (k + 1) 2 = k (k + 1)(2k + 1) + (k + 1) 2 6 1 7 ⎡1 ⎤ ⎡1 ⎤ ⎡1 ⎤ 1 = (k + 1) ⎢ k (2k + 1) + k + 1⎥ = (k + 1) ⎢ k 2 + k + k + 1⎥ = (k + 1) ⎢ k 2 + k + 1⎥ = (k + 1) ⎡⎣ 2k 2 + 7 k + 6 ⎤⎦ 6 6 ⎣6 ⎦ ⎣3 ⎦ ⎣3 ⎦ 6 1 = ⋅ (k + 1)(k + 2)(2k + 3) 6 1 = ⋅ (k + 1) ( ( k + 1) + 1) ( 2 ( k + 1) + 1) 6 Conditions I and II are satisfied; the statement is true.

15. I:

n = 1: 5 − 1 = 4 and

1 ⋅1(9 − 1) = 4 2

II: If 4 + 3 + 2 + " + (5 − k ) =

1 ⋅ k (9 − k ) , then 2

4 + 3 + 2 + " + (5 − k ) + ( 5 − (k + 1) ) = [ 4 + 3 + 2 + " + (5 − k )] + (4 − k ) =

1 k (9 − k ) + (4 − k ) 2

9 1 1 7 1 k − k 2 + 4 − k = − k 2 + k + 4 = − ⋅ ⎡⎣ k 2 − 7k − 8⎤⎦ 2 2 2 2 2 1 1 1 = − ⋅ (k + 1)(k − 8) = ⋅ (k + 1)(8 − k ) = ⋅ (k + 1) [9 − (k + 1) ] 2 2 2 Conditions I and II are satisfied; the statement is true. =

1795

Sequences; Induction; the Binomial Theorem

17. I:

1 n = 1: 1(1 + 1) = 2 and ⋅1(1 + 1)(1 + 2) = 2 3

1 II: If 1 ⋅ 2 + 2 ⋅ 3 + 3 ⋅ 4 + " + k (k + 1) = ⋅ k (k + 1)( k + 2) , then 3 1 ⋅ 2 + 2 ⋅ 3 + 3 ⋅ 4 + " + k (k + 1) + (k + 1)(k + 1 + 1) = [1 ⋅ 2 + 2 ⋅ 3 + 3 ⋅ 4 + " + k (k + 1) ] + (k + 1)(k + 2) 1 ⎡1 ⎤ = ⋅ k (k + 1)(k + 2) + (k + 1)(k + 2) = (k + 1)(k + 2) ⎢ k + 1⎥ 3 3 ⎣ ⎦ 1 = ⋅ (k + 1)(k + 2)(k + 3) 3 1 = ⋅ (k + 1)((k + 1) + 1)((k + 1) + 2) 3

Conditions I and II are satisfied; the statement is true. 19. I:

n = 1: 12 + 1 = 2 is divisible by 2

II: If k 2 + k is divisible by 2 , then (k + 1) 2 + (k + 1) = k 2 + 2k + 1 + k + 1 = (k 2 + k ) + (2k + 2)

Since k 2 + k is divisible by 2 and 2k + 2 is divisible by 2, then (k + 1) 2 + (k + 1) is divisible by 2. Conditions I and II are satisfied; the statement is true. 21. I:

n = 1: 12 − 1 + 2 = 2 is divisible by 2

II: If k 2 − k + 2 is divisible by 2 , then (k + 1) 2 − (k + 1) + 2 = k 2 + 2k + 1 − k − 1 + 2 = (k 2 − k + 2) + (2k ) Since k 2 − k + 2 is divisible by 2 and 2k is divisible by 2, then (k + 1) 2 − (k + 1) + 2 is divisible by 2. Conditions I and II are satisfied; the statement is true. 23. I:

n = 1: If x > 1 then x1 = x > 1.

II: Assume, for some natural number k, that if x > 1 , then x k > 1 . Then x k +1 > 1, for x > 1, x k +1 = x k ⋅ x > 1 ⋅ x = x > 1 ↑ ( x k > 1)

1796

Conditions I and II are satisfied; the statement is true. 25. I:

II:

n = 1: a − b is a factor of a1 − b1 = a − b. If a − b is a factor of a k − b k , show that a − b is a factor of a k +1 − b k +1 . a k +1 − b k +1 = a ⋅ a k − b ⋅ b k = a ⋅ ak − a ⋅ bk + a ⋅ bk − b ⋅ bk

(

)

= a a k − b k + b k ( a − b)

Since a − b is a factor of a k − b k and a − b is a factor of a − b , then a − b is a factor of a k +1 − b k +1 . Conditions I and II are satisfied; the statement is true. 27. I:

n = 1 : (1 + a ) = 1 + a ≥ 1 + 1 ⋅ a 1

II: Assume that there is an integer k for which the inequality holds. We need to show that if

(1 + a ) ≥ 1 + ka then k +1 (1 + a ) ≥ 1 + ( k + 1) a . k

(1 + a )

k +1

= (1 + a ) (1 + a ) k

≥ (1 + ka )(1 + a )

= 1 + ka 2 + a + ka

= 1 + ( k + 1) a + ka 2

≥ 1 + ( k + 1) a

Conditions I and II are satisfied, the statement is true.

Sequences; Induction; the Binomial Theorem 29. II: If 2 + 4 + 6 + " + 2k = k 2 + k + 2 , then 2 + 4 + 6 + " + 2k + 2(k + 1)

1. Pascal Triangle

= k 2 + k + 2 + 2k + 2

⎛n⎞ n! 3. False; ⎜ ⎟ = j j ! n − j )! ( ⎝ ⎠

= [ 2 + 4 + 6 + " + 2k ] + 2k + 2 = (k 2 + 2k + 1) + (k + 1) + 2 = (k + 1)2 + (k + 1) + 2

I: 31. I:

Section 5

2

n = 1: 2 ⋅1 = 2 and 1 + 1 + 2 = 4 ≠ 2 n = 1: a + (1 − 1)d = a and 1 ⋅ a + d

1(1 − 1) =a 2

II: If a + (a + d ) + (a + 2d ) + " + [ a + (k − 1)d ] = ka + d

k (k − 1) 2

then

a + ( a + d ) + ( a + 2d ) + " + [ a + ( k − 1) d ] + ( a + kd )

= [ a + ( a + d ) + ( a + 2d ) + " + [ a + ( k − 1) d ]] + ( a + kd )

k (k − 1) = ka + d + (a + kd ) 2 ⎡ k (k − 1) ⎤ = (k + 1)a + d ⎢ + k⎥ ⎣ 2 ⎦

⎡ k 2 − k + 2k ⎤ = (k + 1)a + d ⎢ ⎥ 2 ⎣ ⎦

⎛5⎞ 5! 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 5 ⋅ 4 = = = 10 5. ⎜ ⎟ = ⎝ 3 ⎠ 3! 2! 3 ⋅ 2 ⋅1 ⋅ 2 ⋅1 2 ⋅1 ⎛7⎞ 7! 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 7 ⋅ 6 = = = 21 7. ⎜ ⎟ = ⎝ 5 ⎠ 5! 2! 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 ⋅ 2 ⋅1 2 ⋅1

⎛ 50 ⎞ 50! 50 ⋅ 49! 50 9. ⎜ ⎟ = = = = 50 49 49!1! 49! ⋅1 1 ⎝ ⎠ ⎛1000 ⎞ 1000! 1 = =1 11. ⎜ ⎟= 1000 ⎝ ⎠ 1000! 0! 1 ⎛ 55 ⎞ 55! ≈ 1.8664 × 1015 13. ⎜ ⎟ = ⎝ 23 ⎠ 23! 32!

⎛ 47 ⎞ 47! 15. ⎜ ⎟ = ≈ 1.4834 × 1013 25 25! 22! ⎝ ⎠

⎡k2 + k ⎤ = (k + 1)a + d ⎢ ⎥ ⎣ 2 ⎦ ⎡ (k + 1)k ⎤ = (k + 1)a + d ⎢ ⎥ ⎣ 2 ⎦

⎡ ( k + 1) ( ( k + 1) − 1) ⎤ = ( k + 1) a + d ⎢ ⎥ 2 ⎣⎢ ⎦⎥

Conditions I and II are satisfied; the statement is true. 33. I:

n = 3 : (3 − 2) ⋅180° = 180° which is the sum of the angles of a triangle. II: Assume that for any integer k, the sum of the angles of a convex polygon with k sides is (k − 2) ⋅180° . A convex polygon with

k + 1 sides consists of a convex polygon with k sides plus a triangle. Thus, the sum of the angles is (k − 2) ⋅180° + 180° = ((k + 1) − 2) ⋅180°. Conditions I and II are satisfied; the statement is true.

1797

Sequences; Induction; the Binomial Theorem ⎛5⎞ ⎛ 5⎞ ⎛5⎞ ⎛ 5⎞ ⎛ 5⎞ ⎛ 5⎞ 17. ( x + 1)5 = ⎜ ⎟ x5 + ⎜ ⎟ x 4 + ⎜ ⎟ x3 + ⎜ ⎟ x 2 + ⎜ ⎟ x1 + ⎜ ⎟ x 0 = x5 + 5 x 4 + 10 x3 + 10 x 2 + 5 x + 1 0 1 2 3 4 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 5⎠ ⎛6⎞ ⎛ 6⎞ ⎛ 6⎞ ⎛ 6⎞ ⎛ 6⎞ ⎛ 6⎞ ⎛ 6⎞ 19. ( x − 2)6 = ⎜ ⎟ x 6 + ⎜ ⎟ x5 (− 2) + ⎜ ⎟ x 4 (− 2) 2 + ⎜ ⎟ x3 (− 2)3 + ⎜ ⎟ x 2 (− 2) 4 + ⎜ ⎟ x(− 2)5 + ⎜ ⎟ x0 (− 2)6 ⎝0⎠ ⎝1⎠ ⎝ 2⎠ ⎝ 3⎠ ⎝ 4⎠ ⎝ 5⎠ ⎝ 6⎠ = x 6 + 6 x5 (− 2) + 15 x 4 ⋅ 4 + 20 x3 (− 8) + 15 x 2 ⋅16 + 6 x ⋅ (−32) + 64 = x 6 − 12 x5 + 60 x 4 − 160 x3 + 240 x 2 − 192 x + 64

⎛ 4⎞ ⎛ 4⎞ ⎛ 4⎞ ⎛ 4⎞ ⎛ 4⎞ 21. (3x + 1) 4 = ⎜ ⎟ (3 x) 4 + ⎜ ⎟ (3 x)3 + ⎜ ⎟ (3 x) 2 + ⎜ ⎟ (3x) + ⎜ ⎟ ⎝0⎠ ⎝1⎠ ⎝ 2⎠ ⎝ 3⎠ ⎝ 4⎠ = 81x 4 + 4 ⋅ 27 x3 + 6 ⋅ 9 x 2 + 4 ⋅ 3 x + 1 = 81x 4 + 108 x3 + 54 x 2 + 12 x + 1

23.

(x

2

+ y2

)

5

⎛5⎞ = ⎜ ⎟ x2 ⎝0⎠

( )

5

⎛5⎞ + ⎜ ⎟ x2 ⎝1⎠

( )

4

⎛ 5⎞ y2 + ⎜ ⎟ x2 ⎝ 2⎠

( ) (y ) 3

2 2

⎛ 5⎞ + ⎜ ⎟ x2 ⎝ 3⎠

( ) (y ) 2

2 3

⎛ 5⎞ + ⎜ ⎟ x2 y 2 ⎝ 4⎠

( )

4

⎛ 5⎞ + ⎜ ⎟ y2 ⎝ 5⎠

( )

5

= x10 + 5 x8 y 2 + 10 x 6 y 4 + 10 x 4 y 6 + 5 x 2 y8 + y10

25.

(

x+ 2

)

6

6 ⎛ 6⎞ 5 1 ⎛ 6⎞ ⎛6⎞ = ⎜ ⎟ x +⎜ ⎟ x 2 +⎜ ⎟ 0 1 ⎝ ⎠ ⎝ ⎠ ⎝ 2⎠ 2 4 ⎛6⎞ ⎛6⎞ +⎜ ⎟ x 2 +⎜ ⎟ x 2 ⎝ 4⎠ ⎝5⎠

( )

( )( )

( )( )

( x) ( 2) 4

( )( )

5

⎛ 6⎞ +⎜ ⎟ ⎝ 6⎠

2

( 2)

⎛ 6⎞ +⎜ ⎟ ⎝ 3⎠

( x) ( 2) 3

3

6

= x3 + 6 2 x5/ 2 + 15 ⋅ 2 x 2 + 20 ⋅ 2 2 x3/ 2 + 15 ⋅ 4 x + 6 ⋅ 4 2 x1/ 2 + 8 = x3 + 6 2 x5/ 2 + 30 x 2 + 40 2 x3/ 2 + 60 x + 24 2 x1/ 2 + 8

27.

⎛5⎞ ⎝0⎠

⎛5⎞ ⎝1⎠

⎛ 5⎞ ⎝ 2⎠

⎛ 5⎞ ⎝ 3⎠

⎛ 5⎞ 4 ⎛ 5⎞ 5 ⎟ ax ( by ) + ⎜ ⎟ ( by ) 4 5 ⎝ ⎠ ⎝ ⎠

( ax + by )5 = ⎜ ⎟ ( ax )5 + ⎜ ⎟ ( ax )4 ⋅ by + ⎜ ⎟ ( ax )3 ( by )2 + ⎜ ⎟ ( ax )2 ( by )3 + ⎜ = a 5 x5 + 5a 4 x 4by + 10a3 x3b 2 y 2 + 10a 2 x 2b3 y 3 + 5axb 4 y 4 + b5 y 5

29. n = 10, j = 4, x = x, a = 3 ⎛10 ⎞ 6 4 10! 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 81x 6 = ⋅ 81x 6 ⎜ ⎟ x ⋅3 = 4! 6! 4 ⋅ 3 ⋅ 2 ⋅1 ⎝4⎠ = 17, 010 x 6

The coefficient of x 6 is 17, 010.

31. n = 12, j = 5, x = 2 x, a = −1 ⎛12 ⎞ 12! 7 5 ⋅128 x 7 (−1) ⎜ ⎟ (2 x) ⋅ (−1) = 5 5! 7! ⎝ ⎠ 12 ⋅11 ⋅10 ⋅ 9 ⋅ 8 = ⋅ (−128) x7 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 = −101,376 x7

The coefficient of x 7 is − 101,376.

1798

33. n = 9, j = 2, x = 2 x, a = 3 ⎛9⎞ 9! 7 2 ⋅128 x7 (9) ⎜ ⎟ (2 x) ⋅ 3 = 2! 7! ⎝ 2⎠ 9 ⋅8 = ⋅128 x7 ⋅ 9 2 ⋅1 = 41, 472 x 7

The coefficient of x 7 is 41,472.

35. n = 7, j = 4, x = x, a = 3 ⎛7⎞ 3 4 7! 7 ⋅6⋅5 ⋅ 81x3 = ⋅ 81x3 = 2835 x3 ⎜ ⎟ x ⋅3 = 4! 3! 3 ⋅ 2 ⋅1 ⎝ 4⎠

Sequences; Induction; the Binomial Theorem 37. n = 9, j = 2, x = 3 x, a = − 2

41. The x 4 term in

⎛9⎞ 9! 7 2 ⋅ 2187 x 7 ⋅ 4 ⎜ ⎟ (3x) ⋅ (− 2) = 2 2! 7! ⎝ ⎠ 9 ⋅8 = ⋅ 8748 x 7 = 314,928 x7 2 ⋅1

j

12 12 12 − j ⎛ 1 ⎞ ⎛12 ⎞ ⎛ ⎞ ∑ ⎜ j ⎟ x 2 ⎜⎝ x ⎟⎠ = ∑ ⎜ j ⎟ x 24−3 j ⎠ ⎠ j =0 ⎝ j =0 ⎝ occurs when: 24 − 3 j = 0

( )

3

10 10 ⎛10 ⎞ 10− j ⎛ − 2 ⎞ ⎛ ⎞ j 10 − 2 j x = ( ) ∑⎜ j ⎟ ∑ ⎜ ⎟ ( −2 ) x ⎜ ⎟ ⎝ x⎠ ⎠ j =0 ⎝ j =0 ⎝ j ⎠ occurs when: 3 10 − j = 4 2 3 − j = −6 2 j=4 The coefficient is ⎛10 ⎞ 10! 10 ⋅ 9 ⋅ 8 ⋅ 7 4 ⋅16 = ⋅16 = 3360 ⎜ ⎟ ( −2 ) = 6! 4! 4 ⋅ 3 ⋅ 2 ⋅1 ⎝4⎠

39. The x 0 term in 12

j

10

24 = 3 j j =8 The coefficient is ⎛12 ⎞ 12! 12 ⋅11 ⋅10 ⋅ 9 ⎜ 8 ⎟ = 8! 4! = 4 ⋅ 3 ⋅ 2 ⋅1 = 495 ⎝ ⎠

2 ⎛ 5⎞ ⎛5⎞ ⎛ 5⎞ ⎛ 5⎞ = ⎜ ⎟15 + ⎜ ⎟14 ⋅10−3 + ⎜ ⎟13 ⋅ 10−3 + ⎜ ⎟12 ⋅ 10−3 ⎝0⎠ ⎝1⎠ ⎝ 2⎠ ⎝ 3⎠ = 1 + 5(0.001) + 10(0.000001) + 10(0.000000001) + ⋅⋅⋅ = 1 + 0.005 + 0.000010 + 0.000000010 + ⋅⋅⋅ = 1.00501 (correct to 5 decimal places)

(

43. (1.001)5 = 1 + 10−3

)

5

(

)

(

)

3

+ ⋅⋅⋅

n ( n − 1) ! ⎛ n ⎞ n! n! = = =n 45. ⎜ ⎟= − n 1 − − − − n 1 !( n n 1 )! n 1 !(1)! ) ( ) ( ) ( n − 1)! ⎝ ⎠ ( ⎛n⎞ n! n! n! n! = = = =1 ⎜ ⎟= n − ⋅ n !( n n )! n ! 0! n ! 1 n! ⎝ ⎠ ⎛n⎞ ⎛n⎞ ⎛ n⎞ 47. Show that ⎜ ⎟ + ⎜ ⎟ + ⋅⋅⋅ + ⎜ ⎟ = 2n ⎝0⎠ ⎝1⎠ ⎝ n⎠ n n 2 = (1 + 1)

⎛n⎞ ⎛n⎞ ⎛ n⎞ ⎛ n⎞ = ⎜ ⎟ ⋅1n + ⎜ ⎟ ⋅1n −1 ⋅1 + ⎜ ⎟ ⋅1n − 2 ⋅12 + ⋅⋅⋅ + ⎜ ⎟ ⋅1n − n ⋅1n ⎝0⎠ ⎝1⎠ ⎝ 2⎠ ⎝ n⎠ ⎛n⎞ ⎛n⎞ ⎛n⎞ = ⎜ ⎟ + ⎜ ⎟ + ⋅⋅⋅ + ⎜ ⎟ ⎝0⎠ ⎝1⎠ ⎝n⎠ 5 4 3 2 2 3 4 5 5 ⎛ 5⎞ ⎛ 1 ⎞ ⎛ 5⎞ ⎛ 1 ⎞ ⎛ 3 ⎞ ⎛ 5 ⎞ ⎛ 1 ⎞ ⎛ 3 ⎞ ⎛ 5⎞ ⎛ 1 ⎞ ⎛ 3 ⎞ ⎛ 5⎞ ⎛ 1 ⎞ ⎛ 3 ⎞ ⎛ 5⎞ ⎛ 3 ⎞ ⎛ 1 3 ⎞ 49. ⎜ ⎟ ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ = ⎜ + ⎟ = (1)5 = 1 ⎝ 0 ⎠ ⎝ 4 ⎠ ⎝ 1 ⎠ ⎝ 4 ⎠ ⎝ 4 ⎠ ⎝ 2 ⎠ ⎝ 4 ⎠ ⎝ 4 ⎠ ⎝ 3⎠ ⎝ 4 ⎠ ⎝ 4 ⎠ ⎝ 4 ⎠ ⎝ 4 ⎠ ⎝ 4 ⎠ ⎝ 5⎠ ⎝ 4 ⎠ ⎝ 4 4 ⎠

1799

Sequences; Induction; the Binomial Theorem

Chapter Review Exercises 1. a1 = (−1)1 21

2. c1 =

12

=

1+ 3 4 2+3 5 3+3 6 4+3 7 5+3 8 = − , a2 = (−1) 2 = , a3 = (−1)3 = − , a4 = (−1) 4 = , a5 = (−1)5 =− 1+ 2 3 2+2 4 3+ 2 5 4+2 6 5+2 7 2 22 4 23 8 24 16 25 32 = 2, c2 = 2 = = 1, c3 = 2 = , c4 = 2 = = 1, c5 = 2 = 1 4 9 16 25 2 3 4 5

3. a1 = 3, a2 =

2 2 4 2 4 8 2 8 16 ⋅ 3 = 2, a3 = ⋅ 2 = , a4 = ⋅ = , a5 = ⋅ = 3 3 3 3 3 9 3 9 27

4. a1 = 2, a2 = 2 − 2 = 0, a3 = 2 − 0 = 2, a4 = 2 − 2 = 0, a5 = 2 − 0 = 2 5.

4

∑ (4k + 2) = ( 4 ⋅1 + 2 ) + ( 4 ⋅ 2 + 2 ) + ( 4 ⋅ 3 + 2 ) + ( 4 ⋅ 4 + 2 ) = ( 6 ) + (10 ) + (14 ) + (18) = 48 k =1

1 1 1 1 13 k +1 ⎛ 1 ⎞ 6. 1 − + − + ⋅⋅⋅ + = ∑ ( −1) ⎜ ⎟ 2 3 4 13 k =1 ⎝k⎠

7.

{ an } = { n + 5 }

Arithmetic

d = (n + 1 + 5) − (n + 5) = n + 6 − n − 5 = 1 Sn =

8.

n n [6 + n + 5] = (n + 11) 2 2

{cn } = { 2n3 }

Examine the terms of the

sequence: 2, 16, 54, 128, 250, ... There is no common difference; there is no common ratio; neither. 9.

{sn } = { 23n } r=

23( n +1)

=

23 n ⎛ 1 − 8n Sn = 8 ⎜⎜ ⎝ 1− 8

Geometric 23n +3

3 3 3 3 11. 3, , , , , ... Geometric 2 4 8 16 ⎛3⎞ ⎜ ⎟ 3 1 1 2 r=⎝ ⎠= ⋅ = 3 2 3 2 ⎛ ⎛ 1 ⎞n ⎞ ⎛ ⎛ 1 ⎞n ⎜ 1− ⎜ ⎟ ⎟ ⎜1− ⎜ ⎟ 2 2 Sn = 3 ⎜ ⎝ ⎠ ⎟ = 3 ⎜ ⎝ ⎠ ⎜ ⎟ ⎜ 1 ⎛1⎞ ⎜⎜ 1 − ⎟ ⎜⎜ ⎜ ⎟ 2 ⎟ ⎝ ⎠ ⎝ ⎝2⎠

12. Neither. There is no common difference or common ratio. 13.

= 23n +3−3n = 23 = 8

23 n ⎞ ⎛ 1 − 8n ⎟⎟ = 8 ⎜⎜ ⎠ ⎝ −7

⎞ 8 n ⎟⎟ = 8 − 1 ⎠ 7

(

)

⎞ ⎟ n⎞ ⎛ ⎟ = 6 ⎜1 − ⎛ 1 ⎞ ⎟ ⎟ ⎜ ⎜⎝ 2 ⎟⎠ ⎟ ⎝ ⎠ ⎟⎟ ⎠

14.

50

50

k =1

k =1

30

30

30

k =1

k =1

k =1

∑ ( k 2 + 2) = ∑ ( k 2 ) + ∑ ( 2) =

10. 0, 4, 8, 12, ... Arithmetic d = 4 − 0 = 4 n n Sn = ( 2(0) + (n − 1)4 ) = ( 4(n − 1) ) = 2n(n − 1) 2 2

⎛ 50 ( 50 + 1) ⎞ ⎟ = 3825 2 ⎝ ⎠

∑ 3k = 3∑ k = 3 ⎜

30 ( 30 + 1)( 2 ⋅ 30 + 1) 6

+ 30(2)

= 9515

15.

40

40

k =1

k =1

40

∑ (− 2k + 8) = ∑ − 2k + ∑ 8 40

k =1 40

k =1

k =1

= − 2∑ k + ∑ 8 ⎛ 40(1 + 40) ⎞ = − 2⎜ ⎟ + 40(8) 2 ⎝ ⎠ = −1640 + 320 = −1320

1800

Sequences; Induction; the Binomial Theorem ⎛ ⎛ 1 ⎞7 ⎞ ⎛ ⎛ 1 ⎞7 ⎞ − 1 ⎜ ⎟ ⎜1− ⎜ ⎟ ⎟ ⎜ ⎟ 7 1 1 3 3 ⎛1⎞ 16. ∑ ⎜ ⎟ = ⎜ ⎝ ⎠ ⎟ = ⎜ ⎝ ⎠ ⎟ ⎜ ⎟ ⎜ 1 2 3 3 3 ⎛ ⎞ ⎟ k =1 ⎝ ⎠ ⎜⎜ 1 − ⎟⎟ ⎜⎜ ⎜ ⎟ ⎟⎟ 3 ⎝ ⎠ ⎝ ⎝3⎠ ⎠ 1⎛ 1 ⎞ = ⎜1 − ⎟ 2 ⎝ 2187 ⎠ 1 2186 1093 = ⋅ = ≈ 0.49977 2 2187 2187 k

17. Arithmetic a1 = 3, d = 4, an = a1 + (n − 1)d a9 = 3 + (9 − 1)4 = 3 + 8(4) = 3 + 32 = 35 18. Geometric

a1 = 1, r =

1 , n = 11; an = a1r n −1 10 11−1

⎛1⎞ a11 = 1 ⋅ ⎜ ⎟ ⎝ 10 ⎠ =

10

⎛1⎞ =⎜ ⎟ ⎝ 10 ⎠

a1 + 9d = 0 a1 + 17d = 8 Subtract the second equation from the first equation and solve for d. −8d = −8 d =1 a1 = −9(1) = −9

an = a1 + ( n − 1) d

= −9 + ( n − 1)(1) = −9 + n − 1

= n − 10 General formula:

1 3 Since r < 1, the series converges.

22. a1 = 3, r =

Sn =

1 10, 000, 000, 000

19. Arithmetic a1 = 2, d = 2, n = 9, an = a1 + (n − 1)d a9 = 2 + (9 − 1) 2 = 2 + 8 2 = 9 2 ≈ 12.7279

20. a7 = a1 + 6d = 31 a20 = a1 + 19d = 96 ; Solve the system of equations: a1 + 6d = 31 a1 + 19d = 96 Subtract the second equation from the first equation and solve for d. −13d = −65

d =5 a1 = 31 − 6(5) = 31 − 30 = 1

an = a1 + ( n − 1) d

1 2 Since r < 1 , the series converges.

Sn =

{an } = {5n − 4}

a1 2 2 4 = = = 1− r ⎛ ⎛ 1 ⎞⎞ ⎛ 3 ⎞ 3 ⎜1 − ⎜ − 2 ⎟ ⎟ ⎜ 2 ⎟ ⎠⎠ ⎝ ⎠ ⎝ ⎝

1 3 , r= 2 2 Since r > 1 , the series diverges.

24. a1 =

1 2 Since r < 1 , the series converges.

25. a1 = 4, r =

= 1 + 5n − 5 = 5n − 4 General formula:

a1 3 3 9 = = = 1 2 1− r ⎛ ⎞ ⎛ ⎞ 2 ⎜1 − ⎟ ⎜ ⎟ ⎝ 3⎠ ⎝ 3⎠

23. a1 = 2, r = −

Sn =

= 1 + ( n − 1)( 5 )

{an } = {n − 10}

26. I:

a1 4 4 = = =8 1− r ⎛ 1 ⎞ ⎛ 1 ⎞ ⎜1 − ⎟ ⎜ ⎟ ⎝ 2⎠ ⎝2⎠ n = 1: 3 ⋅1 = 3 and

3 ⋅1 (1 + 1) = 3 2

21. a10 = a1 + 9d = 0 a18 = a1 + 17 d = 8 ; Solve the system of equations:

1801

Sequences; Induction; the Binomial Theorem 3k (k + 1) , then 2 3 + 6 + 9 + " + 3k + 3(k + 1)

then

3k = (k + 1) + 3(k + 1) 2 ⎛ 3k ⎞ 3(k + 1) ((k + 1) + 1) = (k + 1) ⎜ + 3 ⎟ = 2 ⎝ 2 ⎠

=

II: If 3 + 6 + 9 + " + 3k =

12 + 42 + 7 2 + " + (3k − 2) 2 + ( 3(k + 1) − 2 ) = ⎡⎣12 + 42 + 7 2 + " + (3k − 2) 2 ⎤⎦ + (3k + 1) 2

= [3 + 6 + 9 + " + 3k ] + 3(k + 1)

1 ⋅ (k + 1) ⎡⎣ 6k 2 + 9k + 2 ⎤⎦ 2 1 = ⋅ ⎡⎣ 6k 3 + 6k 2 + 9k 2 + 9k + 2k + 2 ⎤⎦ 2 1 ⎡ 2 = ⋅ ⎣ 6k ( k + 1) + 9k ( k + 1) + 2 ( k + 1) ⎦⎤ 2 1 = ⋅ (k + 1) ⎡⎣ 6k 2 + 12k + 6 − 3k − 3 − 1⎤⎦ 2

n = 1: 2 ⋅ 31−1 = 2 and 31 − 1 = 2 2 + 6 + 18 + " + 2 ⋅ 3k −1 + 2 ⋅ 3k +1−1 = ⎡⎣ 2 + 6 + 18 + " + 2 ⋅ 3k −1 ⎤⎦ + 2 ⋅ 3k

1 ⋅ (k + 1) ⎡⎣ 6(k 2 + 2k + 1) − 3(k + 1) − 1⎤⎦ 2 1 = ⋅ (k + 1) ⎡⎣ 6(k + 1) 2 − 3(k + 1) − 1⎤⎦ 2 Conditions I and II are satisfied; the statement is true. =

= 3k − 1 + 2 ⋅ 3k = 3 ⋅ 3k − 1 = 3k +1 − 1 Conditions I and II are satisfied; the statement is true.

n = 1: (3 ⋅1 − 2) 2 = 1 and

1 ⋅1(6 ⋅12 − 3 ⋅1 − 1) = 1 2

II: If 12 + 42 + " + (3k − 2) 2 =

(

)

1 ⋅ k 6k 2 − 3k − 1 , 2

⎛5⎞ 5! 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 5 ⋅ 4 29. ⎜ ⎟ = = = = 10 ⎝ 2 ⎠ 2! 3! 2 ⋅1 ⋅ 3 ⋅ 2 ⋅1 2 ⋅1

⎛5⎞ ⎛5⎞ ⎛ 5⎞ ⎛ 5⎞ ⎛ 5⎞ ⎛ 5⎞ 30. ( x + 2)5 = ⎜ ⎟ x5 + ⎜ ⎟ x 4 ⋅ 2 + ⎜ ⎟ x3 ⋅ 22 + ⎜ ⎟ x 2 ⋅ 23 + ⎜ ⎟ x1 ⋅ 24 + ⎜ ⎟ ⋅ 25 0 1 2 3 4 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ 5⎠ = x 5 + 5 ⋅ 2 x 4 + 10 ⋅ 4 x3 + 10 ⋅ 8 x 2 + 5 ⋅16 x + 1 ⋅ 32 = x 5 + 10 x 4 + 40 x3 + 80 x 2 + 80 x + 32 ⎛ 4⎞ ⎛ 4⎞ ⎛ 4⎞ ⎛ 4⎞ ⎛ 4⎞ 31. (3x − 4) 4 = ⎜ ⎟ (3x) 4 + ⎜ ⎟ (3x)3 (− 4) + ⎜ ⎟ (3 x) 2 (− 4) 2 + ⎜ ⎟ (3 x)( − 4)3 + ⎜ ⎟ ( − 4) 4 ⎝0⎠ ⎝1⎠ ⎝ 2⎠ ⎝ 3⎠ ⎝ 4⎠ = 81x 4 + 4 ⋅ 27 x3 (− 4) + 6 ⋅ 9 x 2 ⋅16 + 4 ⋅ 3 x(− 64) + 1 ⋅ 256 = 81x 4 − 432 x3 + 864 x 2 − 768 x + 256

1802

)

=

II: If 2 + 6 + 18 + " + 2 ⋅ 3k −1 = 3k − 1 , then

28. I:

(

1 ⋅ k 6k 2 − 3k − 1 + (3k + 1) 2 2 1 = ⋅ ⎡⎣ 6k 3 − 3k 2 − k + 18k 2 + 12k + 2 ⎤⎦ 2 1 ⎡ 3 = ⋅ ⎣ 6k + 15k 2 + 11k + 2 ⎤⎦ 2

Conditions I and II are satisfied; the statement is true.

27. I:

2

Sequences; Induction; the Binomial Theorem 32. n = 9, j = 2, x = x, a = 2 ⎛9⎞ 7 2 9! 9 ⋅8 ⋅ 4 x7 = ⋅ 4 x 7 = 144 x 7 ⎜ ⎟ x ⋅2 = 2 ⋅ 2! 7! 2 1 ⎝ ⎠ 7

The coefficient of x is 144.

33. n = 7, j = 5, x = 2 x, a = 1 ⎛7⎞ 7! 7⋅6 2 5 ⋅ 4 x 2 (1) = ⋅ 4 x 2 = 84 x 2 ⎜ ⎟ (2 x) ⋅1 = 5 ⋅ 5! 2! 2 1 ⎝ ⎠ The coefficient of x 2 is 84.

34. This is an arithmetic sequence with a1 = 80, d = −3, n = 25 a. b.

a25 = 80 + (25 − 1)(−3) = 80 − 72 = 8 bricks 25 (80 + 8) = 25(44) = 1100 bricks 2 1100 bricks are needed to build the steps. S25 =

35. This is an arithmetic sequence with a1 = 30, d = −1, an = 15 15 = 30 + (n − 1)(−1) −15 = − n + 1 −16 = − n n = 16 16 S16 = (30 + 15) = 8(45) = 360 tiles 2 360 tiles are required to make the trapezoid.

36. This is a geometric sequence with 3 a1 = 20, r = . 4 a. After striking the ground the third time, the 3

⎛ 3 ⎞ 135 ≈ 8.44 feet . height is 20 ⎜ ⎟ = 16 ⎝4⎠

b. After striking the ground the n th time, the n

⎛3⎞ height is 20 ⎜ ⎟ feet . ⎝4⎠

c.

If the height is less than 6 inches or 0.5 feet, then:

⎛3⎞ 0.5 ≥ 20 ⎜ ⎟ ⎝4⎠ ⎛3⎞ 0.025 ≥ ⎜ ⎟ ⎝4⎠

n

n

⎛3⎞ log ( 0.025 ) ≥ n log ⎜ ⎟ ⎝4⎠ log ( 0.025 ) ≈ 12.82 n≥ ⎛3⎞ log ⎜ ⎟ ⎝4⎠ The height is less than 6 inches after the 13th strike. d. Since this is a geometric sequence with r < 1 , the distance is the sum of the two

infinite geometric series - the distances going down plus the distances going up. Distance going down: 20 20 Sdown = = = 80 feet. ⎛ 3⎞ ⎛1⎞ 1 − ⎜ ⎟ ⎜ ⎟ ⎝ 4⎠ ⎝4⎠ Distance going up: 15 15 Sup = = = 60 feet. 3 ⎛ ⎞ ⎛1⎞ 1 − ⎜ ⎟ ⎜ ⎟ ⎝ 4⎠ ⎝4⎠ The total distance traveled is 140 feet. 37. This is an ordinary annuity with P = \$200 and n = (12 )( 20 ) = 240 payment periods. The 0.10 . Thus, 12 ⎡ ⎡ 0.10 ⎤ 240 ⎤ − 1⎥ ⎢ ⎢1 + 12 ⎥⎦ ⎥ ≈ \$151,873.77 A = 200 ⎢ ⎣ ⎢ ⎥ 0.10 ⎢ ⎥ 12 ⎢⎣ ⎥⎦

interest rate per period is

38. This is a geometric sequence with a1 = 20, 000, r = 1.04, n = 5 . Find the fifth term of the sequence: a5 = 20, 000(1.04)5−1 = 20, 000(1.04) 4 = 23,397.17 Her salary in the fifth year will be \$23,397.17.

1803

Sequences; Induction; the Binomial Theorem

Chapter Test 1. an =

2 3 4 11 5. − + − + ... + 5 6 7 14 Notice that the signs of each term alternate, with the first term being negative. This implies that the general term will include a power of −1 . Also note that the numerator is always 1 more than the term number and the denominator is 4 more than the term number. Thus, each term is in k ⎛ k +1 ⎞ the form ( −1) ⎜ ⎟ . The last numerator is 11 ⎝k +4⎠ which indicates that there are 10 terms. 2 3 4 11 10 k ⎛ k +1 ⎞ − + − + ... + = ∑ ( −1) ⎜ ⎟ 5 6 7 14 k =1 ⎝k +4⎠

n2 − 1 n+8

12 − 1 0 = =0 1+ 8 9 22 − 1 3 = = 2 + 8 10 32 − 1 8 = = 3 + 8 11 42 − 1 15 5 = = = 4 + 8 12 4 52 − 1 24 = = 5 + 8 13

a1 = a2 a3 a4 a5

3 , The first five terms of the sequence are 0, 10 8 5 24 , , and . 13 11 4

2. a1 = 4; an = 3an −1 + 2

a3 = 3a2 + 2 = 3 (14 ) + 2 = 44

a4 = 3a3 + 2 = 3 ( 44 ) + 2 = 134

a5 = 3a4 + 2 = 3 (134 ) + 2 = 404 The first five terms of the sequence are 4, 14, 44, 134, and 404. 3

∑ ( −1) k =1

k +1⎞ ⎜ 2 ⎟ ⎝ k ⎠

k +1 ⎛

1+1 ⎛ 1 + 1 ⎞

2 +1 ⎛ 2 + 1 ⎞ 3+1 ⎛ 3 + 1 ⎞ ⎜ 2 ⎟ + ( −1) ⎜ 2 ⎟ + ( −1) ⎜ 2 ⎟ ⎝ 1 ⎠ ⎝ 2 ⎠ ⎝ 3 ⎠ 2⎛2⎞ 3⎛3⎞ 4⎛4⎞ = ( −1) ⎜ ⎟ + ( −1) ⎜ ⎟ + ( −1) ⎜ ⎟ ⎝1⎠ ⎝ 4⎠ ⎝9⎠ 3 4 61 = 2− + = 4 9 36

= ( −1)

4.

⎡ 2 k ⎤ ∑ ⎢⎛⎜⎝ 3 ⎞⎟⎠ − k ⎥ ⎥⎦ k =1 ⎢ ⎣ 4

()

⎡ 2

=⎢ ⎣ 3

1

() ⎣

⎤ ⎡ 2

− 1⎥ + ⎢

2

() ⎣

⎤ ⎡ 2

− 2⎥ + ⎢

3

() ⎣

⎤ ⎡ 2

− 3⎥ + ⎢

3 3 ⎦ ⎦ ⎦ 2 4 8 16 = −1+ − 2 + −3+ − 4 3 9 27 81 130 680 = − 10 = − 81 81

1804

12 = 2 and 36 = 3 6 12

The ratio of consecutive terms is not constant. Therefore, the sequence is not geometric.

a2 = 3a1 + 2 = 3 ( 4 ) + 2 = 14

3.

6. 6,12,36,144,... 12 − 6 = 6 and 36 − 12 = 24 The difference between consecutive terms is not constant. Therefore, the sequence is not arithmetic.

3

1 7. an = − ⋅ 4n 2 − 1 ⋅ 4n − 1 ⋅ 4n −1 ⋅ 4 an = 2 n −1 = 2 =4 an −1 − 1 ⋅ 4 − 12 ⋅ 4n −1 2 Since the ratio of consecutive terms is constant, the sequence is geometric with common ratio 1 r = 4 and first term a1 = − ⋅ 41 = −2 . 2 The sum of the first n terms of the sequence is given by 1− rn Sn = a1 ⋅ 1− r 1 − 4n = −2 ⋅ 1− 4 2 = 1 − 4n 3

(

)

8. −2, −10, −18, −26,... 4

− 4⎥

−10 − ( −2 ) = −8 , −18 − ( −10 ) = −8 , −26 − ( −18 ) = −8

The difference between consecutive terms is constant. Therefore, the sequence is arithmetic with common difference d = −8 and first term a1 = −2 .

Sequences; Induction; the Binomial Theorem an = a1 + ( n − 1) d

given by

= −2 + ( n − 1)( −8 ) = −2 − 8n + 8

= 6 − 8n The sum of the first n terms of the sequence is given by n Sn = ( a + an ) 2 n = ( −2 + 6 − 8n ) 2 n = ( 4 − 8n ) 2 = n ( 2 − 4n )

⎤ ⎡ n ⎤ ⎡ ( n − 1) = ⎢− + 7⎥ − ⎢− + 7⎥ 2 2 ⎣ ⎦ ⎣ ⎦

n n −1 = − +7+ −7 2 2 1 =− 2 The difference between consecutive terms is constant. Therefore, the sequence is arithmetic 1 with common difference d = − and first term 2 1 13 . a1 = − + 7 = 2 2 The sum of the first n terms of the sequence is given by n Sn = ( a1 + an ) 2 n ⎛ 13 ⎛ n ⎞⎞ = ⎜ + ⎜ − + 7⎟⎟ 2⎝ 2 ⎝ 2 ⎠⎠

=

n ⎛ 27 n ⎞ − 2 ⎜⎝ 2 2 ⎟⎠

=

n ( 27 − n ) 4

11. an =

2n − 3 2n + 1

an − an −1 = =

n 9. an = − + 7 2 an − an −1

n ⎛ ⎛ 2 ⎞n ⎞ 2⎞ ⎛ 1− 1− ⎜ ⎟ ⎜⎜ ⎜⎝ 5 ⎟⎠ ⎟⎟ 5⎠ 1− rn ⎝ ⎝ ⎠ = 25 ⋅ = 25 ⋅ Sn = a1 ⋅ 2 3 1− r 1− 5 5 n n ⎛ ⎞ ⎛ ⎞ 5 125 ⎛2⎞ ⎛2⎞ = 25 ⋅ ⎜1 − ⎜ ⎟ ⎟ = ⎜1 − ⎟ 3 ⎜⎝ ⎝ 5 ⎠ ⎟⎠ 3 ⎜⎝ ⎜⎝ 5 ⎟⎠ ⎟⎠

=

2n − 3 2 ( n − 1) − 3 2n − 3 2n − 5 − = − 2n + 1 2 ( n − 1) + 1 2n + 1 2n − 1

( 2n − 3)( 2n − 1) − ( 2n − 5 )( 2n + 1) ( 2n + 1)( 2n − 1)

( 4n

2

) (

− 8n + 3 − 4 n 2 − 8n − 5

)

2

4n − 1

8 4n 2 − 1 The difference of consecutive terms is not constant. Therefore, the sequence is not arithmetic. 2n − 3 an 2n − 3 2n − 1 ( 2n − 3)( 2n − 1) = 2n + 1 = ⋅ = an −1 2 ( n − 1) − 3 2n + 1 2n − 5 ( 2n + 1)( 2n − 5 ) 2 ( n − 1) + 1 =

The ratio of consecutive terms is not constant. Therefore, the sequence is not geometric. 12. For this geometric series we have r =

and a1 = 256 . Since r = −

−64 1 =− 256 4

1 1 = < 1 , the series 4 4

converges and we get a 256 256 1024 = 5 = S∞ = 1 = 1− r 1− − 1 5

( 4)

4

8 10. 25,10, 4, ,... 5 8

10 2 4 2 5 8 1 2 = , = , = ⋅ = 25 5 10 5 4 5 4 5 The ratio of consecutive terms is constant. Therefore, the sequence is geometric with common ratio r = 52 and first term a1 = 25 . The sum of the first n terms of the sequence is

1805

Sequences; Induction; the Binomial Theorem

13.

( 3m + 2 )5 = ⎛⎜ 0 ⎞⎟ ( 3m )5 + ⎛⎜ 1 ⎞⎟ ( 3m )4 ( 2 ) + ⎛⎜ 2 ⎞⎟ ( 3m )3 ( 2 )2 + ⎛⎜ 3 ⎞⎟ ( 3m )2 ( 2 )3 + ⎛⎜ 4 ⎞⎟ ( 3m )( 2 )4 + ⎛⎜ 5 ⎞⎟ ( 2 )5 5

5

⎝ ⎠

5

⎝ ⎠

5

⎝ ⎠

4

3

5

5

5

⎝ ⎠

⎝ ⎠

⎝ ⎠

2

= 243m + 5 ⋅ 81m ⋅ 2 + 10 ⋅ 27 m ⋅ 4 + 10 ⋅ 9m ⋅ 8 + 5 ⋅ 3m ⋅16 + 32 = 243m5 + 810m 4 + 1080m3 + 720m 2 + 240m + 32

14. First we show that the statement holds for n = 1 . ⎛ 1⎞ ⎜1 + 1 ⎟ = 1 + 1 = 2 ⎝ ⎠ ⎛ 1⎞⎛ 1 ⎞⎛ 1 ⎞ ⎛ 1 ⎞ The equality is true for n = 1 so Condition I holds. Next we assume that ⎜1 + ⎟ ⎜1 + ⎟ ⎜ 1 + ⎟ ... ⎜ 1 + ⎟ = n + 1 is ⎝ 1⎠⎝ 2 ⎠⎝ 3 ⎠ ⎝ n ⎠ true for some k, and we determine whether the formula then holds for k + 1 . We assume that ⎛ 1⎞⎛ 1 ⎞⎛ 1 ⎞ ⎛ 1 ⎞ ⎜1 + 1 ⎟ ⎜1 + 2 ⎟ ⎜ 1 + 3 ⎟ ... ⎜ 1 + k ⎟ = k + 1 . ⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞⎛ Now we need to show that ⎜1 + ⎟ ⎜1 + ⎟ ⎜ 1 + ⎟ ... ⎜1 + ⎟⎜ 1 + ⎟ = ( k + 1) + 1 = k + 2 . ⎝ 1 ⎠ ⎝ 2 ⎠ ⎝ 3 ⎠ ⎝ k ⎠⎝ k + 1 ⎠ We do this as follows: 1 ⎞ ⎡⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ ⎤ ⎛ 1 ⎞ ⎛ 1⎞⎛ 1 ⎞⎛ 1 ⎞ ⎛ 1 ⎞⎛ ⎜1 + 1 ⎟ ⎜1 + 2 ⎟ ⎜ 1 + 3 ⎟ ... ⎜1 + k ⎟ ⎜ 1 + k + 1 ⎟ = ⎢⎜ 1 + 1 ⎟ ⎜ 1 + 2 ⎟ ⎜ 1 + 3 ⎟ ... ⎜ 1 + k ⎟ ⎥ ⎜ 1 + k + 1 ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠⎦ ⎝ ⎠ 1 ⎛ ⎞ (using the induction assumption) = ( k + 1) ⎜1 + ⎟ ⎝ k +1⎠ 1 = ( k + 1) ⋅1 + ( k + 1) ⋅ = k +1+1 k +1 = k+2 Condition II also holds. Thus, formula holds true for all natural numbers.

15. The yearly values of the car form a geometric sequence with first term a1 = 31, 000 and common ratio r = 0.85 (which represents a 15% loss in value). an = 31, 000 ⋅ ( 0.85 )

n −1

The nth term of the sequence represents the value of the car at the beginning of the nth year. Since we want to know the value after 10 years, we are looking for the 11th term of the sequence. That is, the value of the car at the beginning of the 11th year. a11 = a1 ⋅ r11−1 = 31, 000 ⋅ ( 0.85 ) = 6,103.11 10

After 10 years, the car will be worth \$6,103.11. 16. The weights for each set form an arithmetic sequence with first term a1 = 100 and common difference d = 30 . If we imagine the weightlifter only performed one repetition per set, the total weight lifted in 5 sets would be the sum of the first five terms of the sequence.

1806

an = a1 + ( n − 1) d

a5 = 100 + ( 5 − 1)( 30 ) = 100 + 4 ( 30 ) = 220

Sn =

n ( a + an ) 2

S5 = 5 (100 + 220 ) = 5 ( 320 ) = 800 2

2

Since he performs 10 repetitions in each set, we multiply the sum by 10 to obtain the total weight lifted. 10 ( 800 ) = 8000 The weightlifter will have lifted a total of 8000 pounds after 5 sets.

Sequences; Induction; the Binomial Theorem

Chapter Projects

Project II

Project I – Internet Based Project

1. 2, 4, 8, 16, 32, 64

Answers will vary based on the year that is used. Data used in these solutions will be from 2008.

2. length n Î 2n levels This is a geometric sequence: an = 2n

1. I = net immigration = 887,168 Population for 2008 = 303,824,640 2. r = 0.01416 – 0.00826 = 0.0059

Recursive expression: an = 2an −1 , a0 = 1 3. 256 = 2n 28 = 2n

3. pn = (1 + 0.0059) pn −1 + 887168

n=8

pn = (1.0059) pn −1 + 887168 p0 = 303824640

4. p1 = (1.0059) p0 + 887168 p1 = (1.0059)(303824640) + 887168

Project III 1. Qst = −3 + 2 Pt −1 , Qdt = 18 − 3Pt P0 = 2, b = 2, d = 3, c = 18

p1 = 306,504,373 The population is predicted to be 306,504,373 in 2009.

−a = −3 → a = 3 3 + 18 − 2 Pt −1 21 − 2 Pt −1 = 3 3 2 Pt = 7 − Pt −1 , P0 = 2 3

5. Actual population in 2009: 307,212,123. The formula’s prediction was lower but fairly close. 6. Birth rate: 48.12 per 1000 population (0.04812) Death rate: 12.64 per 1000 population (0.01264) Population for 2008: 31,367,972 I = net immigration = −6587 r = 0.04812 − 0.01264 = 0.03548 pn = (1 + 0.03548) pn −1 − 6587

Pt =

2.

pn = (1.03548) pn −1 − 6587 p0 = 31,367,972 p1 = (1.03548) p0 − 6587

7

p1 = (1.03548)(31367972) − 6587 p1 = 32, 474,321 The population is predicted to be 32,474,321 in 2009. Actual population in 2009: 32,369,558. The formula’s prediction was higher but fairly close.

7. Answers will vary. This appears to support the article. The growth rate for the U.S. is much smaller than the growth rate for Uganda. 8. It could be but one must consider trends in each of the pieces of data to find if the growth rate is increasing or decreasing over time. The same thing must be examined with respect to the net immigration.

Pt 10

−10

Pt −1

−9

3. P1 =

17 3

Qs1 = −3 + 2(2) Qs1 = 1

⎛ 17 ⎞ Qd 1 = 18 − 3 ⎜ ⎟ ⎝ 3⎠ Qd 1 = 1

1807

Sequences; Induction; the Binomial Theorem

P2 =

y1 = 0

29 9

⎛ 17 ⎞ ⎛ 29 ⎞ Qs 2 = −3 + 2 ⎜ ⎟ Qd 2 = 18 − 3 ⎜ ⎟ ⎝ 3⎠ ⎝ 9 ⎠ 25 25 Qs 2 = Qd 2 = 3 3 The market (supply and demand) are getting closer to being the same.

4. The equilibrium price is 4.20. 5. It takes 17 time periods. 6. Qd 17 = 18 − 3(4.20) = 5.40

Qs17 = −3 + 2(4.20) = 5.40 The equilibrium quantity is 5.4.

y2 = 2.5 y3 = 5 y4 = 7.5 y5 = 10 5

∑ ( yr

i

i =1

− yi )

= (0 − 1) + (2.5 − 2) + (5 − 4) + (7.5 − 7) + (10 − 11) = −1 This is the sum of the errors.

5. y = 0.5 x 2 − 0.5 x + 1 The graph passes through all of the points. y6 = 16 y7 = 22 y8 = 29

Project IV 1. 1, 2, 4, 7, 11, 16, 22, 29

y1 = 1

2. It is not arithmetic because there is no common difference. It is not geometric because there is no common ration.

y3 = 4

y2 = 2 y4 = 7 y5 = 11

3. Scatter diagram 15

5

∑ ( yr − yi ) = 0 i =1

i

The sum of the errors is zero. −2

6 −2

4. y = 2.5 x − 2.5 The graph does not pass through any of the points. y6 = 12.5 y7 = 15 y8 = 17.5

6. When trying to obtain the cubic and quartic polynomials of best fit, the cubic and quartic terms have coefficient zero and the polynomial of best fit is given as the quadratic in part e. For the exponential function of best fit, y = (0.59)(1.83) x . y6 = 22.2 y7 = 40.6 y8 = 74.2 The sum of these errors becomes quite large. This error shows that the function does not fit the data very well as x gets larger. 7. The quadratic function is best. 8. The data does not appear to be either logarithmic or sinusoidal in shape, so it does not make sense to try to fit one of those functions to the data.

1808

Counting and Probability

From Chapter 13 of Student’s Solutions Manual for Precalculus Enhanced with Graphing Utilities, Sixth Edition. Michael Sullivan, Michael Sullivan, III. Copyright © 2013 by Pearson Education, Inc. All rights reserved.

1809

Counting and Probability 27. Let A = {those who will purchase a major appliance} and

Section 1

B = {those who will buy a car}

1. union 3. True; the union of two sets includes those elements that are in one or both of the sets. The intersection consists of the elements that are in both sets. Thus, the intersection is a subset of the union. 5. subset; ⊆ 7. n ( A ) + n ( B ) − n ( A ∩ B ) 9. ∅, {a} , {b} , {c} , {d } , {a, b} , {a, c} , {a, d } ,

{b, c} , {b, d } , {c, d } , {a, b, c} , {a, b, d } , {a, c, d } , {b, c, d } , {a, b, c, d }

n(U ) = 500, n( A) = 200, n( B ) = 150, n( A ∩ B ) = 25 n( A ∪ B) = n( A) + n( B) − n( A ∩ B)

= 200 + 150 − 25 = 325 n(purchase neither) = n (U ) − n ( A ∪ B ) = 500 − 325 = 175 n(purchase only a car) = n ( B ) − n ( A ∪ B ) = 150 − 25 = 125

29. Construct a Venn diagram:

11. n( A) = 15, n( B ) = 20, n( A ∩ B) = 10 n( A ∪ B) = n( A) + n( B) − n( A ∩ B)

15. From the figure: n( A) = 15 + 3 + 5 + 2 = 25 17. From the figure: n( A or B ) = n( A ∪ B ) = 15 + 2 + 5 + 3 + 10 + 2 = 37 19. From the figure: n( A but not C ) = n( A) − n( A ∩ C ) = 25 − 7 = 18 21. From the figure: n( A and B and C ) = n( A ∩ B ∩ C ) = 5 23. There are 5 choices of shirts and 3 choices of ties; there are (5)(3) = 15 different arrangements. 25. There are 9 choices for the first digit, and 10 choices for each of the other three digits. Thus, there are (9)(10)(10)(10) = 9000 possible fourdigit numbers.

1810

15

15

5 15

= 15 + 20 − 10 = 25 13. n( A ∪ B ) = 50, n( A ∩ B ) = 10, n( B ) = 20 n( A ∪ B ) = n( A) + n( B ) − n( A ∩ B ) 50 = n( A) + 20 − 10 40 = n( A)

AT&T

IBM

15

10 10

15 GE

(a) 15

(b) 15

(c) 15

(d) 25

(e) 40 31. a. n( widowed or divorced ) = n( widowed ) + n(divorced ) = 2.8 + 9.9 = 12.7 There were 12.7 million (12,700,000) males 18 years old and older who were widowed or divorced. b.

n(married, widowed or divorced) = n(married ) + n( widowed ) + n(divorced) = 64.8 + 2.8 + 9.9 = 77.5 There were 77.5 million (77,500,000) males

Counting and Probability

18 years old and older who were married, widowed, or divorced. 33. There are 8 choices for the DOW stocks, 15 choices for the NASDAQ stocks, and 4 choices for the global stocks. Thus, there are (8)(15)(4) = 480 different portfolios. 35. Answers will vary.

Section 2

23. {abc, abd , abe, acb, acd , ace, adb, adc, ade, aeb, aec, aed , bac, bad , bae, bca, bcd , bce, bda, bdc, bde, bea, bec, bed , cab, cad , cae, cba, cbd , cbe, cda, cdb, cde, cea, ceb, ced , dab, dac, dae, dba, dbc, dbe, dca, dcb, dce, dea, deb, dec, eab, eac, ead , eba, ebc, ebd , eca, ecb, ecd , eda, edb, edc} P (5,3) =

1. 1; 1 3. permutation 5. P (n, r ) =

n! (n − r )!

7. P (6, 2) =

6! 6! 6 ⋅ 5 ⋅ 4! = = = 30 (6 − 2)! 4! 4!

9. P (4, 4) =

4! 4! 4 ⋅ 3 ⋅ 2 ⋅1 = = = 24 (4 − 4)! 0! 1

11. P (7, 0) =

7! 7! = =1 (7 − 0)! 7!

8! 8! 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4! 13. P (8, 4) = = = = 1680 (8 − 4)! 4! 4! 8! 8! 8 ⋅ 7 ⋅ 6! 15. C (8, 2) = = = = 28 (8 − 2)! 2! 6! 2! 6! ⋅ 2 ⋅1

5! 5! 5 ⋅ 4 ⋅ 3 ⋅ 2! = = = 60 (5 − 3)! 2! 2!

25. {123, 124, 132, 134, 142, 143, 213, 214, 231, 234, 241, 243, 312, 314, 321, 324, 341, 342, 412, 413, 421, 423, 431, 432} P (4,3) =

4! 4! 4 ⋅ 3 ⋅ 2 ⋅1 = = = 24 (4 − 3)! 1! 1

27. {abc, abd , abe, acd , ace, ade, bcd , bce, bde, cde} C (5,3) =

5! 5 ⋅ 4 ⋅ 3! = = 10 (5 − 3)! 3! 2 ⋅1 ⋅ 3!

29. {123, 124, 134, 234} C (4,3) =

4! 4 ⋅ 3! = =4 (4 − 3)! 3! 1! 3!

31. There are 4 choices for the first letter in the code and 4 choices for the second letter in the code; there are (4)(4) = 16 possible two-letter codes.

7! 7! 7 ⋅ 6 ⋅ 5 ⋅ 4! = = = 35 (7 − 4)! 4! 3! 4! 4! ⋅ 3 ⋅ 2 ⋅1

33. There are two choices for each of three positions; there are (2)(2)(2) = 8 possible threedigit numbers.

19. C (15, 15) =

15! 15! 15! = = =1 (15 − 15)!15! 0!15! 15! ⋅1

21. C (26, 13) =

26! 26! = = 10, 400, 600 (26 − 13)!13! 13!13!

35. To line up the four people, there are 4 choices for the first position, 3 choices for the second position, 2 choices for the third position, and 1 choice for the fourth position. Thus there are (4)(3)(2)(1) = 24 possible ways four people can be lined up.

17. C (7, 4) =

37. Since no letter can be repeated, there are 5 choices for the first letter, 4 choices for the second letter, and 3 choices for the third letter. Thus, there are (5)(4)(3) = 60 possible threeletter codes. 39. There are 26 possible one-letter names. There are (26)(26) = 676 possible two-letter names.

1811

Counting and Probability

There are (26)(26)(26) = 17,576 possible threeletter names. Thus, there are 26 + 676 + 17,576 = 18,278 possible companies that can be listed on the New York Stock Exchange. 41. A committee of 4 from a total of 7 students is given by: 7! 7! 7 ⋅ 6 ⋅ 5 ⋅ 4! C (7, 4) = = = = 35 (7 − 4)! 4! 3! 4! 3 ⋅ 2 ⋅1 ⋅ 4! 35 committees are possible. 43. There are 2 possible answers for each question. Therefore, there are 210 = 1024 different possible arrangements of the answers.

≈ 1.157 × 1076

45. There are 5 choices for the first position, 4 choices for the second position, 3 choices for the third position, 2 choices for the fourth position, and 1 choice for the fifth position. Thus, there are (5)(4)(3)(2)(1) = 120 possible arrangements of the books.

57. There are 9 choices for the first position, 8 choices for the second position, 7 for the third position, etc. There are 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 = 9! = 362,880 possible batting orders.

47. The 1st person can have any of 365 days, the 2nd person can have any of the remaining 364 days. Thus, there are (365)(364) = 132,860 possible ways two people can have different birthdays.

59. The team must have 1 pitcher and 8 position players (non-pitchers). For pitcher, choose 1 player from a group of 4 players, i.e., C(4, 1). For position players, choose 8 players from a group of 11 players, i.e., C(11, 8). Thus, the number different teams possible is C (4,1) ⋅ C (11,8) = 4 ⋅165 = 660.

49. Choosing 2 boys from the 4 boys can be done C(4,2) ways. Choosing 3 girls from the 8 girls can be done in C(8,3) ways. Thus, there are a total of: 4! 8! C (4,2) ⋅ C (8,3) = ⋅ (4 − 2)! 2! (8 − 3)! 3! 4! 8! = ⋅ 2! 2! 5! 3! 4 ⋅ 3 ⋅ 2! 8 ⋅ 7 ⋅ 6 ⋅ 5! = ⋅ 2 ⋅ 1 ⋅ 2! 5! 3!

= 6 ⋅ 56 = 336

51. This is a permutation with repetition. There are 9! = 90, 720 different words. 2! 2!

1812

55. There are C(100, 22) ways to form the first committee. There are 78 senators left, so there are C(78, 13) ways to form the second committee. There are C(65, 10) ways to form the third committee. There are C(55, 5) ways to form the fourth committee. There are C(50, 16) ways to form the fifth committee. There are C(34, 17) ways to form the sixth committee. There are C(17, 17) ways to form the seventh committee. The total number of committees = C (100, 22) ⋅ C (78,13) ⋅ C (65,10) ⋅ C (55,5) ⋅ C (50,16) ⋅ C (34,17) ⋅ C (17,17)

53. a.

C (7, 2) ⋅ C (3,1) = 21 ⋅ 3 = 63

b.

C (7,3) ⋅ C (3, 0) = 35 ⋅1 = 35

c.

C (3,3) ⋅ C (7, 0) = 1 ⋅1 = 1

61. Choose 2 players from a group of 6 players. Thus, there are C (6, 2) = 15 different teams possible. 63. a.

b.

If numbers can be repeated, there are (50)(50)(50) = 125,000 different lock combinations. If no number can be repeated, then there are 50 ⋅ 49 ⋅ 48 = 117, 600 different lock combinations. Answers will vary. Typical combination locks require two full clockwise rotations to the first number, followed by a full counter-clockwise rotation past the first number to the second number, followed by a clockwise rotation to the third number (not past the second). This is not clear from the given directions. Perhaps a better name for a combination lock would be a permutation lock since the order in which the numbers are entered matters.

Counting and Probability

Section 3 1. equally likely 3. False; probability may equal 0. In such cases, the corresponding event will never happen. 5. Probabilities must be between 0 and 1, inclusive. Thus, 0, 0.01, 0.35, and 1 could be probabilities. 7. All the probabilities are between 0 and 1. The sum of the probabilities is 0.2 + 0.3 + 0.1 + 0.4 = 1. This is a probability model. 9. All the probabilities are between 0 and 1. The sum of the probabilities is 0.3 + 0.2 + 0.1 + 0.3 = 0.9. This is not a probability model. 11. The sample space is: S = { HH , HT , TH , TT } .

Each outcome is equally likely to occur; so P( E ) =

n( E ) . The probabilities are: n( S ) 1

1

1

1

4

4

4

4

P ( HH ) = , P ( HT ) = , P (TH ) = , P (TT ) =

.

13. The sample space of tossing two fair coins and a fair die is: S = {HH 1, HH 2, HH 3, HH 4, HH 5, HH 6, HT 1, HT 2, HT 3, HT 4, HT 5, HT 6, TH 1, TH 2, TH 3, TH 4, TH 5, TH 6, TT 1, TT 2, TT 3, TT 4, TT 5, TT 6} There are 24 equally likely outcomes and the

probability of each is

1 . 24

15. The sample space for tossing three fair coins is: S = {HHH , HHT , HTH , HTT , THH , THT , TTH , TTT } There are 8 equally likely outcomes and the

probability of each is

1 . 8

17. The sample space is: S = {1 Yellow, 1 Red, 1 Green, 2 Yellow, 2 Red, 2 Green, 3 Yellow, 3 Red, 3 Green, 4 Yellow, 4 Red, 4 Green} There are 12 equally likely events and the 1 . The probability of probability of each is 12

19. The sample space is: S = {1 Yellow Forward, 1 Yellow Backward, 1 Red Forward, 1 Red Backward, 1 Green Forward, 1 Green Backward, 2 Yellow Forward, 2 Yellow Backward, 2 Red Forward, 2 Red Backward, 2 Green Forward, 2 Green Backward, 3 Yellow Forward, 3 Yellow Backward, 3 Red Forward, 3 Red Backward, 3 Green Forward, 3 Green Backward, 4 Yellow Forward, 4 Yellow Backward, 4 Red Forward, 4 Red Backward, 4 Green Forward, 4 Green Backward} There are 24 equally likely events and the probability of each is 1 . The probability of 24 getting a 1, followed by a Red or Green, followed by a Backward is: P(1 Red Backward) + P(1 Green Backward)

=

1 1 1 + = 24 24 12

21. The sample space is: S = {1 1 Yellow, 1 1 Red, 1 1 Green, 1 2 Yellow, 1 2 Red, 1 2 Green, 1 3 Yellow, 1 3 Red, 1 3 Green, 1 4 Yellow, 1 4 Red, 1 4 Green, 2 1 Yellow, 2 1 Red, 2 1 Green, 2 2 Yellow, 2 2 Red, 2 2 Green, 2 3 Yellow, 2 3 Red, 2 3 Green, 2 4 Yellow, 2 4 Red, 2 4 Green, 3 1 Yellow, 3 1 Red, 3 1 Green, 3 2 Yellow, 3 2 Red, 3 2 Green, 3 3 Yellow, 3 3 Red, 3 3 Green, 3 4 Yellow, 3 4 Red, 3 4 Green, 4 1 Yellow, 4 1 Red, 4 1 Green, 4 2 Yellow, 4 2 Red, 4 2 Green, 4 3 Yellow, 4 3 Red, 4 3 Green, 4 4 Yellow, 4 4 Red, 4 4 Green} There are 48 equally likely events and the 1 . The probability of probability of each is 48 getting a 2, followed by a 2 or 4, followed by a Red or Green is P( 2 2 Red) + P (2 4 Red) + P (2 2 Green) + P (2 4 Green) 1 1 1 1 1 = + + + = 48 48 48 48 12 23. A, B, C, F 25. B

getting a 2 or 4 followed by a Red is 1 1 1 P(2 Red) + P(4 Red) = + = . 12 12 6

1813

Counting and Probability 27. Let P (tails) = x, then P (heads) = 4 x x + 4x = 1 5x = 1 1 x= 5 1 4 P (tails) = , P(heads) = 5 5 29. P (2) = P (4) = P(6) = x P (1) = P (3) = P (5) = 2 x P (1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1 2x + x + 2x + x + 2x + x = 1 9x = 1 1 x= 9 1 P (2) = P (4) = P (6) = 9 2 P(1) = P(3) = P(5) = 9 31. P( E ) =

n( E ) n{1, 2,3} 3 = = n( S ) 10 10

33. P( E ) =

n( E ) n{2, 4, 6,8,10} 5 1 = = = n( S ) 10 10 2

n(white) 5 5 1 = = = 35. P(white) = n( S ) 5 + 10 + 8 + 7 30 6

37. The sample space is: S = {BBB, BBG, BGB, GBB, BGG, GBG, GGB, GGG} n(3 boys) 1 P (3 boys) = = n( S ) 8 39. The sample space is: S = {BBBB, BBBG, BBGB, BGBB, GBBB, BBGG, BGBG, GBBG, BGGB, GBGB, GGBB, BGGG, GBGG, GGBG, GGGB, GGGG} n(1 girl, 3 boys) 4 1 P (1 girl, 3 boys) = = = n( S ) 16 4 n(sum of two dice is 7) n( S ) n{1,6 or 2,5 or 3,4 or 4,3 or 5,2 or 6,1} 6 1 = = = n( S ) 36 6

41. P (sum of two dice is 7) =

1814

n(sum of two dice is 3) n( S ) n{1,2 or 2,1} 2 1 = = = n( S ) 36 18

43. P (sum of two dice is 3) =

45. P ( A ∪ B ) = P ( A) + P( B) − P( A ∩ B) = 0.25 + 0.45 − 0.15 = 0.55 47. P ( A ∪ B ) = P ( A) + P ( B) = 0.25 + 0.45 = 0.70 49. P ( A ∪ B ) = P ( A) + P( B) − P( A ∩ B) 0.85 = 0.60 + P( B ) − 0.05 P ( B ) = 0.85 − 0.60 + 0.05 = 0.30 51. P (theft not cleared) = 1 − P ( theft cleared) = 1 − 0.13 = 0.87 53. P (does not own cat) = 1 − P (owns cat) = 1 − 0.34 = 0.66 55. P (never gambled online) = 1 − P (gambled online) = 1 − 0.05 = 0.95 57. P (white or green) = P(white) + P(green) n(white) + n(green) = n( S ) 9+8 17 = = 9 + 8 + 3 20 59. P (not white) = 1 − P (white) n(white) = 1− n( S ) 9 11 = 1− = 20 20 61. P (strike or one) = P (strike) + P (one) n(strike) + n(one) = n( S ) 3 +1 4 1 = = = 8 8 2

Counting and Probability 63. There are 30 households out of 100 with an income of \$30,000 or more. 30 3 n( E ) n(30, 000 or more) P( E ) = = = = n( S ) n(total households) 100 10 65. There are 40 households out of 100 with an income of less than \$20,000. n( E ) n(less than \$20,000) 40 2 P( E ) = = = = n( S ) n(total households) 100 5 67. a.

P (1 or 2) = P (1) + P (2) = 0.24 + 0.33 = 0.57

b.

P (1 or more) = 1 − P ( none ) = 1 − 0.05 = 0.95

c.

P (3 or fewer) = 1 − P ( 4 or more ) = 1 − 0.17 = 0.83

d.

P (3 or more) = P(3) + P (4 or more) = 0.21 + 0.17 = 0.38

e.

P (fewer than 2) = P (0) + P (1) = 0.05 + 0.24 = 0.29

f.

P (fewer than 1) = P (0) = 0.05

g.

P(1, 2, or 3) = P(1) + P(2) + P (3) = 0.24 + 0.33 + 0.21 = 0.78

h.

P(2 or more) = P(2) + P(3) + P(4 or more) = 0.33 + 0.21 + 0.17 = 0.71

69. a.

P (freshman or female) = P (freshman) + P (female) − P (freshman and female)

= =

b.

n(freshman) + n(female) − n(freshman and female)

n( S ) 18 + 15 − 8 25 = 33 33

P (sophomore or male) = P (sophomore) + P ( male) − P (sophomore and male)

= =

n(sophomore) + n( male) − n(sophomore and male) n( S )

15 + 18 − 8 25 = 33 33

71.

P(at least 2 with same birthday) = 1 − P (none with same birthday) n(different birthdays) = 1− n( S ) 365 ⋅ 364 ⋅ 363 ⋅ 362 ⋅ 361 ⋅ 360 ⋅⋅⋅ 354 = 1− 36512 ≈ 1 − 0.833 = 0.167

73. The sample space for picking 5 out of 10 numbers in a particular order contains 10! 10! P (10,5) = = = 30, 240 possible (10 − 5)! 5! outcomes. One of these is the desired outcome. Thus, the probability of winning is: n( E ) n(winning) P( E ) = = n( S ) n(total possible outcomes) 1 = ≈ 0.000033069 30, 240

Chapter Review Exercises 1. ∅, {Dave} , {Joanne} , {Erica} , {Dave, Joanne} ,

{Dave, Erica} , {Joanne, Erica} , {Dave, Joanne,Erica}

2. n( A) = 8, n( B) = 12, n( A ∩ B) = 3 n( A ∪ B) = n( A) + n( B) − n( A ∩ B)

= 8 + 12 − 3 = 17 3. n( A) = 12, n( A ∪ B) = 30, n( A ∩ B) = 6 n( A ∪ B ) = n( A) + n( B) − n( A ∩ B) 30 = 12 + n( B ) − 6 n( B ) = 30 − 12 + 6 = 24 4. From the figure: n( A) = 20 + 2 + 6 + 1 = 29 5. From the figure: n( A or B ) = 20 + 2 + 6 + 1 + 5 + 0 = 34 6. From the figure: n( A and C ) = n( A ∩ C ) = 1 + 6 = 7 7. From the figure: n(not in B) = 20 + 1 + 4 + 20 = 45

1815

Counting and Probability 8. From the figure: n(neither in A nor in C ) = n( A ∪ C ) = 20 + 5 = 25 9. From the figure: n(in B but not in C ) = 2 + 5 = 7 10. P (8,3) =

11. C (8,3) =

8! 8! 8 ⋅ 7 ⋅ 6 ⋅ 5! = = = 336 (8 − 3)! 5! 5! 8! 8! 8 ⋅ 7 ⋅ 6 ⋅ 5! = = = 56 (8 − 3)! 3! 5! 3! 5! ⋅ 3 ⋅ 2 ⋅1

12. There are 2 choices of material, 3 choices of color, and 10 choices of size. The complete assortment would have: 2 ⋅ 3 ⋅10 = 60 suits. 13. There are two possible outcomes for each game or 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 27 = 128 outcomes for 7 games. 14. Since order is significant, this is a permutation. 9! 9! 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5! P (9, 4) = = = = 3024 (9 − 4)! 5! 5! ways to seat 4 people in 9 seats. 15. Choose 4 runners –order is significant: 8! 8! 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4! P (8, 4) = = = = 1680 (8 − 4)! 4! 4! ways a squad can be chosen. 16. Choose 2 teams from 14–order is not significant: 14! 14! 14 ⋅13 ⋅12! C (14, 2) = = = = 91 (14 − 2)! 2! 12! 2! 12! ⋅ 2 ⋅1 ways to choose 2 teams. 17. There are 8 ⋅10 ⋅10 ⋅10 ⋅10 ⋅10 ⋅ 2 = 1, 600, 000 possible phone numbers. 18. There are 24 ⋅ 9 ⋅10 ⋅10 ⋅10 = 216, 000 possible license plates. 19. There are two choices for each digit, so there are 28 = 256 different numbers. (Note this allows numbers with initial zeros, such as 011.) 20. Since there are repeated colors: 10! 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 = = 12, 600 4! ⋅ 3! ⋅ 2! ⋅1! 4 ⋅ 3 ⋅ 2 ⋅1 ⋅ 3 ⋅ 2 ⋅1 ⋅ 2 ⋅1 ⋅ 1 different vertical arrangements.

1816

21. a.

C (9, 4) ⋅ C (9,3) ⋅ C (9, 2) = 126 ⋅ 84 ⋅ 36 = 381, 024 committees can be formed.

b.

C (9, 4) ⋅ C (5,3) ⋅ C (2, 2) = 126 ⋅10 ⋅1 = 1260 committees can be formed.

22. a.

365 ⋅ 364 ⋅ 363 ⋅⋅⋅⋅348 = 8.634628387 × 1045

b.

P (no one has same birthday) 365 ⋅ 364 ⋅ 363 ⋅⋅⋅ 348 = ≈ 0.6531 365 18

c.

P (at least 2 have same birthday) = 1 − P (no one has same birthday) = 1 − 0.6531 = 0.3469

23. a. b.

P (unemployed) = 0.088

P(not unemployed) = 1 − P(unemployed)

= 1 − 0.088 = 0.912 24. P(\$1 bill)=

n(\$1 bill) 4 = 9 n( S )

25. Let S be all possible selections, so n( S ) = 100 . Let D be a card that is divisible by 5, so n( D) = 20. Let PN be a card that is 1 or a prime number, so n( PN ) = 26 . n( D) 20 1 = = = 0.2 n( S ) 100 5 n( PN ) 26 13 P ( PN ) = = = = 0.26 n( S ) 100 50 P( D) =

26. Let S be all possible selections, let T be a car that needs a tune-up, and let B be a car that needs a brake job. a.

P ( Tune-up or Brake job )

= P (T ∪ B )

= P (T ) + P ( B ) − P (T ∩ B ) = 0.6 + 0.1 − 0.02 = 0.68

Counting and Probability b.

P ( Tune-up but not Brake job )

= P ( Tune-up ) − P ( Tune-up and Brake job ) = P (T ) − P (T ∩ B )

P ( Neither Tune-up nor Brake job )

= 1 − P (Tune-up or Brake job) = 1 − ( P (T ) + P( B ) − P(T ∩ B) ) = 1 − ( 0.6 + 0.1 − 0.02 ) = 0.32

1. From the figure: n ( physics ) = 4 + 2 + 7 + 9 = 22 2. From the figure: n ( biology or chemistry or physics ) = 22 + 8 + 2 + 4 + 9 + 7 + 15 = 67 Therefore, n ( none of the three ) = 70 − 67 = 3

3. From the figure: n ( only biology and chemistry ) = n ( biol. and chem.) − n ( biol. and chem. and phys.)

= (8 + 2 ) − 2 =8

4. From the figure: n ( physics or chemistry ) = 4 + 2 + 7 + 9 + 15 + 8 = 45

5. 7! = 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 = 5040 10! 10! = − 10 6 ! ( ) 4!

10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4! 4! = 10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6 ⋅ 5 = 151, 200 =

11 ⋅10 ⋅ 9 ⋅ 8 ⋅ 7 ⋅ 6! 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 ⋅ 6! 11 ⋅10 ⋅ 9 ⋅ 8 ⋅ 7 = 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 = 462

8. Since the order in which the colors are selected doesn’t matter, this is a combination problem. We have n = 21 colors and we wish to select r = 6 of them. 21! 21! C ( 21, 6 ) = = 6!( 21 − 6 ) ! 6!15! 21 ⋅ 20 ⋅19 ⋅18 ⋅17 ⋅16 ⋅15! 6!15! 21 ⋅ 20 ⋅19 ⋅18 ⋅17 ⋅16 = 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 = 54, 264 There are 54,264 ways to choose 6 colors from the 21 available colors. =

Chapter Test

6. P (10, 6 ) =

11! 11! = 5!(11 − 5 ) ! 5!6!

=

= 0.6 − 0.02 = 0.58

c.

7. C (11,5 ) =

9. Because the letters are not distinct and order matters, we use the permutation formula for nondistinct objects. We have four different letters, two of which are repeated (E four times and D two times). n! 8! = n1 !n2 !n3 !n4 ! 4!2!1!1! 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4! 4!⋅ 2 ⋅1 8⋅7 ⋅6⋅5 = 2 = 4⋅7⋅6⋅5 = 840 There are 840 distinct arrangements of the letters in the word REDEEMED. =

10. Since the order of the horses matters and all the horses are distinct, we use the permutation formula for distinct objects. 8! 8! 8 ⋅ 7 ⋅ 6! P ( 8, 2 ) = = = = 8 ⋅ 7 = 56 8 − 2 ! 6! ( ) 6!

There are 56 different exacta bets for an 8-horse race. 11. We are choosing 3 letters from 26 distinct letters and 4 digits from 10 distinct digits. The letters and numbers are placed in order following the

1817

Counting and Probability

format LLL DDDD with repetitions being allowed. Using the Multiplication Principle, we get 26 ⋅ 26 ⋅ 23 ⋅10 ⋅10 ⋅10 ⋅10 = 155, 480, 000 Note that there are only 23 possibilities for the third letter. There are 155,480,000 possible license plates using the new format. 12. Let A = Kiersten accepted at USC, and B = Kiersten accepted at FSU. Then, we get P ( A ) = 0.60 , P ( B ) = 0.70 , and P ( A ∩ B ) = 0.35 .

a.

Here we need to use the Addition Rule. P ( A ∪ B ) = P ( A) + P ( B ) − P ( A ∩ B )

= 0.60 + 0.70 − 0.35 = 0.95 Kiersten has a 95% chance of being admitted to at least one of the universities. b. Here we need the Complement of an event.

( )

P B = 1 − P ( B ) = 1 − 0.70 = 0.30

Kiersten has a 30% chance of not being admitted to FSU. 13. a.

b.

Since the bottle is chosen at random, all bottles are equally likely to be selected. Thus, 5 5 1 P ( Coke ) = = = = 0.25 8 + 5 + 4 + 3 20 4 There is a 25% chance that the selected bottle contains Coke. 8+3 11 = = 0.55 8 + 5 + 4 + 3 20 There is a 55% chance that the selected bottle contains either Pepsi or IBC. P ( Pepsi ∪ IBC ) =

14. Since the ages cover all possibilities and the age groups are mutually exclusive, the sum of all the probabilities must equal 1. 0.03 + 0.23 + 0.29 + 0.25 + 0.01 = 0.81 1 − 0.81 = 0.19 The given probabilities sum to 0.81. This means the missing probability (for 18-20) must be 0.19. 15. The number of different selections of 6 numbers is the number of ways we can choose 5 white balls and 1 red ball, where the order of the white balls is not important. This requires the use of the Multiplication Principle and the combination formula. Thus, the total number of distinct ways to pick the 6 numbers is given by

1818

n ( white balls ) ⋅ n ( red ball )

= C ( 53,5 ) ⋅ C ( 42,1) =

53! 42! 53! 42! ⋅ = ⋅ 5!⋅ ( 53 − 5 ) ! 1!⋅ ( 42 − 1) ! 5!⋅ 48! 1!⋅ 41!

⎛ 53 ⋅ 52 ⋅ 51 ⋅ 50 ⋅ 49 ⎞ ⎛ 42 ⋅ 41! ⎞ =⎜ ⎟⎜ ⎟ ⎝ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅1 ⎠ ⎝ 41! ⎠ 53 ⋅ 52 ⋅ 51 ⋅ 50 ⋅ 49 ⋅ 42 = = 120,526, 770 5⋅ 4 ⋅3⋅ 2 Since each possible combination is equally likely, the probability of winning on a \$1 play is 1 P ( win on \$1 play ) = 120,526, 770 ≈ 0.0000000083

16. The number of elements in the sample space can be obtained by using the Multiplication Principle: 6 ⋅ 6 ⋅ 6 ⋅ 6 ⋅ 6 = 7, 776 Consider the rolls as a sequence of 5 slots. The number of ways to position 2 fours in 5 slots is C ( 5, 2 ) . The remaining three slots can be filled

with any of the five remaining numbers from the die. Repetitions are allowed so this can be done in 5 ⋅ 5 ⋅ 5 = 125 different ways. Therefore, the total number of ways to get exactly 2 fours is 5! 5 ⋅ 4 ⋅125 C ( 5, 2 ) ⋅125 = ⋅125 = = 1250 2!⋅ 3! 2 The probability of getting exactly 2 fours on 5 rolls of a die is given by 1250 P ( exactly 2 fours ) = ≈ 0.1608 . 7776

Counting and Probability

Chapter Projects

4

16 ⎛2⎞ 2. P ( symbol received correctly ) = ⎜ ⎟ = 81 ⎝3⎠

Project I 1. Research will vary. See answer to part (c). 2. P ( win | not switched ) = P ( win | switched ) =

1 3

8

2 3

Results of simulations will vary. 3. In the Monty Hall Game, a curtain is selected by the contestant and left unopened. The host then reveals the contents behind one of the unselected curtains. In this situation, the host knows the contents behind the curtain being opened. The grand prize will never be revealed by the host.

In Deal or No Deal, a suitcase is selected by the contestant and left unopened. The contestant then chooses another unselected suitcase to open. In this situation, the content within the suitcase being opened is not known. Since the contestant selects the case to open, the grand prize may be revealed (and eliminated). 4. P(winning grand prize) = 5. Answers will vary. 6. Answers will vary.

3. # of received symbols with 2 bit errors: C (8, 2) = 28

1 26

256 ⎛2⎞ P ( received correctly ) = ⎜ ⎟ = 6561 ⎝3⎠ P ( received incorrectly ) = 1 − P ( received correctly ) 6305 6561

=

4. Let k = # of errors, n = 8 = length of symbol. Probability of k errors : ⎛n⎞ k n−k P (n, k ) = ⎜ ⎟ ( p ) (1 − p ) k ⎝ ⎠ k

8− k

⎛ 8⎞⎛ 1 ⎞ ⎛ 2 ⎞ P(8, k ) = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ k ⎠⎝ 3 ⎠ ⎝ 3 ⎠ Since this parity code only detects odd numbers of errors, P (error detected) = P (8,1) + P (8,3) + P (8,5) + P (8, 7) 1

7

3

⎛8⎞⎛ 1 ⎞ ⎛ 2 ⎞ ⎛8⎞⎛ 1 ⎞ ⎛ 2 ⎞ = ⎜ ⎟⎜ ⎟ ⎜ ⎟ + ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝1⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3⎠⎝ 3 ⎠ ⎝ 3 ⎠

5

5 3 7 1 ⎛8⎞⎛ 1 ⎞ ⎛ 2 ⎞ ⎛8⎞⎛ 1 ⎞ ⎛ 2 ⎞ + ⎜ ⎟⎜ ⎟ ⎜ ⎟ + ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝5⎠⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 7⎠⎝ 3 ⎠ ⎝ 3 ⎠

= 0.156464 + 0.272992 + 0.068044 + 0.002423

Project II 1. 0 bit errors: 1011

1 bit errors: 0011 1111 1001 1010 2 bit errors: 0111 0001 0010 1101 1110 1000 3 bit errors: 0110 0101 0000 1100 4 bit errors: 0100

= 0.499923 To find the probability that an error occurred but is not detected, we need to assume that an even number of errors occurred: P (error occured, but not detected) = P (8, 2) + P (8, 4) + P(8, 6) + P (8,8) 2

6

4

⎛8⎞⎛ 1 ⎞ ⎛ 2 ⎞ ⎛8⎞⎛ 1 ⎞ ⎛ 2 ⎞ = ⎜ ⎟⎜ ⎟ ⎜ ⎟ + ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ 2⎠⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 4⎠⎝ 3 ⎠ ⎝ 3 ⎠

4

6 2 8 0 ⎛ 8⎞⎛ 1 ⎞ ⎛ 2 ⎞ ⎛8⎞⎛ 1 ⎞ ⎛ 2 ⎞ + ⎜ ⎟⎜ ⎟ ⎜ ⎟ + ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ 6⎠⎝ 3 ⎠ ⎝ 3 ⎠ ⎝8⎠⎝ 3 ⎠ ⎝ 3 ⎠

= 0.273402 + 0.170364 + 0.016985 + 0.000151 = 0.460951

Project III

1819

Counting and Probability Project IV e. Answers will vary, depending on the L2 generated by the calculator. f. The data accumulates around y = 0.5. Project V

One simulation might be: Woman Woman told you has about Boy-Boy

Probability 1 4 1 4 1 4 1 4

Older boy

Boy-Boy Younger boy Boy-Girl Younger boy Girl-Boy

Older boy

We leave out the combinations where she would have to tell you about a girl. Thus, the probability that she has 2 boys is

Man has

1 1 1 + = . 4 4 2

Boy-Boy Older boy Girl-Boy Older boy

1 2 1 2

Thus the probability he has two boys is probabilities are the same.

1820

1 . The 2

A Preview of Calculus: The Limit, Derivative, and Integral of a Function Section 1 1.

⎧3x − 2 f ( x) = ⎨ ⎩3

if x ≠ 2 if x = 2

⎛ x2 − 4 x ⎞ 11. lim ⎜⎜ ⎟⎟ x →4 ⎝ x−4 ⎠

⎛ x2 − 4 x ⎞ lim ⎜⎜ ⎟⎟ = 4 x →4 ⎝ x−4 ⎠

(

)

(

)

13. lim e x + 1 x →0

3. lim f ( x ) x →c

5. True

( )

7. lim 4 x3 x →2

( )

lim 4 x3 = 32

x →2

⎛ x +1 ⎞ 9. lim ⎜ 2 ⎟ x →0 ⎝ x + 1 ⎠

lim e x + 1 = 2

x →0

⎛ cos x − 1 ⎞ 15. lim ⎜ ⎟ x →0 ⎝ x ⎠

⎛ cos x − 1 ⎞ lim ⎜ ⎟=0 x →0 ⎝ x ⎠

17. lim f ( x) = 3 x→2

The value of the function gets close to 3 as x gets close to 2. 19. lim f ( x) = 4 x→2

⎛ x +1 ⎞ lim ⎜ 2 ⎟ =1 ⎝ x +1⎠

x →0

The value of the function gets close to 4 as x gets close to 2. 21. lim f ( x) does not exist because as x gets close x →3

to 3, but is less than 3, f ( x) gets close to 3. However, as x gets close to 3, but is greater than 3, f ( x) gets close to 6. From Chapter 14 of Student’s Solutions Manual for Precalculus Enhanced with Graphing Utilities, Sixth Edition. Michael Sullivan, Michael Sullivan, III. Copyright © 2013 by Pearson Education, Inc. All rights reserved.

1821

A Preview of Calculus 23.

f ( x) = 3 x + 1

29.

lim ( sin x ) = 1

lim (3x + 1) = 13

x →4

x→

The value of the function gets close to 13 as x gets close to 4. 25.

π 2

The value of the function gets close to 1 as x gets close to π2 .

f ( x) = 1 − x 2

31.

(

f ( x) = sin x

)

f ( x) = e x

lim 1 − x 2 = −3

x →2

27.

( )

lim e x = 1

The value of the function gets close to −3 as x gets close to 2.

x →0

The value of the function gets close to 1 as x gets close to 0.

f ( x) = 2 x

33.

f ( x) =

1 x

lim 2 x = 6

x →−3

The value of the function gets close to 6 as x gets close to −3 .

1822

⎛1⎞ lim ⎜ ⎟ = −1 x⎠ The value of the function gets close to −1 as x gets close to −1 . x →−1 ⎝

A Preview of Calculus

35.

⎪⎧ x 2 f ( x) = ⎨ ⎪⎩2 x

x≥0 x0

lim f ( x) = 0

lim f ( x ) = 0

The value of the function gets close to 0 as x gets close to 0.

The value of the function gets close to 0 as x gets close to 0.

x →0

37.

41.

x ≤1 ⎧3x f ( x) = ⎨ ⎩x +1 x > 1

x →0

⎛ x3 − x 2 + x − 1 ⎞ 43. lim ⎜⎜ 4 ⎟ x →1 x − x 3 + 2 x − 2 ⎟ ⎝ ⎠

⎛ x3 − x 2 + x − 1 ⎞ lim ⎜⎜ 4 ⎟ ≈ 0.67 x →1 x − x 3 + 2 x − 2 ⎟ ⎝ ⎠

lim f ( x) does not exist x →1

The value of the function does not approach a single value as x approaches 1. For x < 1 , the function approaches the value 3, while for x > 1 the function approaches the value 2.

39.

⎧x ⎪ f ( x) = ⎨1 ⎪3x ⎩

⎛ x3 − 2 x 2 + 4 x − 8 ⎞ 45. lim ⎜⎜ ⎟⎟ x →2 x2 + x − 6 ⎝ ⎠

x0

⎛ x3 − 2 x 2 + 4 x − 8 ⎞ lim ⎜⎜ ⎟⎟ = 1.60 x →2 x2 + x − 6 ⎝ ⎠

lim f ( x) = 0 x →0

The value of the function gets close to 0 as x gets close to 0.

1823

A Preview of Calculus

47.

⎛ x3 + 2 x 2 + x ⎞ lim ⎜⎜ 4 ⎟ x →−1 x + x 3 + 2 x + 2 ⎟ ⎝ ⎠

x 2 − 4) 02 − 4 − 4 ⎛ x 2 − 4 ⎞ xlim( →0 = = = −1 25. lim ⎜⎜ 2 ⎟= x →0 x + 4 ⎟ 4 x 2 + 4) 02 + 4 ⎝ ⎠ xlim( →0

(

27. lim (3 x − 2)5 / 2 = lim (3 x − 2) x →2

x →2

= ( 3(2) − 2 )

1. product 3. c 5. False; the function may not be defined at 5.

31.

7. lim ( 5 ) = 5 x →1

9. lim ( x ) = 4 x→4

lim ( 5 x ) = 5 ( −2 ) = −10

x →−2

( )

4

x →1

lim (3x 2 − 5 x) = 3(−1) 2 − 5(−1) = 8

35.

x →−1

19. lim (5 x 4 − 3x 2 + 6 x − 9) = 5(1) 4 − 3(1) 2 + 6(1) − 9 x →1

= 5−3+6−9 = −1

x →1

x →1

)

23. lim 5 x + 4 = lim (5 x + 4) x →1

x →1

= 5(1) + 4 = 9 =3

1824

)

= 12 + 1 + 1 = 3

x →2

(

(

= lim x 2 + x + 1

15. lim (3 x + 2) = 3(2) + 2 = 8

21. lim ( x 2 + 1)3 = lim ( x 2 + 1)

⎛ x 2 − x − 12 ⎞ ⎛ ( x − 4)( x + 3) ⎞ lim ⎜⎜ ⎟⎟ = lim ⎜ ⎟ 2 x →−3 x →− 3 ⎝ ( x − 3)( x + 3) ⎠ ⎝ x −9 ⎠ ⎛ x−4⎞ = lim ⎜ ⎟ x →−3 ⎝ x − 3 ⎠ −3 − 4 − 7 7 = = = −3 − 3 − 6 6

⎛ x3 − 1 ⎞ ⎛ ( x − 1)( x 2 + x + 1) ⎞ 33. lim ⎜⎜ ⎟⎟ = lim ⎜⎜ ⎟⎟ x →1 x −1 ⎝ x − 1 ⎠ x →1 ⎝ ⎠

13. lim 5 x 4 = 4 ( 2 ) = 5 (16 ) = 80

17.

5/ 2

⎛ x2 − 4 ⎞ ⎛ ( x − 2)( x + 2) ⎞ 29. lim ⎜⎜ 2 ⎟ = lim ⎜ ⎟ x →2 x − 2 x ⎟ x→2 ⎝ x ( x − 2) ⎠ ⎝ ⎠ ⎛ x+2⎞ = lim ⎜ ⎟ x→2 ⎝ x ⎠ 2+2 4 = = =2 2 2

Section 2

x →2

5/ 2

= 45/ 2 = 32

⎛ x3 + 2 x 2 + x ⎞ lim ⎜⎜ 4 ⎟ = 0.00 x →−1 x + x 3 + 2 x + 2 ⎟ ⎝ ⎠

11.

)

3

(

)

3

= 12 + 1 = 23 = 8

⎛ ( x + 1) 2 lim ⎜⎜ 2 x →−1 ⎝ x −1

⎞ ⎛ ( x + 1) 2 ⎞ ⎟⎟ = xlim ⎜⎜ ⎟⎟ ⎠ →−1 ⎝ ( x − 1)( x + 1) ⎠ ⎛ x +1⎞ = lim ⎜ ⎟ x →−1 ⎝ x − 1 ⎠ −1 + 1 0 = = =0 −1 − 1 − 2

A Preview of Calculus

37.

⎛ x3 − x 2 + x − 1 ⎞ lim ⎜⎜ 4 ⎟ x →1 x − x 3 + 2 x − 2 ⎟ ⎝ ⎠ ⎛ x 2 ( x − 1) + 1( x − 1) ⎞ = lim ⎜⎜ 3 ⎟ x →1 x ( x − 1) + 2( x − 1) ⎟ ⎝ ⎠

⎛ x2 − 9 ⎞ ⎛ f ( x) − f (3) ⎞ 45. lim ⎜ = lim ⎜⎜ ⎟ ⎟ x →3 ⎝ x −3 ⎠ x →3 ⎝ x − 3 ⎟⎠ ⎛ ( x − 3)( x + 3) ⎞ = lim ⎜ ⎟ x →3 ⎝ x−3 ⎠ = lim ( x + 3)

⎛ ( x − 1)( x 2 + 1) ⎞ ⎛ x2 + 1 ⎞ = lim ⎜⎜ = lim ⎟ ⎜ ⎟⎟ ⎜ 3 x →1 ( x − 1)( x 3 + 2) ⎟ ⎝ ⎠ x →1 ⎝ x + 2 ⎠ =

12 + 1 3

1 +2

=

2 3

47.

⎛ x3 − 2 x 2 + 4 x − 8 ⎞ 39. lim ⎜⎜ ⎟⎟ x →2 x2 + x − 6 ⎝ ⎠ 2 ⎛ x ( x − 2) + 4( x − 2) ⎞ = lim ⎜⎜ ⎟⎟ x→2 ⎝ ( x + 3)( x − 2) ⎠ ⎛ ( x − 2)( x 2 + 4) ⎞ ⎛ x2 + 4 ⎞ = lim ⎜⎜ ⎟⎟ = lim ⎜⎜ ⎟⎟ x→2 ⎝ ( x + 3)( x − 2) ⎠ x →2 ⎝ x + 3 ⎠ =

41.

x →3

= 3+3 = 6

22 + 4 8 = 2+3 5

⎛ x3 + 2 x 2 + x ⎞ lim ⎜⎜ 4 ⎟ x →−1 x + x 3 + 2 x + 2 ⎟ ⎝ ⎠ ⎛ x( x 2 + 2 x + 1) ⎞ = lim ⎜⎜ 3 ⎟ x →−1 x ( x + 1) + 2( x + 1) ⎟ ⎝ ⎠

⎛ x 2 + 2 x − (−1) ⎞ ⎛ f ( x ) − f (−1) ⎞ lim ⎜ = lim ⎜⎜ ⎟⎟ ⎟ x →−1 ⎝ x − (−1) ⎠ x →−1 ⎝ x +1 ⎠ ⎛ x2 + 2 x + 1 ⎞ = lim ⎜⎜ ⎟⎟ x →−1 ⎝ x +1 ⎠ ⎛ ( x + 1) 2 = lim ⎜⎜ x →−1 ⎝ x +1 = lim ( x + 1)

⎞ ⎟⎟ ⎠

x →−1

= −1 + 1 = 0

⎛ 3x3 − 2 x 2 + 4 − 4 ⎞ ⎛ f ( x) − f (0) ⎞ 49. lim ⎜ = lim ⎜⎜ ⎟⎟ ⎟ x →0 ⎝ x−0 x ⎠ x →0 ⎝ ⎠ ⎛ 3x3 − 2 x 2 ⎞ = lim ⎜⎜ ⎟⎟ x →0 x ⎝ ⎠ = lim (3 x 2 − 2 x) = 0 x →0

⎛ x( x + 1) 2 ⎞ = lim ⎜⎜ ⎟ x →−1 ( x + 1)( x 3 + 2) ⎟ ⎝ ⎠ ⎛ x( x + 1) ⎞ −1(−1 + 1) = lim ⎜ 3 ⎟= x →−1 ⎝ x + 2 ⎠ (−1)3 + 2 −1(0) 0 = = =0 −1 + 2 1

⎛ f ( x) − f (2) ⎞ ⎛ (5 x − 3) − 7 ⎞ 43. lim ⎜ ⎟ = xlim ⎜ ⎟ x →2 ⎝ → 2 x−2 x−2 ⎠ ⎠ ⎝ ⎛ 5 x − 10 ⎞ = lim ⎜ ⎟ x→2 ⎝ x − 2 ⎠ ⎛ 5( x − 2) ⎞ = lim ⎜ ⎟ x→2 ⎝ x − 2 ⎠ = lim ( 5 ) = 5 x→2

1825

A Preview of Calculus ⎛1 ⎞ ⎛ 1− x ⎞ ⎜ x −1 ⎟ ⎜ x ⎟ ⎛ f ( x ) − f (1) ⎞ 51. lim ⎜ = lim ⎜ ⎟ = lim ⎜ ⎟ ⎟ x →1 ⎝ x −1 ⎠ x →1 ⎜ x − 1 ⎟ x →1 ⎜ x − 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ −1( x − 1) ⎞ ⎛ −1 ⎞ = lim ⎜⎜ ⎟⎟ = lim ⎜ ⎟ x →1 x →1 ⎝ x ⎠ 1 x x − ( ) ⎝ ⎠ −1 = = −1 1 ⎛ sin x ⎞ ⎜ ⎟ ⎛ tan x ⎞ ⎛ sin x 1 ⎞ = lim ⎜ cos x ⎟ = lim ⎜ ⋅ 53. lim ⎜ ⎟ ⎟ x →0 ⎝ x ⎠ x →0 x →0 ⎝ x cos x ⎠ x ⎜⎜ ⎟⎟ ⎝ ⎠ ⎛ ⎛ sin x ⎞ ⎞ ⎛ ⎛ 1 ⎞⎞ = ⎜ lim ⎜ ⎟ ⎟ ⋅ ⎜ xlim ⎜ ⎟⎟ x → 0 → 0 ⎝ x ⎠⎠ ⎝ ⎝ cos x ⎠ ⎠ ⎝ ⎛ lim 1 ⎞ ⎟ = 1⋅ 1 = 1 = 1 ⋅ ⎜ x →0 ⎜ lim ( cos x ) ⎟ 1 ⎝ x →0 ⎠ ⎛ 3sin x + cos x − 1 ⎞ 55. lim ⎜ ⎟ x →0 ⎝ 4x ⎠ ⎛ 3sin x cos x − 1 ⎞ = lim ⎜ + ⎟ x →0 ⎝ 4 x 4x ⎠ ⎛ 3sin x ⎞ ⎛ cos x − 1 ⎞ = lim ⎜ + lim ⎜ ⎟ ⎟ x →0 ⎝ 4 x ⎠ x →0 ⎝ 4x ⎠ 3 ⎛ sin x ⎞ 1 ⎛ cos x − 1 ⎞ = lim ⎜ ⎟ + lim ⎜ ⎟ 4 x → 0 ⎝ x ⎠ 4 x →0 ⎝ x ⎠ 3 1 3 = ⋅1 + ⋅ 0 = 4 4 4

17. 19. 21. 23.

11. True 13. Domain: [ −8, −6 ) ∪ ( −6, 4 ) ∪ ( 4, 6] or

{ x | −8 ≤ x < −6 or

1826

− 6 < x < 4 or 4 < x ≤ 6}

lim f ( x) = 1

x→2 −

lim f ( x) = 0 since lim− f ( x) = lim+ f ( x) = 0 x→4

x→4

27. f is not continuous at −6 because f ( −6 ) does

not exist, nor does the lim f ( x ) . x →−6

29. f is continuous at 0 because f (0) = lim− f ( x) = lim+ f ( x) = 3 x →0

x →0

31. f is not continuous at 4 because f ( 4 ) does not

exist. 33. lim+ (2 x + 3) = 2(1) + 3 = 5 x →1

(

)

35. lim− 2 x3 + 5 x = 2(1)3 + 5(1) = 2 + 5 = 7 x →1

37.

lim+ ( sin x ) = sin

x→

π 2

π =1 2

⎛ x2 − 4 ⎞ lim+ ⎜⎜ ⎟⎟ = lim+ x →2 ⎝ x − 2 ⎠ x →2 = lim+ x →2

⎛ ( x + 2)( x − 2) ⎞ ⎜ ⎟ x−2 ⎝ ⎠ ( x + 2)

= 2+2 = 4

41.

9. continuous, c

lim f ( x) = 2

x →−4 −

x →4

3. True

7. one-sided

lim f ( x) = ∞

x →−6 −

x →4

f ( 0 ) = 02 = 0 , f ( 2 ) = 5 − 2 = 3

5. True

f (− 8) = 0; f (− 4) = 2

25. lim f ( x) does exist.

39.

Section 3 1.

15. x-intercepts: –8, –5, –3

⎛ x2 − 1 ⎞ ⎛ ( x + 1)( x − 1) ⎞ lim − ⎜⎜ 3 ⎟⎟ = lim − ⎜ ⎟ 2 x →−1 ⎝ x + 1 ⎠ x →−1 ⎝ ( x + 1)( x − x + 1) ⎠ ⎛ x −1 ⎞ = lim − ⎜ 2 ⎟ x →−1 ⎝ x − x + 1 ⎠ 2 −1 − 1 = =− 2 3 (−1) − (−1) + 1

A Preview of Calculus

43.

45.

⎛ ( x + 2)( x − 1) ⎞ ⎜⎜ ⎟⎟ ⎝ x ( x + 2) ⎠ ⎛ x −1 ⎞ = lim + ⎜ ⎟ x →−2 ⎝ x ⎠ − 2 −1 − 3 3 = = = −2 −2 2

⎛ x2 + x − 2 ⎞ lim + ⎜⎜ 2 ⎟⎟ = lim + x →−2 ⎝ x + 2 x ⎠ x →−2

2.

⎛ x( x 2 + 3) ⎞ = lim− ⎜⎜ ⎟⎟ x →0 ⎝ x( x − 3) ⎠ ⎛ x2 + 3 ⎞ 3 = lim− ⎜⎜ = −1 ⎟⎟ = x →0 ⎝ x − 3 ⎠ −3 Since lim− f ( x) ≠ f (0) , the function is not

f ( x) = x3 − 3 x 2 + 2 x − 6; c = 2

1. 2. 3.

x →0

continuous at c = 0 .

f (2) = 23 − 3 ⋅ 22 + 2 ⋅ 2 − 6 = − 6 lim f ( x) = 23 − 3 ⋅ 22 + 2 ⋅ 2 − 6 = − 6

x → 2−

lim f ( x) = 23 − 3 ⋅ 22 + 2 ⋅ 2 − 6 = − 6

x → 2+

55.

Thus, f ( x ) is continuous at c = 2 . 47.

f ( x) =

x2 + 5 ; c=3 x−6

f (3) =

2.

lim− f ( x) =

x →3

2.

⎛ x3 + 3 x ⎞ lim− f ( x) = lim− ⎜⎜ 2 ⎟⎟ x →0 x →0 ⎝ x − 3x ⎠

⎧ x3 + 3 x if x ≠ 0 ⎪ f ( x) = ⎨ x 2 − 3 x ; ⎪1 if x = 0 ⎩ f (0) = 1

c=0

⎛ x2 + 3 ⎞ 3 = lim− ⎜⎜ = −1 ⎟⎟ = x →0 ⎝ x − 3 ⎠ −3 ⎛ x3 + 3x ⎞ lim+ f ( x ) = lim+ ⎜⎜ 2 ⎟⎟ x →0 x →0 ⎝ x − 3x ⎠ ⎛ x( x 2 + 3) ⎞ = lim+ ⎜⎜ ⎟⎟ x →0 ⎝ x( x − 3) ⎠

⎛ x2 + 3 ⎞ 3 = lim+ ⎜⎜ = −1 ⎟⎟ = x →0 ⎝ x − 3 ⎠ −3 The function is continuous at c = 0 .

x3 + 3 x

; c=0 x2 − 3x Since f ( x ) is not defined at c = 0 , the function is not continuous at c = 0 .

1.

f (0) = −1

3.

x+3 ; c=3 49. f ( x) = x −3 Since f ( x ) is not defined at c = 3 , the function is not continuous at c = 3 .

53.

1.

32 + 5 14 14 = =− 3 − 6 −3 3

lim+ f ( x) =

f ( x) =

c=0

⎛ x( x 2 + 3) ⎞ = lim− ⎜⎜ ⎟⎟ x →0 ⎝ x( x − 3) ⎠

32 + 5 14 14 = =− x →3 3 − 6 −3 3 Thus, f ( x ) is continuous at c = 3 .

51.

⎧ x3 + 3 x if x ≠ 0 ⎪ f ( x) = ⎨ x 2 − 3 x ; ⎪−1 if x = 0 ⎩

32 + 5 14 14 = =− 3 − 6 −3 3

1.

3.

⎛ x3 + 3 x ⎞ lim− f ( x) = lim− ⎜⎜ 2 ⎟⎟ x →0 x →0 ⎝ x − 3x ⎠

57.

⎧ x3 − 1 ⎪ 2 ⎪⎪ x − 1 f ( x) = ⎨2 ⎪ 3 ⎪ ⎪⎩ x + 1

if x < 1 if x = 1 ;

c =1

if x > 1

1.

f (1) = 2

2.

⎛ x3 − 1 ⎞ lim− f ( x) = lim− ⎜⎜ 2 ⎟⎟ x →1 x →1 ⎝ x −1 ⎠ ⎛ ( x − 1)( x 2 + x + 1) ⎞ = lim− ⎜⎜ ⎟⎟ x →1 ⎝ ( x − 1)( x + 1) ⎠ ⎛ x2 + x + 1 ⎞ 3 = lim− ⎜⎜ ⎟⎟ = x →1 ⎝ x +1 ⎠ 2

1827

A Preview of Calculus

Since lim− f ( x) ≠ f (1) , the function is not x →1

continuous at c = 1 . ⎧ ⎪2e x ⎪⎪ 59. f ( x) = ⎨2 ⎪ 3 2 ⎪ x + 2x ⎪⎩ x 2 1. f (0) = 2

2. 3.

2x + 5

Therefore, f ( x ) is continuous everywhere except at x = 2 and x = − 2 . f ( x) is discontinuous at x = 2 and x = − 2 .

c=0

if x > 0

71.

x →0

( )

⎛ x3 + 2 x 2 lim+ f ( x) = lim+ ⎜⎜ 2 x →0 x →0 ⎝ x

⎞ ⎟⎟ ⎠

⎛ x 2 ( x + 2) ⎞ = lim+ ⎜⎜ ⎟⎟ x →0 x2 ⎝ ⎠ = lim+ ( x + 2) = 0 + 2 = 2 x →0

The function is continuous at c = 0 . 61. The domain of f ( x) = 2 x + 3 is all real numbers, and f ( x ) is a polynomial function. Therefore, f ( x ) is continuous everywhere. 63. The domain of f ( x) = 3 x 2 + x is all real

numbers, and f ( x ) is a polynomial function. Therefore, f ( x ) is continuous everywhere. 65. The domain of f ( x) = 4sin x is all real numbers, and trigonometric functions are continuous at every point in their domains. Therefore, f ( x) is continuous everywhere. 67. The domain of f ( x) = 2 tan x is all real numbers π , and 2 trigonometric functions are continuous at every point in their domains. Therefore, f ( x ) is continuous everywhere except where kπ x= where k is an odd integer. f ( x ) is 2 kπ discontinuous at x = where k is an odd 2 integer.

except odd integer multiples of

1828

2x + 5 = . The domain of x 2 − 4 ( x − 2)( x + 2) f ( x ) is all real numbers except x = 2 and

f ( x) =

x = − 2 , and f ( x) is a rational function. if x < 0 if x = 0 ;

lim f ( x) = lim− 2e x = 2e0 = 2 ⋅1 = 2

x →0−

69.

x −3 . The domain of ln x f ( x) is (0, 1) or (1, ∞) . Thus, f ( x) is continuous on the interval (0, ∞) except at x = 1 . f ( x) is discontinuous at x = 1 . f ( x) =

73. R( x) = R is

x −1 2

x −1

{x

=

x −1 . The domain of ( x − 1)( x + 1)

x ≠ −1, x ≠ 1 } . Thus R is

discontinuous at both –1 and 1. ⎛ ⎞ x −1 lim − R ( x ) = lim − ⎜ ⎟ x →−1 x →−1 ⎝ ( x − 1)( x + 1) ⎠ ⎛ 1 ⎞ = lim − ⎜ ⎟ = −∞ x →−1 ⎝ x + 1 ⎠

since when 1 1 x < −1, < 0, and as x approaches − 1, x +1 x +1 becomes unbounded. ⎛ 1 ⎞ lim + R ( x) = lim + ⎜ ⎟ = ∞ since when x →−1 x →−1 ⎝ x + 1 ⎠ 1 1 > 0, and as x approaches − 1, x > −1, x +1 x +1 becomes unbounded. ⎛ 1 ⎞ 1 lim R ( x) = lim ⎜ ⎟ = . Note there is a hole x →1 x →1 ⎝ x + 1 ⎠ 2

A Preview of Calculus ⎛ 1⎞ in the graph at ⎜1, ⎟ . ⎝ 2⎠

77. R ( x) = =

x3 − x 2 + x − 1 x 4 − x3 + 2 x − 2 ( x − 1)( x 2 + 1) ( x − 1)( x 3 + 2)

=

x 2 ( x − 1) + 1( x − 1) x3 ( x − 1) + 2( x − 1)

x2 + 1

=

x3 + 2

, x ≠1

There is a vertical asymptote where x3 + 2 = 0 . x = − 3 2 is a vertical asymptote. There is a hole

( )

in the graph at x = 1 (at the point 1, 23 ). 79. R ( x) =

x3 − 2 x 2 + 4 x − 8 x2 + x − 6

=

x 2 ( x − 2) + 4( x − 2) ( x + 3)( x − 2)

( x − 2)( x 2 + 4) x 2 + 4 ,x≠2 = ( x + 3)( x − 2) x+3 There is a vertical asymptote where x + 3 = 0 . x = −3 is a vertical asymptote. There is a hole =

75. R ( x) = R is

x2 + x

{x

2

x −1

=

x( x + 1) . The domain of ( x − 1)( x + 1)

x ≠ −1, x ≠ 1 } . Thus R is

discontinuous at both –1 and 1. ⎛ x( x + 1) ⎞ ⎛ x ⎞ = lim− ⎜ lim− R( x) = lim− ⎜ ⎟ ⎟ = −∞ x →1 x →1 ⎝ ( x − 1)( x + 1) ⎠ x →1 ⎝ x − 1 ⎠ since when x x 0 < x < 1, < 0, and as x approaches 1, x −1 x −1 x becomes unbounded. lim+ R ( x ) = lim+ =∞ x →1 x →1 x − 1 since when x x x > 1, > 0, and as x approaches 1, x −1 x −1 becomes unbounded. ⎛ x ⎞ −1 1 lim R ( x) = lim ⎜ = . Note there ⎟= x →−1 x →−1 ⎝ x − 1 ⎠ −2 2 1⎞ ⎛ is a hole in the graph at ⎜ −1, ⎟ . 2⎠ ⎝

( )

in the graph at x = 2 (at the point 2, 85 ). 81. R( x) = =

x3 + 2 x 2 + x x 4 + x3 + 2 x + 2 x( x + 1) 2 3

( x + 1)( x + 2)

=

=

x( x 2 + 2 x + 1) x3 ( x + 1) + 2( x + 1)

x( x + 1) x3 + 2

, x ≠ −1

There is a vertical asymptote where x3 + 2 = 0 . x = − 3 2 is a vertical asymptote. There is a hole in the graph at x = −1 (at the point ( −1, 0 ) ). 83. R ( x) =

x3 − x 2 + x − 1 x 4 − x3 + 2 x − 2 3.1

–4.7

4.7

–3.1

1829

A Preview of Calculus

85. R( x) =

x3 − 2 x 2 + 4 x − 8

9.

x + x−6 10

–9.4

9.4

x →1

–25

87. R( x) =

f ( x) = 3x + 5 at (1, 8) ⎛ f ( x) − f (1) ⎞ ⎛ 3x + 5 − 8 ⎞ mtan = lim ⎜ = lim ⎜ ⎟ ⎟ x →1 ⎝ x −1 ⎠ x →1 ⎝ x − 1 ⎠ ⎛ 3x − 3 ⎞ ⎛ 3( x − 1) ⎞ = lim ⎜ ⎟ = lim ⎜ ⎟ x →1 ⎝ x − 1 ⎠ x →1 ⎝ x − 1 ⎠ = lim ( 3) = 3

2

Tangent Line: y − 8 = 3( x − 1) y − 8 = 3x − 3 y = 3x + 5

x3 + 2 x 2 + x x 4 + x3 + 2 x + 2 1

–4.7

4.7

–1

11.

f ( x) = x 2 + 2 at (−1, 3) ⎛ x2 + 2 − 3 ⎞ ⎛ f ( x) − f (−1) ⎞ mtan = lim ⎜ = lim ⎜⎜ ⎟ ⎟ x →−1 ⎝ x +1 ⎠ x →−1 ⎝ x + 1 ⎠⎟

89. Answers will vary. Three possible functions are: f ( x ) = x 2 ; g ( x ) = sin ( x ) ; h ( x ) = e x .

⎛ x2 − 1 ⎞ ⎛ ( x + 1)( x − 1) ⎞ = lim ⎜⎜ ⎟⎟ = lim ⎜ ⎟ x →−1 x →− 1 x +1 ⎝ ⎠ ⎝ x +1 ⎠ = lim ( x − 1) = −1 − 1 = − 2 x →−1

Section 4 1. Slope 5; containing point ( 2, −4 ) y − y1 = m ( x − x1 )

y − ( −4 ) = 5 ( x − 2 ) y + 4 = 5 x − 10 y = 5 x − 14

3. tangent line 5. velocity 7. True

1830

Tangent Line: y − 3 = − 2( x − (−1)) y − 3 = − 2x − 2 y = − 2x +1

A Preview of Calculus

13.

f ( x) = 3x 2 at (2, 12)

17.

⎛ 3x 2 − 12 ⎞ ⎛ f ( x) − f (2) ⎞ = mtan = lim ⎜ lim ⎟⎟ ⎟ x → 2 ⎜⎜ x→2 ⎝ x−2 ⎠ ⎝ x−2 ⎠ = lim

(

3 x2 − 4 x−2

f ( x ) − f (−1) x2 − 2 x + 3 − 6 = lim x →−1 x →−1 x +1 x +1 2 x − 2x − 3 ( x + 1)( x − 3) = lim = lim x →−1 x →−1 x +1 x +1 = lim ( x − 3) = −1 − 3 = − 4

mtan = lim

) = lim ⎛ 3( x + 2)( x − 2) ⎞

⎜ x−2 ⎝ = lim ( 3( x + 2) ) = 3(2 + 2) = 12 x→2

x→2

f ( x) = x 2 − 2 x + 3 at (−1, 6)

⎟ ⎠

x →−1

Tangent Line: y − 6 = − 4( x − (−1)) y − 6 = − 4x − 4

x→2

Tangent Line: y − 12 = 12( x − 2) y − 12 = 12 x − 24

y = − 4x + 2

y = 12 x − 12

15.

19.

2

f ( x) = 2 x + x at (1, 3)

⎛ 2x2 + x − 3 ⎞ ⎛ f ( x) − f (1) ⎞ = lim ⎜⎜ mtan = lim ⎜ ⎟ ⎟ x →1 ⎝ x −1 x − 1 ⎟⎠ ⎠ x →1 ⎝ ⎛ (2 x + 3)( x − 1) ⎞ = lim ⎜ (2 x + 3) ⎟ = lim x →1 ⎝ x −1 ⎠ x →1 = 2(1) + 3 = 5 Tangent Line: y − 3 = 5( x − 1) y − 3 = 5x − 5 y = 5x − 2

f ( x) = x3 + x at (2, 10) ⎛ x3 + x − 10 ⎞ ⎛ f ( x) − f (2) ⎞ = mtan = lim ⎜ lim ⎟⎟ ⎟ x → 2 ⎜⎜ x→2 ⎝ x−2 ⎠ ⎝ x−2 ⎠ ⎛ x3 − 8 + x − 2 ⎞ = lim ⎜⎜ ⎟⎟ x→2 x−2 ⎝ ⎠ ⎛ ( x − 2)( x 2 + 2 x + 4) + ( x − 2 ) ⋅1 ⎞ = lim ⎜ ⎟ ⎟ x→2 ⎜ x−2 ⎝ ⎠ ⎛ ( x − 2)( x 2 + 2 x + 4 + 1) ⎞ = lim ⎜⎜ ⎟⎟ x→2 x−2 ⎝ ⎠ = lim ( x 2 + 2 x + 5) = 4 + 4 + 5 = 13 x→2

Tangent Line: y − 10 = 13( x − 2) y − 10 = 13 x − 26 y = 13 x − 16

1831

A Preview of Calculus 21.

f ( x) = − 4 x + 5 at 3

27.

⎛ f ( x) − f (3) ⎞ f ′(3) = lim ⎜ ⎟ x →3 ⎝ x −3 ⎠ ⎛ − 4 x + 5 − (−7) ⎞ = lim ⎜ ⎟ x →3 ⎝ x−3 ⎠

⎛ f ( x) − f (−1) ⎞ f ′(−1) = lim ⎜ ⎟ x →−1 ⎝ x − (−1) ⎠ ⎛ x3 + 4 x − (−5) ⎞ = lim ⎜⎜ ⎟⎟ x →−1 x +1 ⎝ ⎠ ⎛ x3 + 1 + 4 x + 4 ⎞ = lim ⎜⎜ ⎟⎟ x →−1 x +1 ⎝ ⎠

⎛ − 4 x + 12 ⎞ = lim ⎜ ⎟ x →3 ⎝ x −3 ⎠ ⎛ − 4( x − 3) ⎞ = lim ⎜ ⎟ x →3 ⎝ x−3 ⎠ = lim (− 4) = − 4

⎛ ( x + 1)( x 2 − x + 1) + 4( x + 1) ⎞ = lim ⎜⎜ ⎟⎟ x →−1 x +1 ⎝ ⎠ ⎛ ( x + 1)( x 2 − x + 1 + 4) ⎞ = lim ⎜⎜ ⎟⎟ x →−1 x +1 ⎝ ⎠

x →3

23.

f ( x) = x 2 − 3 at 0 ⎛ f ( x) − f (0) ⎞ f ′(0) = lim ⎜ ⎟ x →0 ⎝ x−0 ⎠ ⎛ x − 3 − (−3) ⎞ = lim ⎜⎜ ⎟⎟ x →0 x ⎝ ⎠

= lim ( x 2 − x + 5) x →−1

= (−1) 2 − (−1) + 5 = 7

2

⎛x ⎞ = lim ⎜⎜ ⎟⎟ x →0 ⎝ x ⎠ = lim ( x ) = 0 2

29.

f ( x) = 2 x 2 + 3x at 1

⎛ f ( x) − f (1) ⎞ f ′(1) = lim ⎜ ⎟ x →1 ⎝ x −1 ⎠ ⎛ 2 x 2 + 3x − 5 ⎞ = lim ⎜⎜ ⎟⎟ x →1 x −1 ⎝ ⎠ ⎛ (2 x + 5)( x − 1) ⎞ = lim ⎜ ⎟ x →1 ⎝ x −1 ⎠ = lim (2 x + 5) = 7 x →1

f ( x) = x3 + x 2 − 2 x at 1 ⎛ f ( x) − f (1) ⎞ f ′(1) = lim ⎜ ⎟ x →1 ⎝ x −1 ⎠ ⎛ x3 + x 2 − 2 x − 0 ⎞ = lim ⎜⎜ ⎟⎟ x →1 x −1 ⎝ ⎠ 2 ⎛ x( x + x − 2) ⎞ = lim ⎜⎜ ⎟⎟ x →1 x −1 ⎝ ⎠ ⎛ x( x + 2)( x − 1) ⎞ = lim ⎜ ⎟ x →1 ⎝ x −1 ⎠ = lim ( x( x + 2) )

x →0

25.

f ( x) = x3 + 4 x at − 1

x →1

= 1(1 + 2) = 3

31.

f ( x) = sin x at 0 ⎛ f ( x) − f (0) ⎞ f ′(0) = lim ⎜ ⎟ x →0 ⎝ x−0 ⎠ ⎛ sin x − 0 ⎞ = lim ⎜ ⎟ x →0 ⎝ x − 0 ⎠ ⎛ sin x ⎞ = lim ⎜ ⎟ =1 x →0 ⎝ x ⎠

33. Use nDeriv:

1832

A Preview of Calculus 35. Use nDeriv:

4 3 πr at r = 2 3 ⎛ V (r ) − V (2) ⎞ V ′(2) = lim ⎜ ⎟ r →2 ⎝ r−2 ⎠ ⎛ 4 3 32 ⎞ ⎜ πr − 3 π ⎟ = lim ⎜ 3 ⎟ r →2 r −2 ⎜⎜ ⎟⎟ ⎝ ⎠ ⎛⎛4 ⎞⎞ 3 ⎜ ⎜ 3 π(r − 8) ⎟ ⎟ ⎠⎟ = lim ⎜ ⎝ r →2 ⎜ r−2 ⎟ ⎜ ⎟ ⎝ ⎠ ⎛⎛4 ⎞⎞ 2 ⎜ ⎜ 3 π(r − 2)(r + 2r + 4) ⎟ ⎟ ⎠⎟ = lim ⎜ ⎝ r →2 ⎜ r−2 ⎟ ⎜ ⎟ ⎝ ⎠ 4 ⎛ ⎞ = lim ⎜ π(r 2 + 2r + 4) ⎟ r →2 ⎝ 3 ⎠ 4 = π(4 + 4 + 4) = 16π 3 At the instant r = 2 feet, the volume of the sphere is increasing at a rate of 16π cubic feet per foot.

45. V (r ) =

37. Use nDeriv:

39. Use nDeriv:

41. Use nDeriv:

47. a. 43. V (r ) = 3π r 2

at r = 3

⎛ V (r ) − V (3) ⎞ V ′(3) = lim ⎜ ⎟ r →3 ⎝ r −3 ⎠ ⎛ 3π r 2 − 27π ⎞ = lim ⎜⎜ ⎟⎟ r →3 r −3 ⎝ ⎠ ⎛ 3π(r 2 − 9) ⎞ = lim ⎜⎜ ⎟⎟ r →3 ⎝ r −3 ⎠ ⎛ 3π(r − 3)(r + 3) ⎞ = lim ⎜ ⎟ r →3 ⎝ r −3 ⎠

b.

−16t 2 + 96t = 0 −16t (t − 6) = 0 t = 0 or t = 6 The ball strikes the ground after 6 seconds. Δ s s (2) − s (0) = Δt 2−0 −16(2) 2 + 96(2) − 0 2 128 = = 64 feet/sec 2 =

= lim ( 3π(r + 3) ) r →3

= 3π(3 + 3) = 18π At the instant r = 3 feet, the volume of the cylinder is increasing at a rate of 18π cubic feet per foot.

1833

A Preview of Calculus

c.

⎛ s ( t ) − s ( t0 ) ⎞ s ′ ( t0 ) = lim ⎜⎜ ⎟⎟ t → t0 t − t0 ⎝ ⎠

b.

(

⎛ −16t 2 + 96t − −16t0 2 + 96t0 = lim ⎜ t → t0 ⎜ t − t0 ⎝

945 − 987 2 − 42 = 2 = − 21 feet/sec

) ⎞⎟

=

⎟ ⎠

⎞ ⎟⎟ ⎠ ⎛ −16 t 2 − t0 2 + 96 ( t − t0 ) ⎞ ⎟ = lim ⎜ ⎟ t →t0 ⎜ t − t0 ⎝ ⎠ ⎛ −16 ( t − t0 )( t + t0 ) + 96 ( t − t0 ) ⎞ = lim ⎜⎜ ⎟⎟ t →t0 t − t0 ⎝ ⎠ ⎛ ( t − t0 ) ( −16 ( t + t0 ) + 96 ) ⎞ ⎟ = lim ⎜ ⎟ t →t0 ⎜ t t − 0 ⎝ ⎠

⎛ −16t 2 + 16t0 2 + 96t − 96t0 = lim ⎜ t →t0 ⎜ t − t0 ⎝

(

)

c.

= ( −16 ( t0 + t0 ) + 96 ) = −32t0 + 96 ft/sec The instantaneous speed at time t is −32t + 96 feet per second.

d.

s ′(2) = −32(2) + 96 = − 64 + 96 = 32 feet/sec

e.

s ′(t ) = 0 −32t + 96 = 0 −32t = −96 t = 3 seconds

f.

s (3) = −16(3) 2 + 96(3) = −144 + 288 = 144 feet

g.

s ′(6) = −32(6) + 96 = −192 + 96 = −96 feet/sec

49. a.

Δ s s (4) − s (1) = Δt 4 −1 917 − 987 3 −70 = 3 1 = − 23 feet/sec 3 =

1834

Δ s s (2) − s (1) = Δt 2 −1 969 − 987 1 − 18 = = − 18 feet/sec 1 =

d.

= lim ( −16 ( t + t0 ) + 96 ) t →t0

Δ s s (3) − s (1) = Δt 3 −1

e.

s ( t ) = −2.631t 2 − 10.269t + 999.933

⎛ s ( t ) − s (1) ⎞ s ′ (1) = lim ⎜ ⎟ t →1 t −1 ⎠ ⎝ ⎛ − 2.631t 2 − 10.269 t + 12.9 ⎞ = lim ⎜⎜ ⎟⎟ t →1 t −1 ⎝ ⎠ ⎛ − 2.631t 2 + 2.631t − 12.9 t + 12.9 ⎞ = lim ⎜⎜ ⎟⎟ t →1 t −1 ⎝ ⎠ ⎛ − 2.631t (t − 1) − 12.9 (t − 1) ⎞ = lim ⎜ ⎟ t →1 ⎝ t −1 ⎠ ⎛ (− 2.631t − 12.9) (t − 1) ⎞ = lim ⎜ ⎟ t →1 ⎝ t −1 ⎠ = lim (− 2.631t − 12.9) t →1

= − 2.631(1) − 12.9 = −15.531 feet/sec At the instant when t = 1 , the instantaneous speed of the ball is –15.531 feet / sec.

A Preview of Calculus

Section 5 1. A = lw 3.

b

∫a f ( x ) dx

5. A ≈ f (1) ⋅1 + f (2) ⋅1 = 1 ⋅1 + 2 ⋅1 = 1 + 2 = 3 7. A ≈ f (0) ⋅ 2 + f (2) ⋅ 2 + f (4) ⋅ 2 + f (6) ⋅ 2 = 10 ⋅ 2 + 6 ⋅ 2 + 7 ⋅ 2 + 5 ⋅ 2 = 20 + 12 + 14 + 10 = 56 9. a.

Graph f ( x) = 3 x :

b.

A ≈ f (0)(2) + f (2)(2) + f (4)(2) = 0(2) + 6(2) + 12(2) = 0 + 12 + 24 = 36

c.

A ≈ f (2)(2) + f (4)(2) + f (6)(2) = 6(2) + 12(2) + 18(2) = 12 + 24 + 36 = 72

d.

A ≈ f (0)(1) + f (1)(1) + f (2)(1) + f (3)(1) + f (4)(1) + f (5)(1) = 0(1) + 3(1) + 6(1) + 9(1) + 12(1) + 15(1) = 0 + 3 + 6 + 9 + 12 + 15 = 45

e.

A ≈ f (1)(1) + f (2)(1) + f (3)(1) + f (4)(1) + f (5)(1) + f (6)(1) = 3(1) + 6(1) + 9(1) + 12(1) + 15(1) + 18(1) = 3 + 6 + 9 + 12 + 15 + 18 = 63

f.

The actual area is the area of a triangle: A =

1 (6)(18) = 54 2

1835

A Preview of Calculus 11. a.

b.

A ≈ f (0)(1) + f (1)(1) + f (2)(1) = 9(1) + 6(1) + 3(1) = 9 + 6 + 3 = 18

c.

A ≈ f (1)(1) + f (2)(1) + f (3)(1) = 6(1) + 3(1) + 0(1) = 6+3+ 0 = 9

d.

e.

f.

1836

Graph f ( x) = −3 x + 9 :

⎛1⎞ ⎛ 1 ⎞⎛ 1 ⎞ ⎛1⎞ ⎛ 3 ⎞⎛ 1 ⎞ ⎛1⎞ A ≈ f (0) ⎜ ⎟ + f ⎜ ⎟ ⎜ ⎟ + f (1) ⎜ ⎟ + f ⎜ ⎟ ⎜ ⎟ + f (2) ⎜ ⎟ + 2 2 2 2 2 2 ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ 2⎠ ⎛ 1 ⎞ 15 ⎛ 1 ⎞ ⎛ 1 ⎞ 9 ⎛ 1 ⎞ ⎛ 1 ⎞ 3 ⎛ 1 ⎞ = 9 ⎜ ⎟ + ⎜ ⎟ + 6 ⎜ ⎟ + ⎜ ⎟ + 3⎜ ⎟ + ⎜ ⎟ ⎝ 2⎠ 2 ⎝ 2⎠ ⎝ 2⎠ 2⎝ 2⎠ ⎝ 2⎠ 2⎝ 2⎠ 9 15 9 3 3 63 = + +3+ + + = = 15.75 2 4 4 2 4 4 ⎛ 1 ⎞⎛ 1 ⎞ ⎛1⎞ ⎛ 3 ⎞⎛ 1 ⎞ ⎛1⎞ A ≈ f ⎜ ⎟ ⎜ ⎟ + f (1) ⎜ ⎟ + f ⎜ ⎟ ⎜ ⎟ + f (2) ⎜ ⎟ + ⎝ 2 ⎠⎝ 2 ⎠ ⎝2⎠ ⎝ 2 ⎠⎝ 2 ⎠ ⎝ 2⎠ 15 ⎛ 1 ⎞ ⎛ 1 ⎞ 9 ⎛ 1 ⎞ ⎛ 1 ⎞ 3 ⎛ 1 ⎞ ⎛ 1 ⎞ = ⎜ ⎟ + 6 ⎜ ⎟ + ⎜ ⎟ + 3⎜ ⎟ + ⎜ ⎟ + 0 ⎜ ⎟ 2 ⎝ 2⎠ ⎝ 2⎠ 2⎝ 2⎠ ⎝ 2⎠ 2⎝ 2⎠ ⎝ 2⎠ 15 9 3 3 45 = +3+ + + + 0 = = 11.25 4 4 2 4 4

The actual area is the area of a triangle: A =

⎛ 5 ⎞⎛ 1 ⎞ f ⎜ ⎟⎜ ⎟ ⎝ 2 ⎠⎝ 2 ⎠

⎛ 5 ⎞⎛ 1 ⎞ ⎛1⎞ f ⎜ ⎟ ⎜ ⎟ + f ( 3) ⎜ ⎟ ⎝ 2 ⎠⎝ 2 ⎠ ⎝ 2⎠

1 27 (3)(9) = = 13.5 2 2

A Preview of Calculus

13. a.

b. c.

Graph f ( x) = x 2 + 2, [0, 4] :

A ≈ f (0)(1) + f (1)(1) + f (2)(1) + f (3)(1) = 2(1) + 3(1) + 6(1) + 11(1) = 2 + 3 + 6 + 11 = 22 ⎛1⎞ ⎛ 1 ⎞⎛ 1 ⎞ ⎛1⎞ ⎛ 3 ⎞⎛ 1 ⎞ ⎛1⎞ ⎛ 5 ⎞⎛ 1 ⎞ ⎛1⎞ A ≈ f (0) ⎜ ⎟ + f ⎜ ⎟ ⎜ ⎟ + f (1) ⎜ ⎟ + f ⎜ ⎟⎜ ⎟ + f (2) ⎜ ⎟ + f ⎜ ⎟⎜ ⎟ + f (3) ⎜ ⎟ + ⎝2⎠ ⎝ 2 ⎠⎝ 2 ⎠ ⎝2⎠ ⎝ 2 ⎠⎝ 2 ⎠ ⎝ 2⎠ ⎝ 2 ⎠⎝ 2 ⎠ ⎝ 2⎠ ⎛ 1 ⎞ 9 ⎛ 1 ⎞ ⎛ 1 ⎞ 17 ⎛ 1 ⎞ ⎛ 1 ⎞ 33 ⎛ 1 ⎞ ⎛ 1 ⎞ 57 ⎛ 1 ⎞ = 2 ⎜ ⎟ + ⎜ ⎟ + 3 ⎜ ⎟ + ⎜ ⎟ + 6 ⎜ ⎟ + ⎜ ⎟ + 11⎜ ⎟ + ⎜ ⎟ ⎝ 2⎠ 4⎝ 2⎠ ⎝ 2⎠ 4 ⎝ 2⎠ ⎝ 2⎠ 4 ⎝ 2⎠ ⎝ 2⎠ 4 ⎝ 2⎠ 9 3 17 33 11 57 51 = 1+ + + + 3 + + + = = 25.5 8 2 8 8 2 8 2 4

d.

A = ∫0 ( x 2 + 2)dx

e.

Use fnInt function:

15. a.

b.

⎛ 7 ⎞⎛ 1 ⎞ f ⎜ ⎟⎜ ⎟ ⎝ 2 ⎠⎝ 2 ⎠

Graph f ( x) = x3 , [0, 4] :

A ≈ f (0)(1) + f (1)(1) + f (2)(1) + f (3)(1) = 0(1) + 1(1) + 8(1) + 27(1) = 0 + 1 + 8 + 27 = 36

1837

A Preview of Calculus

c.

A = ∫0 x3 dx

e.

Use fnInt function:

b. c.

d.

⎛ 7 ⎞⎛ 1 ⎞ f ⎜ ⎟⎜ ⎟ ⎝ 2 ⎠⎝ 2 ⎠

4

d.

17. a.

1838

⎛1⎞ ⎛ 1 ⎞⎛ 1 ⎞ ⎛1⎞ ⎛ 3 ⎞⎛ 1 ⎞ ⎛1⎞ ⎛ 5 ⎞⎛ 1 ⎞ ⎛1⎞ A ≈ f (0) ⎜ ⎟ + f ⎜ ⎟ ⎜ ⎟ + f (1) ⎜ ⎟ + f ⎜ ⎟⎜ ⎟ + f (2) ⎜ ⎟ + f ⎜ ⎟⎜ ⎟ + f (3) ⎜ ⎟ + ⎝2⎠ ⎝ 2 ⎠⎝ 2 ⎠ ⎝2⎠ ⎝ 2 ⎠⎝ 2 ⎠ ⎝ 2⎠ ⎝ 2 ⎠⎝ 2 ⎠ ⎝ 2⎠ ⎛ 1 ⎞ 1 ⎛ 1 ⎞ ⎛ 1 ⎞ 27 ⎛ 1 ⎞ ⎛ 1 ⎞ 125 ⎛ 1 ⎞ ⎛ 1 ⎞ 343 ⎛ 1 ⎞ = 0 ⎜ ⎟ + ⎜ ⎟ + 1⎜ ⎟ + ⎜ ⎟ + 8 ⎜ ⎟ + ⎜ ⎟ + 27 ⎜ ⎟ + ⎜ ⎟ ⎝ 2⎠ 8⎝ 2⎠ ⎝ 2⎠ 8 ⎝ 2⎠ ⎝ 2⎠ 8 ⎝ 2⎠ ⎝2⎠ 8 ⎝2⎠ 1 1 27 125 27 343 = 0+ + + +4+ + + = 49 16 2 16 16 2 16

Graph f ( x) =

1 , [1, 5] : x

1 1 1 1 1 1 25 A ≈ f (1)(1) + f (2)(1) + f (3)(1) + f (4)(1) = 1(1) + (1) + (1) + (1) = 1 + + + = 2 3 4 2 3 4 12 ⎛1⎞ ⎛ 3 ⎞⎛ 1 ⎞ ⎛1⎞ ⎛ 5 ⎞⎛ 1 ⎞ ⎛1⎞ ⎛ 7 ⎞⎛ 1 ⎞ ⎛1⎞ A ≈ f (1) ⎜ ⎟ + f ⎜ ⎟ ⎜ ⎟ + f (2) ⎜ ⎟ + f ⎜ ⎟ ⎜ ⎟ + f (3) ⎜ ⎟ + f ⎜ ⎟ ⎜ ⎟ + f (4) ⎜ ⎟ + ⎝2⎠ ⎝ 2 ⎠⎝ 2 ⎠ ⎝2⎠ ⎝ 2 ⎠⎝ 2 ⎠ ⎝ 2⎠ ⎝ 2 ⎠⎝ 2 ⎠ ⎝ 2⎠ ⎛ 1⎞ 2⎛ 1⎞ 1⎛ 1⎞ 2⎛ 1⎞ 1⎛ 1⎞ 2⎛ 1⎞ 1⎛ 1⎞ 2⎛ 1⎞ = 1⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ ⎝ 2 ⎠ 3⎝ 2 ⎠ 2⎝ 2 ⎠ 5 ⎝ 2 ⎠ 3⎝ 2 ⎠ 7 ⎝ 2 ⎠ 4⎝ 2 ⎠ 9 ⎝ 2 ⎠ 1 1 1 1 1 1 1 1 4609 = + + + + + + + = ≈ 1.829 2 3 4 5 6 7 8 9 2520 ⌠

A = ⎮⎮

5

⌡1

1 dx x

⎛ 9 ⎞⎛ 1 ⎞ f ⎜ ⎟⎜ ⎟ ⎝ 2 ⎠⎝ 2 ⎠

A Preview of Calculus e.

19. a.

b. c.

Use fnInt function:

Graph f ( x) = e x , [−1, 3] :

A ≈ ( f (−1) + f (0) + f (1) + f (2) ) (1) ≈ (0.3679 + 1 + 2.7183 + 7.3891)(1) ≈ 11.475 ⎛ ⎛ 1⎞ ⎛1⎞ ⎛3⎞ ⎛ 5 ⎞⎞ ⎛ 1 ⎞ A ≈ ⎜ f (−1) + f ⎜ − ⎟ + f (0) + f ⎜ ⎟ + f (1) + f ⎜ ⎟ + f (2) + f ⎜ ⎟ ⎟ ⋅ ⎜ ⎟ 2 2 2 2 ⎠⎠ ⎝ 2 ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎝ ≈ ( 0.3679 + 0.6065 + 1 + 1.6487 + 2.7183 + 4.4817 + 7.3891 + 12.1825 )( 0.5 ) = 30.3947 ( 0.5 ) ≈ 15.197 3

d.

A = ⌠⌡−1 e x dx

e.

Use fnInt function:

21. a.

Graph f ( x) = sin x , [0, π] :

1839

A Preview of Calculus

b.

⎛ 2 2 ⎞⎛ π ⎞ ⎛π⎞ ⎛π⎞ ⎛ 3π ⎞ ⎞ ⎛ π ⎞ ⎛ +1+ A ≈ ⎜ f (0) + f ⎜ ⎟ + f ⎜ ⎟ + f ⎜ ⎟ ⎟ ⎜ ⎟ = ⎜⎜ 0 + ⎟⎜ ⎟ 2 2 ⎟⎠ ⎝ 4 ⎠ ⎝4⎠ ⎝2⎠ ⎝ 4 ⎠⎠⎝ 4 ⎠ ⎝ ⎝ ⎛π ⎞ = 1 + 2 ⎜ ⎟ ≈ 1.896 ⎝4⎠

(

c.

)

⎛ ⎛π⎞ ⎛π⎞ ⎛ 3π ⎞ ⎛ π⎞ ⎛ 5π ⎞ ⎛ 3π ⎞ ⎛ 7π ⎞ ⎞ ⎛ π ⎞ A ≈ ⎜ f (0) + f ⎜ ⎟ + f ⎜ ⎟ + f ⎜ ⎟ + f ⎜ ⎟ + f ⎜ ⎟ + f ⎜ ⎟ + f ⎜ ⎟ ⎟ ⎜ ⎟ ⎝8⎠ ⎝4⎠ ⎝ 8 ⎠ ⎝2⎠ ⎝ 8 ⎠ ⎝ 4 ⎠ ⎝ 8 ⎠ ⎠⎝ 8 ⎠ ⎝ ⎛π ⎞ ≈ ( 0 + 0.3827 + 0.7071 + 0.9239 + 1 + 0.9239 + 0.7071 + 0.3827 ) ⎜ ⎟ ⎝8⎠ ⎛π ⎞ = 5.0274 ⎜ ⎟ ≈ 1.974 ⎝8⎠ π

d.

A = ∫0 sin xdx

e.

Use fnInt function:

23. a.

The integral represents the area under the graph of f ( x) = 3x + 1 from x = 0 to x = 4 .

b.

b.

c. c.

27. a.

The integral represents the area under the graph of f ( x) = sin x from x = 0 to x =

25. a.

1840

The integral represents the area under the graph of f ( x) = x 2 − 1 from x = 2 to x = 5 .

π 2

.

A Preview of Calculus b.

b.

c. c.

29. a.

The integral represents the area under the graph of f ( x) = e x from x = 0 to x = 2 .

31. Using left endpoints: n = 2: 0 + 0.5 = 0.5 n = 4: 0 + 0.125 + 0.25 + 0.375 = 0.75 n = 10 : n = 100 :

10 ( 0 + 0.18) = 0.9 2 100 0 + 0.0002 + 0.0004 + 0.0006 + ⋅⋅⋅ + 0.0198 = ( 0 + 0.0198 ) = 0.99 2

0 + 0.02 + 0.04 + 0.06 + ⋅⋅⋅ + 0.18 =

Using right endpoints: n = 2 : 0.5 + 1 = 1.5 n = 4 : 0.125 + 0.25 + 0.375 + 0.5 = 1.25 10 n = 10 : 0.02 + 0.04 + 0.06 + ⋅⋅⋅ + 0.20 = ( 0.02 + 0.20 ) = 1.1 2 100 n = 100 : 0.0002 + 0.0004 + 0.0006 + ⋅⋅⋅ + 0.02 = ( 0.0002 + 0.02 ) = 1.01 2

1841

A Preview of Calculus

Chapter Review Exercises

(

8.

)

1. lim 3 x 2 − 2 x + 1 = 3(2) 2 − 2(2) + 1 x →2

= 12 − 4 + 1 = 9

2.

(

)

lim x 2 + 1

x →−2

2

(

))

(

= lim x 2 + 1 x →−2

(

)

= (− 2) 2 + 1

3. lim

x →3

2

2

= 52 = 25

9.

x 2 + 7 = lim ( x 2 + 7)

⎛ x2 − 9 ⎞ ⎛ ( x − 3)( x + 3) ⎞ lim ⎜⎜ 2 ⎟ = lim ⎜ ⎟ x →−3 x − x − 12 ⎟ x →−3 ⎝ ( x − 4)( x + 3) ⎠ ⎝ ⎠ ⎛ x − 3 ⎞ −3 − 3 = lim ⎜ ⎟= x →−3 ⎝ x − 4 ⎠ −3 − 4 −6 6 = = −7 7

⎛ x2 − 1 ⎞ lim − ⎜⎜ 3 ⎟⎟ = lim − x →−1 ⎝ x − 1 ⎠ x →−1

⎛ ( x + 1)( x − 1) ⎞ ⎜ ⎟ 2 ⎝ ( x − 1)( x + x + 1) ⎠ ⎛ x +1 ⎞ = lim − ⎜ 2 ⎟ x →−1 ⎝ x + x + 1 ⎠ −1 + 1 = 2 (−1) + (−1) + 1 0 = =0 1

x →3

= 32 + 7 = 16 = 4

4. lim− 1 − x 2 = x →1

lim (1 − x 2 )

x →1−

= 1 − 12 = 0 =0

(

5. lim (5 x + 6)3/ 2 = lim (5 x + 6) x →2

x→2

= ( 5(2) + 6 )

)

3/ 2

3/ 2

= 163/ 2 = 64

( x 2 + x + 2) ⎛ x 2 + x + 2 ⎞ xlim →−1 6. lim ⎜⎜ 2 ⎟⎟ = x →−1 lim ( x 2 − 9) ⎝ x −9 ⎠ x →−1 =

(−1) 2 + (−1) + 2

(−1) 2 − 9 2 1 = =− −8 4

⎛ ⎞ x −1 ⎛ x −1 ⎞ 7. lim ⎜ 3 ⎟ = lim ⎜ ⎟ x →1 ⎝ x − 1 ⎠ x →1 ( x − 1)( x 2 + x + 1) ⎝ ⎠ 1 ⎛ ⎞ = lim ⎜ 2 ⎟ x →1 ⎝ x + x + 1 ⎠ 1 1 = 2 = 1 +1+1 3

⎛ ⎞ x3 − 8 10. lim ⎜⎜ 3 ⎟ x →2 x − 2 x 2 + 4 x − 8 ⎟ ⎝ ⎠ ⎛ ( x − 2)( x 2 + 2 x + 4) ⎞ = lim ⎜⎜ 2 ⎟⎟ x→2 ⎝ x ( x − 2) + 4( x − 2) ⎠ ⎛ ( x − 2)( x 2 + 2 x + 4) ⎞ = lim ⎜⎜ ⎟⎟ 2 x→2 ⎝ ( x − 2)( x + 4) ⎠ ⎛ x 2 + 2 x + 4 ⎞ ⎛ 22 + 2(2) + 4 ⎞ = lim ⎜⎜ ⎟⎟ = ⎜⎜ ⎟ 2 x→2 22 + 4 ⎟⎠ ⎝ x +4 ⎠ ⎝ 12 3 = = 8 2

11.

⎛ x 4 − 3 x3 + x − 3 ⎞ lim ⎜⎜ 3 ⎟ x →3 x − 3 x 2 + 2 x − 6 ⎟ ⎝ ⎠ ⎛ x3 ( x − 3) + 1( x − 3) ⎞ = lim ⎜⎜ 2 ⎟ x →3 x ( x − 3) + 2( x − 3) ⎟ ⎝ ⎠ 3 ⎛ ( x − 3)( x + 1) ⎞ = lim ⎜⎜ ⎟ x →3 ( x − 3)( x 2 + 2) ⎟ ⎝ ⎠ 3 ⎛ x +1 ⎞ = lim ⎜⎜ 2 ⎟ x →3 x + 2 ⎟ ⎝ ⎠ =

1842

33 + 1 2

3 +2

=

28 11

A Preview of Calculus

12.

f ( x) = 3x 4 − x 2 + 2; c = 5

1. 2.

3.

f (5) = 3(5) 4 − 52 + 2 = 1852

lim f ( x) = 3(5) 4 − 52 + 2 = 1852

x →5−

4

2

lim f ( x) = 3(5) − 5 + 2 = 1852

x → 5+

Thus, f ( x) is continuous at c = 5 . x2 − 4 ; c = −2 13. f ( x) = x+2 Since f ( x) is not defined at c = − 2 , the function is not continuous at c = − 2 .

14.

⎧ x2 − 4 if x ≠ − 2 ⎪ f ( x) = ⎨ x + 2 ; ⎪4 = − if x 2 ⎩

1. 2.

c = −2

f (− 2) = 4

⎛ x2 − 4 ⎞ lim − f ( x) = lim − ⎜⎜ ⎟⎟ x →−2 x →−2 ⎝ x+2 ⎠ ⎛ ( x − 2)( x + 2) ⎞ = lim − ⎜ ⎟ x →−2 ⎝ x+2 ⎠ = lim − ( x − 2) = − 4

16. Domain: { x − 6 ≤ x < 2 or 2 < x < 5 or 5 < x ≤ 6}

18. x-intercepts: 1, 6 19. y-intercept: 4 20. 21. 22. 23. 24.

x →−2

continuous at c = − 2 . ⎧x −4 ⎪ f ( x) = ⎨ x + 2 ⎪− 4 ⎩ 2

15.

1.

lim f ( x) = − 2

x →−4+

lim f ( x ) = −∞

x→2 −

lim f ( x) = ∞

x →2 +

x →0

lim f ( x ) = 4 ≠ lim+ f ( x) = 1

x →0−

26.

x →0

f is not continuous at 0 because lim f ( x ) = 4 ≠ lim+ f ( x) = 1

x →0−

27.

x →0

f is continuous at 4 because x→4

if x ≠ − 2

;

c = −2

if x = − 2

28. R ( x ) = R is

f (− 2) = − 4

⎛ x −4⎞ lim − f ( x) = lim − ⎜⎜ ⎟⎟ x →−2 x →−2 ⎝ x+2 ⎠ ⎛ ( x − 2)( x + 2) ⎞ = lim − ⎜ ⎟ x →−2 ⎝ x+2 ⎠ = lim − ( x − 2) = − 4 x →−2

⎛ x −4⎞ lim + f ( x) = lim + ⎜⎜ ⎟⎟ x →−2 x →−2 ⎝ x+2 ⎠ ⎛ ( x − 2)( x + 2) ⎞ = lim + ⎜ ⎟ x →−2 ⎝ x+2 ⎠ = lim + ( x − 2) = − 4 2

3.

lim f ( x) = 4

x →−4 −

f (4) = lim− f ( x) = lim+ f ( x)

2

2.

f (− 6) = 2; f (− 4) = 1

25. lim f ( x) does not exist because

x →−2

Since lim − f ( x) ≠ f (−2) , the function is not

( −∞, ∞ ) , or all real numbers

17. Range:

x →−2

The function is continuous at c = − 2 .

x→4

x+4 2

x − 16

{x

=

x+4 . The domain of ( x − 4)( x + 4)

x ≠ − 4, x ≠ 4 } . Thus R is

discontinuous at both –4 and 4. ⎛ ⎞ x+4 lim R ( x) = lim− ⎜ ⎟ x → 4− x → 4 ⎝ ( x − 4)( x + 4) ⎠ = lim− x→4

( x−1 4 ) = −∞

since when x < 4,

1 < 0 , and as x x−4

1 becomes unbounded. x−4 lim R ( x ) = lim+ 1 = ∞ since when x−4 x → 4+ x→4 1 x > 4, > 0, and as x approaches 4, 1 x −4 x−4 becomes unbounded. Thus, there is a vertical asymptote at x = 4 . 1 lim R ( x) = lim 1 = − . x →−4 x →−4 x − 4 8

approaches 4,

( )

( )

1843

A Preview of Calculus

(

)

Thus, there is a hole in the graph at − 4, − 1 . 8

31.

f ( x) = x 2 + 2 x − 3 at (−1, − 4)

⎛ f ( x) − f (−1) ⎞ mtan = lim ⎜ ⎟ x →−1 ⎝ x +1 ⎠ ⎛ x 2 + 2 x − 3 − (− 4) ⎞ = lim ⎜⎜ ⎟⎟ x →−1 x +1 ⎝ ⎠ 2 ⎛ x + 2x +1 ⎞ = lim ⎜⎜ ⎟⎟ x →−1 ⎝ x +1 ⎠

29. R( x) =

⎛ ( x + 1) 2 ⎞ = lim ⎜⎜ ⎟⎟ x →−1 ⎝ x +1 ⎠ = lim ( x + 1) x →−1

x3 − 2 x 2 + 4 x − 8

= −1 + 1 = 0 Tangent Line: y − (− 4) = 0( x − (−1)) y+4=0 y = −4

x 2 − 11x + 18 x ( x − 2) + 4( x − 2) = ( x − 9)( x − 2) 2

=

( x − 2)( x 2 + 4) ( x − 9)( x − 2)

x2 + 4 ,x≠2 x −9 Undefined at x = 2 and x = 9 . There is a vertical asymptote where x − 9 = 0 . x = 9 is a vertical asymptote. There is a hole in the graph 8⎞ ⎛ at x = 2 , the point ⎜ 2, − ⎟ . 7⎠ ⎝ =

30.

f ( x) = 2 x 2 + 8 x at (1, 10)

⎛ 2 x + 8 x − 10 ⎞ ⎛ f ( x ) − f (1) ⎞ mtan = lim ⎜ ⎟⎟ ⎟ = lim ⎜⎜ x →1 x −1 x −1 ⎝ ⎠ x →1 ⎝ ⎠ x x + − 2( 5)( 1) ⎛ ⎞ = lim ⎜ ( 2( x + 5) ) ⎟ = lim x →1 ⎝ x −1 ⎠ x →1 = 2(1 + 5) = 12 Tangent Line: y − 10 = 12( x − 1) y − 10 = 12 x − 12 y = 12 x − 2 2

32.

f ( x) = x3 + x 2 at (2, 12)

⎛ x3 + x 2 − 12 ⎞ ⎛ f ( x) − f (2) ⎞ mtan = lim ⎜ = lim ⎜⎜ ⎟⎟ ⎟ x →2 ⎝ x−2 x−2 ⎠ x→2 ⎝ ⎠ ⎛ x3 − 2 x 2 + 3x 2 − 12 ⎞ = lim ⎜⎜ ⎟⎟ x →2 x−2 ⎝ ⎠ ⎛ x 2 ( x − 2) + 3( x − 2)( x + 2) ⎞ = lim ⎜⎜ ⎟⎟ x →2 x−2 ⎝ ⎠ ⎛ ( x − 2)( x 2 + 3x + 6) ⎞ = lim ⎜⎜ ⎟⎟ x →2 x−2 ⎝ ⎠ = lim ( x 2 + 3 x + 6) = 4 + 6 + 6 = 16 x →2

1844

A Preview of Calculus Tangent Line: y − 12 = 16( x − 2) y − 12 = 16 x − 32 y = 16 x − 20

35.

f ( x) = 2 x 2 + 3 x + 2 at 1 ⎛ f ( x) − f (1) ⎞ f ′(1) = lim ⎜ ⎟ x →1 ⎝ x −1 ⎠ ⎛ 2 x 2 + 3x + 2 − 7 ⎞ = lim ⎜⎜ ⎟⎟ x →1 x −1 ⎝ ⎠ ⎛ 2 x 2 + 3x − 5 ⎞ = lim ⎜⎜ ⎟⎟ x →1 x −1 ⎝ ⎠ ⎛ (2 x + 5)( x − 1) ⎞ = lim ⎜ ⎟ x →1 ⎝ x −1 ⎠ = lim (2 x + 5) = 7 x →1

33.

f ( x) = − 4 x 2 + 5 at 3

36. Use nDeriv:

⎛ f ( x) − f (3) ⎞ f ′(3) = lim ⎜ ⎟ x →3 ⎝ x −3 ⎠ ⎛ − 4 x 2 + 5 − (−31) ⎞ = lim ⎜⎜ ⎟⎟ x →3 x −3 ⎝ ⎠ ⎛ − 4 x 2 + 36 ⎞ = lim ⎜⎜ ⎟⎟ x →3 ⎝ x−3 ⎠ ⎛ − 4( x 2 − 9) ⎞ = lim ⎜⎜ ⎟⎟ x →3 ⎝ x −3 ⎠ ⎛ − 4( x − 3)( x + 3) ⎞ = lim ⎜ ⎟ x →3 ⎝ x −3 ⎠ = lim ( ( − 4)( x + 3) ) x →3

37. Use nDeriv:

38. a.

−16(t 2 − 6t − 7) = 0

= − 4(6) = − 24

34.

−16(t + 1)(t − 7) = 0 t = −1 or t = 7 The ball strikes the ground after 7 seconds in the air.

2

f ( x) = x − 3 x at 0

⎛ f ( x) − f (0) ⎞ f ′(0) = lim ⎜ ⎟ x →0 ⎝ x−0 ⎠ ⎛ x2 − 3x − 0 ⎞ = lim ⎜⎜ ⎟⎟ x →0 x ⎝ ⎠ x ( x − 3) ⎛ ⎞ = lim ⎜ ⎟ x →0 ⎝ x ⎠ = lim ( x − 3) = −3

−16t 2 + 96t + 112 = 0

b.

−16t 2 + 96t + 112 = 112 −16t 2 + 96t = 0 −16t (t − 6) = 0 t = 0 or t = 6 The ball passes the rooftop after 6 seconds.

x →0

c.

Δ s s (2) − s (0) = 2−0 Δt −16(2) 2 + 96(2) + 112 − 112 2 128 = = 64 feet/sec 2 =

1845

A Preview of Calculus

d.

⎛ s ( t ) − s ( t0 ) ⎞ s ′ ( t0 ) = lim ⎜⎜ ⎟⎟ t → t0 t − t0 ⎝ ⎠ ⎛ −16t 2 + 16t0 2 + 96t − 96t0 = lim ⎜ t →t0 ⎜ t − t0 ⎝

(

)

e. ⎞ ⎟⎟ ⎠

⎛ −16 t 2 − t0 2 + 96 ( t − t0 ) ⎞ ⎟ = lim ⎜ ⎟ t →t0 ⎜ t − t0 ⎝ ⎠

x → 25

⎛ −16 ( t − t0 )( t + t0 ) + 96 ( t − t0 ) ⎞ = lim ⎜⎜ ⎟⎟ t →t0 t − t0 ⎝ ⎠ ⎛ ( t − t0 ) ( −16 ( t + t0 ) + 96 ) ⎞ ⎟ = lim ⎜ ⎟ t →t0 ⎜ t − t0 ⎝ ⎠

⎛ R ( x ) − R (25) ⎞ R ′ ( 25 ) = lim ⎜ ⎟ x → 25 ⎝ x − 25 ⎠ ⎛ −0.25 x 2 + 100.014 x − 2344 ⎞ = lim ⎜⎜ ⎟⎟ x → 25 x − 25 ⎝ ⎠ − − + x 25 0.25 x 93.76 ⎛( )( )⎞ = lim ⎜ ⎟ x → 25 − x 25 ⎝ ⎠ = lim ( −0.25 x + 93.76 )

= −0.25 ( 25 ) + 93.76 = \$87.51 /watch

40. a.

Graph f ( x) = 2 x + 3 :

= lim ( −16 ( t + t0 ) + 96 ) t →t0

= ( −16 ( t0 + t0 ) + 96 ) = −32t0 + 96 ft/sec The instantaneous speed at time t is −32t + 96 feet per second.

e.

s ′(2) = −32(2) + 96 = − 64 + 96 = 32 feet/sec

f.

s ′(t ) = 0 −32t + 96 = 0 −32t = −96 t = 3 seconds

g.

b.

= (3 + 5 + 7 + 9)(1) = 24(1) = 24

c.

s ′(6) = −32(6) + 96 = −192 + 96 s ′(7) = −32(7) + 96 = −224 + 96 = −128 feet/sec

1846

39. a.

Δ R 8775 − 2340 6435 = = ≈ \$61.29/watch 130 − 25 105 Δx

b.

Δ R 6975 − 2340 4635 = = ≈ \$71.31/watch 90 − 25 65 Δx

c.

Δ R 4375 − 2340 2035 = = ≈ \$81.40/watch 50 − 25 25 Δx

d.

R ( x ) = −0.25 x 2 + 100.01x − 1.24

A ≈ ( f (1) + f (2) + f (3) + f (4) ) (1) = (5 + 7 + 9 + 11)(1) = 32(1) = 32

= −96 feet/sec

h.

A ≈ ( f (0) + f (1) + f (2) + f (3) ) (1)

d.

⎛ A ≈ ⎜ f (0) + ⎝

⎛1⎞ f ⎜ ⎟ + f (1) + ⎝2⎠

⎛5⎞ + f ⎜ ⎟ + f (3) + ⎝2⎠

⎛3⎞ f ⎜ ⎟ + f (2) ⎝2⎠

⎛ 7 ⎞⎞ ⎛ 1 ⎞ f ⎜ ⎟⎟ ⋅ ⎜ ⎟ ⎝ 2 ⎠⎠ ⎝ 2 ⎠

⎛1⎞ = (3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) ⎜ ⎟ ⎝2⎠ ⎛1⎞ = 52 ⎜ ⎟ = 26 ⎝2⎠

A Preview of Calculus

e.

⎛ A≈⎜ ⎝

⎛1⎞ f ⎜ ⎟ + f (1) + ⎝2⎠

⎛3⎞ f ⎜ ⎟ + f (2) + ⎝2⎠

⎛5⎞ f⎜ ⎟ ⎝2⎠

e.

Use fnInt function:

⎞ ⎛1⎞ ⎛7⎞ + f (3) + f ⎜ ⎟ + f (4) ⎟ ⋅ ⎜ ⎟ 2 ⎝ ⎠ ⎠ ⎝2⎠ ⎛1⎞ = (4 + 5 + 6 + 7 + 8 + 9 + 10 + 11) ⎜ ⎟ ⎝2⎠ ⎛1⎞ = 60 ⎜ ⎟ = 30 ⎝2⎠

f.

41. a.

42. a.

Graph f ( x) =

1 x2

, [1, 4] :

The actual area is the area of a trapezoid: 1 56 A = (3 + 11)(4) = = 28 2 2 Graph f ( x) = 4 − x 2 , [−1, 2] :

b.

A ≈ ( f (1) + f (2) + f (3) ) (1) ⎛ 1 1⎞ = ⎜ 1 + + ⎟ (1) ⎝ 4 9⎠ 49 49 = (1) = ≈ 1.36 36 36

b.

A ≈ ( f (−1) + f (0) + f (1) ) (1)

c.

= (3 + 4 + 3)(1) = 10(1) = 10

c.

⎛ A ≈ ⎜ f (−1) + ⎝

⎛ 1⎞ f ⎜ − ⎟ + f (0) + ⎝ 2⎠

=

d.

77 ⎛ 1 ⎞ 77 = 9.625 ⎜ ⎟= 4 ⎝2⎠ 8

A = ⌠⎮

2

⌡ −1

⎛3⎞ f ⎜ ⎟ + f (2) + ⎝2⎠

⎛5⎞ f⎜ ⎟ ⎝2⎠

⎛ 7 ⎞⎞ ⎛ 1 ⎞ + f (3) + f ⎜ ⎟ ⎟ ⋅ ⎜ ⎟ ⎝ 2 ⎠⎠ ⎝ 2 ⎠ ⎛ 4 1 4 1 4 ⎞⎛ 1 ⎞ = ⎜1 + + + + + ⎟ ⎜ ⎟ ⎝ 9 4 25 9 49 ⎠ ⎝ 2 ⎠ = 1.02

⎛1⎞ f⎜ ⎟ ⎝2⎠

⎛ 3 ⎞⎞ ⎛ 1 ⎞ + f (1) + f ⎜ ⎟ ⎟ ⋅ ⎜ ⎟ ⎝ 2 ⎠⎠ ⎝ 2 ⎠ 15 7 ⎞⎛ 1 ⎞ ⎛ 15 = ⎜3+ + 4 + + 3+ ⎟⎜ ⎟ 4 4 4 ⎠⎝ 2 ⎠ ⎝

⎛ A ≈ ⎜ f (1) + ⎝

4

d. e.

⌠ ⎛ 1 A = ⎮⎮ ⎜ 2 ⌡1 ⎝ x

⎞ ⎟ dx ⎠

Use fnInt function:

( 4 − x ) dx 2

43. a.

The integral represents the area under the graph of f ( x) = 9 − x 2 from x = −1 to x =3.

1847

A Preview of Calculus b.

(

)

lim − x 2 + 3x − 5 = − ( 3) + 3 ( 3) − 5 x →3

2

= −9 + 9 − 5 = −5

c.

44. a.

Use fnInt function:

2. For this problem, direct substitution does not work because it would yield the indeterminate 0 form . However, notice that this is a one-sided 0 limit from the right. As we approach 2 from the right, we will have x values such that x > 2 . Therefore, we have x − 2 = x − 2 and get the

following: x−2 x−2 lim = lim x→2+ 3x − 6 x →2+ 3x − 6 x−2 = lim x →2+ 3 ( x − 2 )

The integral represents the area under the graph of f ( x) = e x from x = −1 to x = 1 .

= lim

b.

x →2+

1 3

1 3 Remember that we can cancel the common factor ( x − 2 ) because we are interested in what =

happens near 2, not actually at 2.

c.

Use fnInt function: 3.

lim

x →−6

7 − 3 x = 7 − 3 ( −6 ) = 7 + 18 = 25 = 5

Chapter Test 1. Here we are taking the limit of a polynomial. Therefore, we evaluate the polynomial expression for the given value.

1848

A Preview of Calculus 4. Note that direct substitution will yield the 0 indeterminate form . For rational functions, 0 this means that there is a common factor that can be cancelled before taking the limit. ( x − 5)( x + 1) x2 − 4 x − 5 lim = lim 3 x →−1 x →− 1 x +1 ( x + 1) x 2 − x + 1

(

lim tan x π x→ tan x 4 6. lim = π 1 + cos 2 x lim 1 + cos 2 x x→ 4

x→

=

)

x −5 x2 − x + 1 lim ( x − 5 )

=

(

1

1 1+ 2 2 = 3

)

lim x 2 − x + 1

x →−1

−1 − 5

π 4

1 + cos

=

x →−1

=

4

tan

= lim

x →−1

π

2

π

=

4

=

1 ⎛ 2⎞ 1+ ⎜ ⎟ ⎝ 2 ⎠

2

1 3 2

( −1) − ( −1) + 1

=

2

−6 = −2 3

7. To be continuous at a point x = c , we need to show that lim f ( x ) = lim f ( x ) = f ( c ) . x →c −

2⎤

5. lim ⎡( 3 x )( x − 2 ) = lim ( 3x ) ⋅ lim ( x − 2 ) ⎦ x →5 x →5 ⎣ x →5

= lim 3 ⋅ lim x ⋅ ⎡ lim ( x − 2 ) ⎤ ⎢⎣ x →5 ⎥⎦ x →5 x →5 = 3 ⋅ 5 ⋅ (5 − 2) = 15 ( 3) = 135

2

2

x − 9 42 − 9 = =1 x→4− x →4− x + 3 4+3 lim f ( x ) = lim ( kx + 5 ) = 4k + 5 lim f ( x ) = lim

2 2

x →c +

x→4+

2

x →4+

2

4 −9 =1 4+3 Therefore, we need to solve 4k + 5 = 1 4 k = −4 k = −1 f ( 4) =

8. To find the limit, we look at at the values of f when x is close to 3, but more than 3. From the graph, we conclude that lim f ( x ) = −3 . x →3+

9. To find the limit, we look at at the values of f when x is close to 3, but less than 3. From the graph, we conclude that lim f ( x ) = 5 . x →3−

10. To find the limit, we look at at the values of f when x is close to −2 on either side. From the graph, we see that the limits from the left and right are the same and conclude that lim f ( x ) = 2 . x →−2

1849

A Preview of Calculus 11. For a limit to exist, the limit from the left and the limit from the right must both exist and be equal. From the graph we see that the limit exists since lim f ( x ) = lim f ( x ) = 2 = lim f ( x ) x →1−

x →1+

x →1

Note that the limit need not be equal to the function value at the point c. In fact, the function does not even need to be defined at c. We simply need the left and right limits to exist and be the same. 12. a. The graph has a hole at x = −2 and the function is undefined. Thus, the function is not continuous at x = −2 . b. The function is defined at x = 1 and lim f ( x ) exists, but lim f ( x ) ≠ f (1) so x →1

x →1

the function is not continuous at x = 1 . c. The function is defined at x = 3 , but there is a gap in the graph. That is, lim f ( x ) ≠ lim f ( x ) so the two-sided x →3 −

x →3+

limit as x → 3 does not exist. Thus, the function is not continuous at x = 3 . d. From the graph we can see that lim f ( x ) = lim f ( x ) = f ( 4 ) . Therefore, x→4−

x→4+

the function is continuous at x = 4 . x3 + 6 x 2 − 4 x − 24 13. R ( x ) = x 2 + 5 x − 14 Begin by factoring the numerator and denominator, but do not cancel any common factors yet. x3 + 6 x 2 − 4 x − 24 R ( x) = x 2 + 5 x − 14 x2 ( x + 6) − 4 ( x + 6) = ( x + 7 )( x − 2 )

( x + 6) ( x2 − 4) = ( x + 7 )( x − 2 ) ( x + 6 )( x + 2 )( x − 2 ) = ( x + 7 )( x − 2 )

From the denominator, we can see that the function is undefined at the values x = −7 and x = 2 because these values make the denominator equal 0. To determine whether an asymptote or hole occur at these restricted values, we need to write the function in lowest terms by canceling

1850

common factors. ( x + 6 )( x + 2 )( x − 2 ) ( x + 6 )( x + 2 ) R ( x) = = ( x + 7) ( x + 7 )( x − 2 ) (where x ≠ 2 ) Since x = −7 still makes the denominator equal to 0, there will be a vertical asymptote at x = −7 . Since x = 2 no longer makes the denominator equal to 0 when the expression is in lowest terms, there will be a hole in the graph at x = 2 . 14. a.

f ' ( 2) = lim

x →2

= lim

f ( x ) − f ( 2) x−2

( 4x

2

) (

− 11x − 3 − 4 ( 2 ) − 11( 2 ) − 3

x →2

2

)

x−2

4 x 2 − 11x − 3 − ( −9 ) = lim x →2 x−2 2 4 x − 11x + 6 = lim x →2 x−2 x − ( 2 )( 4 x − 3) = lim x →2 x−2 = lim ( 4 x − 3) x →2

= 4 ( 2) − 3 = 5

b. The derivative evaluated at x = 2 is the slope of the tangent line to the graph of f at x = 2 . From part (a), we have mtan = 5 . Using the

slope and the given point, ( 2, −9 ) , we can find the equation of the tangent line. y − y1 = m ( x − x1 ) y − ( −9 ) = 5 ( x − 2 )

y + 9 = 5 x − 10 y = 5 x − 19 Therefore, the equation of the tangent line to the graph of f at x = 2 is y = 5 x − 19 .

A Preview of Calculus y

c.

5

−5

5

x

−5

( )

( )

1 A ≈ [ f ( 0 ) + f 12 + f (1) + f 32 + f ( 2 ) 2

15. a.

x 0

y = f ( x) 2

y = 16 − 0 = 4

( x, y ) ( 0, 4 )

3

y = 16 − 32 = 7

( 2, 2 3 ) ( 3, 7 )

4

y = 16 − 42 = 0

( 4, 0 )

2 y = 16 − 22 = 2 3

b. Each subinterval will have length b−a 4−0 1 = = 8 8 2 Since u = left endpoint , we have 1 3 a = u0 = 0, u1 = , u2 = 1, u3 = , 2 2 5 7 u4 = 2, u5 = , u6 = 3, u7 = 2 2

( )

( )

+ f 52 + f ( 3) + f 72 ] 1 = [4 + 3.969 + 3.873 + 3.708 + 3.464 2 + 3.123 + 2.646 + 1.937] 1 = ( 26.72 ) 2 = 13.36 square units

c. The desired region is one quarter of a circle with radius r = 4 . The area of this region is 1 2 A = ⎡π ( 4 ) ⎤ ⎦ 4⎣ 1 = (16π ) = 4π ≈ 12.566 square units 4 The estimate in part (b) is slightly larger than the actual area. Since the function is decreasing over the entire interval, the largest value of the function on a subinterval always occurs at the left endpoint. Therefore, we would expect our estimate to be larger than the actual value. 4

(

)

16. Area = ∫1 − x 2 + 5 x + 3 dx 17. The average rate of change is given by s ( 6 ) − s ( 3) 137 − 31 a.r.c. = = 6−3 6−3 106 1 = = 35 ≈ 35.33 ft. per sec. 3 3

1851

A Preview of Calculus

Chapter Projects Project I Total Midyear Population for the World: 1950-2050

t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

1852

Year 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979

Population 2,555,360,972 2,593,139,857 2,635,192,901 2,680,522,529 2,728,486,476 2,779,929,940 2,832,880,780 2,888,699,042 2,945,196,478 2,997,522,100 3,039,585,530 3,080,367,474 3,136,451,432 3,205,956,565 3,277,024,728 3,346,002,675 3,416,184,968 3,485,881,292 3,557,690,668 3,632,294,522 3,707,475,887 3,784,957,162 3,861,537,222 3,937,599,035 4,013,016,398 4,086,150,193 4,157,827,615 4,229,922,943 4,301,953,661 4,376,897,872

t 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

Year 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Population 4,452,584,592 4,528,511,458 4,608,410,617 4,689,840,421 4,769,886,824 4,851,592,622 4,934,892,988 5,020,809,215 5,107,404,183 5,194,105,912 5,281,653,820 5,365,480,276 5,449,369,636 5,531,014,635 5,611,269,983 5,691,759,210 5,770,701,020 5,849,885,301 5,927,556,529 6,004,170,056 6,079,603,571 6,153,801,961 6,226,933,918 6,299,763,405 6,372,797,742 6,446,131,400

Source: U.S. Bureau of the Census, International Database. Note: Data updated 9-30-2004 http://www.census.gov/ipc/www/worldpop.html

A Preview of Calculus 1.

6.

Rate of growth 108

y=

1.13 × 1010 1 + 3.65e−0.029t

2. 7 × 109

t (Years)

60

7.

y

60

The growth rate is largest in 1994. Then, the growth rate begins to decrease. The y value on the graph for this time is 5.65 × 109 .

t

3.

8.

⎛ 1.13 × 1010 ⎞ lim f ( t ) = lim ⎜ ⎟ t →∞ t →∞ 1 + 3.65e −0.029t ⎝ ⎠

1.13 × 1010 = 1.13 × 1010 1+ 0 The carrying capacity of the Earth is 1.13 × 1010 people. =

9.

4.

y (1961) ≈ 3, 039,585,530 + 64, 288,855 = 3,103,874,385 The actual population in 1961 was 3,080,367,474.

5.

If the population exceeds the carrying capacity, the population will begin to die off very quickly due to hunger and disease in particular. There will not be enough agricultural growth to keep up with the increase in population. Urban sprawl will cause agricultural growth to diminish since land will be taken away.

Project II 1.

D (m) 0.006

The instantaneous growth is slowing down. Thus, Malthus’ contention is not true.

20°, 40%

0.001 20

40

t (Years)

1853

A Preview of Calculus 2. t = 1 to t = 2 : ΔD 0.0014 − 0.0011 = = 0.003 m/hr 2 −1 Δt

4. t = 1 to t = 2: ΔD 0.002 − 0.0017 = = 0.0003m / hr 2 −1 Δt

t = 2 to t = 5 : ΔD 0.0017 − 0.0014 = = 0.001 m/hr 5−2 Δt

t = 2 to t = 5: ΔD 0.0023 − 0.002 = = 0.0001m / hr Δt 5−2

t = 24 to t = 48 : ΔD 0.004 − 0.003 = = 0.000042 m/hr 48 − 24 Δt

t = 24 to t = 48: ΔD 0.006 − 0.004 = = 0.000083m / hr 48 − 24 Δt

3.

5. There are no differences until the time span from t = 24 to t = 48. The highest rate is for 80 % RH from t = 24 to t = 48. The lowest rate is for 20% for the time span t = 24 to t = 48.

D (m) 0.006

20°, 80%

6. 0.001 20

40

t (Years)

D (t ) = −0.0000006t 2 + 0.0001t + 0.0014

−0.0000006(t + h) 2 + 0.0001(t + h) + 0.0014 − ( −0.0000006t 2 + 0.0001t + 0.0014) h →0 h 2 2 −0.0000006(t + 2th + h ) + 0.0001(t + h) + 0.0014 + 0.0000006t 2 − 0.0001t − 0.0014 = lim h →0 h 2 −0.0000012th − 0.0000006h + 0.0001h = lim h →0 h = lim (−0.0000012t − 0.000006h + 0.0001)

7. D '(t ) = lim

h →0

= −0.0000012t + 0.0001 D '(2) = 0.0000976 m/hr ≈ 0.0001 m/hr D '(24) = 0.0000712 m/hr

8.

D(t ) = −0.0000003t 2 + 0.0001t + 0.0017

1854

A Preview of Calculus

−0.0000003(t + h) 2 + 0.0001(t + h) + 0.0017 − (−0.0000003t 2 + 0.0001t + 0.0017) h →0 h 2 2 −0.0000003(t + 2th + h ) + 0.0001(t + h) + 0.0017 + 0.0000003t 2 − 0.0001t − 0.0017 = lim h →0 h 2 −0.0000006th − 0.0000003h + 0.0001h = lim h →0 h = lim (−0.0000006t − 0.000003h + 0.0001)

D '(t ) = lim

h →0

= −0.0000006t + 0.0001 D '(2) = 0.0000988m / hr ≈ 0.0001m / hr D '(24) = 0.0000856m / hr ≈ 0.00009m / hr The instantaneous rate of change is the same at t = 2, but the rates are different for t = 24.

Project III 1.

2.

X P (total profit) 0 -24000 25 250 60 8500 102 18150 150 20160 190 19610 223 14985 249 9625 The profit-maximizing level is 150 bicycles.

y

(\$)

50,000

200

x (bicycles)

3.

C ( x) = 0.002 x3 − 0.6 x 2 + 157 x + 24068

1855

A Preview of Calculus 4.

R ( x) = −1.8 x 2 + 638 x + 7205

5. P ( x) = R ( x) − C ( x) = (−1.8 x 2 + 638 x + 7205) − (0.002 x3 − 0.6 x 2 + 157 x + 24068) = −0.002 x3 − 2.4 x 2 + 795 x + 31273

6. y

y

(\$)

(\$)

y2 = C ( x )

50,000

50,000

y1 = R (x )

y3 = P (x ) 100

x

200

x

The answer in (a) was 150 and this one is 147. Those are very close. 7.

9.

\$/Bicycle 500

\$/Bicycle 500 y 5 = R ' (x )

P ' (x )

y4 = C ' ( x )

275 8.

x

275

x

10.

This is the same result as in part (f). 11. Marginal revenue is the rate of change of the revenue as another bicycle is produced. Marginal cost gives the change in cost that making the next bicycle will cause. When the rate of change of the revenue is the same as the rate of change of the cost, the two changes offset each other, thus maximizing the cost.

1856

Appendix Review Section 1

31.

1. variable

33. x > 0

3. strict

35. x < 2

5. True. 7. False; the absolute value of a real number is nonnegative. 0 = 0 which is not a positive

37. x ≤ 1 39. Graph on the number line: x ≥ −2

number. 9. A ∪ B = {1, 3, 4,5, 9} ∪ {2, 4, 6, 7,8}

= {1, 2,3, 4, 5, 6, 7,8, 9}

( A ∪ B) ∩ C

= ({1, 3, 4,5, 9} ∪ {2, 4, 6, 7,8} ) ∩ {1,3, 4, 6} = {1, 2,3, 4,5, 6, 7,8,9} ∩ {1,3, 4, 6} = {1, 3, 4, 6}

41. Graph on the number line: x > −1

43. d (C , D ) = d (0,1) = 1 − 0 = 1 = 1 45. d ( D, E ) = d (1,3) = 3 − 1 = 2 = 2 47. d ( A, E ) = d (−3,3) = 3 − (−3) = 6 = 6

51. 5 xy + 2 = 5(− 2)(3) + 2 = −30 + 2 = − 28

17. A ∩ B = {1, 3, 4, 5, 9} ∩ {2, 4, 6, 7, 8} = {4} = {0, 1, 2, 3, 5, 6, 7, 8, 9}

19. A ∪ B = {0, 2, 6, 7, 8} ∪ {0, 1, 3, 5, 9}

53.

2(− 2) − 4 4 2x = = = x − y − 2 − 3 −5 5

55.

3x + 2 y 3(− 2) + 2(3) − 6 + 6 0 = = = =0 2+ y 2+3 5 5

= {0, 1, 2, 3, 5, 6, 7, 8, 9} 0.25 −2.5

23.

1 >0 2

0

49. x + 2 y = − 2 + 2 ⋅ 3 = − 2 + 6 = 4

15. A = {0, 2, 6, 7, 8}

21.

0

−2

−1

11. A ∩ B = {1, 3, 4,5, 9} ∩ {2, 4, 6, 7,8} = {4} 13.

2 < 0.67 3

−1 0

3 4

1

5 2

57.

x + y = 3 + (− 2) = 1 = 1

59.

x + y = 3 + −2 = 3+ 2 = 5

61.

x 3 3 = = =1 x 3 3

63.

4 x − 5 y = 4(3) − 5(− 2)

25. −1 > −2

= 12 + 10

27. π > 3.14

= 22 = 22

1 = 0.5 29. 2 From Appendix A of Student’s Solutions Manual for Precalculus Enhanced with Graphing Utilities, Sixth Edition. Michael Sullivan, Michael Sullivan, III. Copyright © 2013 by Pearson Education, Inc. All rights reserved.

1857

Appendix: Review

65.

4x − 5 y

= 4(3) − 5(− 2) = 12 − − 10 = 12 − 10 = 2 =2

67.

x2 − 1 x Part (c) must be excluded. The value x = 0 must be excluded from the domain because it causes division by 0.

x x 69. 2 = x − 9 ( x − 3)( x + 3) Part (a) must be excluded. The values x = −3 and x = 3 must be excluded from the domain because they cause division by 0.

71.

73.

75.

77.

1858

x2 x2 + 1 None of the given values are excluded. The domain is all real numbers. 2

2

x + 5 x − 10 x + 5 x − 10 = 3 x ( x − 1)( x + 1) x −x Parts (b), (c), and (d) must be excluded. The values x = 0, x = 1, and x = −1 must be excluded from the domain because they cause division by 0. 4 x −5 x = 5 must be exluded because it makes the denominator equal 0. Domain = { x x ≠ 5}

x x+4 x = −4 must be excluded sine it makes the denominator equal 0. Domain = { x x ≠ −4}

83. (− 4) 2 = (− 4)(− 4) = 16 85. 4−2 =

1 1 = 42 16

87. 3−6 ⋅ 34 = 3−6 + 4 = 3−2 = 89.

(3 )

−2 −1

1 1 = 32 9

−2 −1 = 3( )( ) = 32 = 9

91.

25 = 52 = 5

93.

( −4 )2

= −4 = 4

95.

(8x )

= 82 x 3

97.

(x

) = (x ) ⋅( y )

99.

x2 y3 x = x 2 −1 y 3− 4 = x1 y −1 = 4 y xy

101.

3 2

( )

2 −1 2

y

32 x y 3 z

−2

105. 2 xy −1 =

= 64 x 6

2 2

(− 2)3 x 4 ( y z ) 2

⎛ 3x −1 ⎞ 103. ⎜ −1 ⎟ ⎝ 4y ⎠

2

−1 2

= x 4 y −2 =

x4 y2

− 8x4 y2 z 2 9 x y3 z − 8 4 −1 2 −3 2 −1 x y z = 9 − 8 3 −1 1 x y z = 9 8 x3 z =− 9y =

⎛ 3y ⎞ =⎜ ⎟ ⎝ 4x ⎠

−2

2

42 x 2 16 x 2 ⎛ 4x ⎞ =⎜ ⎟ = 2 2 = 3 y 9 y2 ⎝ 3y ⎠

2x 2 ( 2) = = −4 y ( −1)

107. x 2 + y 2 = ( 2 ) + ( −1) = 4 + 1 = 5 2

109.

( xy )2 = ( 2 ⋅ ( −1) )

2

2

5 5 5 79. C = ( F − 32) = (32 − 32) = (0) = 0°C 9 9 9

111.

x2 = x = 2 = 2

5 5 5 81. C = ( F − 32) = (77 − 32) = (45) = 25°C 9 9 9

113.

x2 + y 2 =

= ( −2 ) = 4 2

( 2 )2 + ( −1)2

= 4 +1 = 5

Appendix: Review

115. x y = 2−1 =

b.

1 2

117. If x = 2, 2 x 3 − 3 x 2 + 5 x − 4 = 2 ⋅ 23 − 3 ⋅ 22 + 5 ⋅ 2 − 4 = 16 − 12 + 10 − 4 = 10 If x = 1, 3

2

3

2

2 x − 3 x + 5 x − 4 = 2 ⋅ 1 − 3 ⋅1 + 5 ⋅1 − 4 = 2−3+5− 4 =0

119.

4

(666) 4 ⎛ 666 ⎞ 4 =⎜ ⎟ = 3 = 81 4 222 ⎝ ⎠ (222)

141. We want the difference between x and 4 to be at least 6 units. Since we don’t care whether the value for x is larger or smaller than 4, we take the absolute value of the difference. We want the inequality to be non-strict since we are dealing with an ‘at least’ situation. Thus, we have x−4 ≥ 6 143. a.

121. (8.2) ≈ 304, 006.671 b.

123. (6.1) −3 ≈ 0.004

145. a.

= 0.001 ≤ 0.01 A radius of 2.999 centimeters is acceptable.

131. C = π d

139. a.

b.

If x = 1000, C = 4000 + 2 x

= 4000 + 2(1000) = 4000 + 2000 = \$6000 The cost of producing 1000 watches is \$6000.

x − 3 = 2.89 − 3 = − 0.11

3 2 x 4

137. V = x3

x − 3 = 2.999 − 3 = − 0.001

129. A = lw

4 135. V = π r 3 3

x − 110 = 104 − 110 = − 6 = 6 > 5

104 volts is not acceptable.

127. (− 8.11) −4 ≈ 0.000

133. A =

x − 110 = 108 − 110 = − 2 = 2 ≤ 5

108 volts is acceptable.

6

125. (− 2.8)6 ≈ 481.890

If x = 2000, C = 4000 + 2 x = 4000 + 2(2000) = 4000 + 4000 = \$8000 The cost of producing 2000 watches is \$8000.

= 0.11 ≤/ 0.01 A radius of 2.89 centimeters is not acceptable.

147.

1 = 0.333333 ... > 0.333 3 1 is larger by approximately 0.0003333 ... 3

149. No. For any positive number a, the value

a is 2

smaller and therefore closer to 0. 151. Answers will vary.

Section 2 1. right; hypotenuse 3. C = 2π r

1859

Appendix: Review 5. True 7. False; the volume of a sphere of radius r is given 4 by V = π r 3 . 3

23. 62 = 32 + 42 36 = 9 + 16 36 = 25 false The given triangle is not a right triangle.

9. True; Two corresponding angles are equal.

25. A = l ⋅ w = 4 ⋅ 2 = 8 in 2

11.

a = 5, b = 12, 2

2

c = a +b

2

= 52 + 122 = 25 + 144 = 169 ⇒ c = 13

13.

a = 10, b = 24, c 2 = a 2 + b2 = 102 + 242 = 100 + 576 = 676 ⇒ c = 26

15.

a = 7, b = 24, c 2 = a 2 + b2 = 7 2 + 242 = 49 + 576 = 625 ⇒ c = 25

27. A =

1 1 b ⋅ h = (2)(4) = 4 in 2 2 2

29. A = π r 2 = π (5) 2 = 25π m 2 C = 2π r = 2π (5) = 10π m 31. V = l w h = 8 ⋅ 4 ⋅ 7 = 224 ft 3 S = 2lw + 2lh + 2wh = 2 ( 8 )( 4 ) + 2 ( 8 )( 7 ) + 2 ( 4 )( 7 ) = 64 + 112 + 56 = 232 ft 2 4 3 4 256 π r = π⋅ 43 = π cm3 3 3 3 S = 4π r 2 = 4π ⋅ 42 = 64π cm 2

33. V =

35. V = π r 2 h = π(9) 2 (8) = 648π in 3 S = 2π r 2 + 2π rh = 2π ( 9 ) + 2π ( 9 )( 8 ) 2

2

2

2

17. 5 = 3 + 4 25 = 9 + 16 25 = 25 The given triangle is a right triangle. The hypotenuse is 5. 19. 62 = 42 + 52 36 = 16 + 25 36 = 41 false The given triangle is not a right triangle. 21. 252 = 7 2 + 242 625 = 49 + 576 625 = 625 The given triangle is a right triangle. The hypotenuse is 25.

= 162π + 144π = 306π in 2

37. The diameter of the circle is 2, so its radius is 1. A = π r 2 = π(1) 2 = π square units 39. The diameter of the circle is the length of the diagonal of the square. d 2 = 22 + 22 = 4+4 =8 d = 8=2 2 d 2 2 = = 2 2 2 The area of the circle is: r=

A = π r2 = π

( 2)

2

= 2π square units

41. Since the triangles are similar, the lengths of corresponding sides are proportional. Therefore,

1860

Appendix: Review

we get 8 x = 4 2 8⋅ 2 =x 4 4=x In addition, corresponding angles must have the same angle measure. Therefore, we have A = 90° , B = 60° , and C = 30° . 43. Since the triangles are similar, the lengths of corresponding sides are proportional. Therefore, we get 30 x = 20 45 30 ⋅ 45 =x 20 135 = x or x = 67.5 2 In addition, corresponding angles must have the same angle measure. Therefore, we have A = 60° , B = 95° , and C = 25° . 45. The total distance traveled is 4 times the circumference of the wheel. Total Distance = 4C = 4(π d ) = 4π ⋅ 16

= 64π ≈ 201.1 inches ≈ 16.8 feet

47. Area of the border = area of EFGH – area of ABCD = 102 − 62 = 100 − 36 = 64 ft 2 49. Area of the window = area of the rectangle + area of the semicircle. 1 A = (6)(4) + ⋅ π⋅ 22 = 24 + 2π ≈ 30.28 ft 2 2 Perimeter of the window = 2 heights + width + one-half the circumference. 1 P = 2(6) + 4 + ⋅ π (4) = 12 + 4 + 2π 2 = 16 + 2π ≈ 22.28 feet 51. We can form similar triangles using the Great Pyramid’s height/shadow and Thales’ height/shadow:

{

{

h

126 240

114

This allows us to write

h 2 = 240 3 2 ⋅ 240 h= = 160 3 The height of the Great Pyramid is 160 paces.

53. Convert 20 feet to miles, and solve the Pythagorean Theorem to find the distance: 1 mile = 0.003788 miles 20 feet = 20 feet ⋅ 5280 feet d 2 = (3960 + 0.003788) 2 − 39602 = 30 sq. miles d ≈ 5.477 miles d 20 ft

3960

3960

54. Convert 6 feet to miles, and solve the Pythagorean Theorem to find the distance: 1 mile 6 feet = 6 feet ⋅ = 0.001136 miles 5280 feet d 2 = (3960 + 0.001136) 2 − 39602 = 9 sq. miles d ≈ 3 miles d 6 ft

3960

3960

55. Convert 100 feet to miles, and solve the Pythagorean Theorem to find the distance: 1 mile 100 feet = 100 feet ⋅ = 0.018939 miles 5280 feet d 2 = (3960 + 0.018939) 2 − 39602 ≈ 150 sq. miles d ≈ 12.2 miles Convert 150 feet to miles, and solve the Pythagorean Theorem to find the distance: 1 mile 150 feet = 150 feet ⋅ = 0.028409 miles 5280 feet d 2 = (3960 + 0.028409) 2 − 39602 ≈ 225 sq. miles

2

d ≈ 15.0 miles 3

57. Let l = length of the rectangle and w = width of the rectangle.

1861

Appendix: Review

Notice that (l + w) 2 − (l − w) 2 = [(l + w) + (l − w)][(l + w) − (l − w)] = (2l )(2 w) = 4lw = 4 A

17.

1 4

So A = [(l + w) 2 − (l − w) 2 ]

Since (l − w) 2 ≥ 0 , the largest area will occur when l – w = 0 or l = w; that is, when the rectangle is a square. But 1000 = 2l + 2 w = 2(l + w) 500 = l + w = 2l 250 = l = w The largest possible area is 2502 = 62500 sq ft. A circular pool with circumference = 1000 feet 500 yields the equation: 2π r = 1000 ⇒ r =

19. x 2 + 2 x − 5 Not a monomial; the expression contains more than one term. This expression is a trinomial. 21. 3x 2 − 5 23. 5

27. 2 y 3 − 2 29.

Section 3 31.

1. 4; 3 3.

x

3

Polynomial; Degree: 3

x2 + 5 Not a polynomial; the polynomial in x3 − 1 the denominator has a degree greater than 0. ( x 2 + 4 x + 5) + (3 x − 3) = x 2 + (4 x + 3 x) + (5 − 3)

−8

= x2 + 7 x + 2

(

5. False; x3 + a 3 = ( x + a ) x 2 − ax + a 2

)

33. ( x3 − 2 x 2 + 5 x + 10) − (2 x 2 − 4 x + 3) = x3 − 2 x 2 + 5 x + 10 − 2 x 2 + 4 x − 3

7. 3x ( x − 2 )( x + 2 ) 9. True; x 2 + 4 is prime over the set of real numbers.

= x3 + (− 2 x 2 − 2 x 2 ) + (5 x + 4 x) + (10 − 3) = x3 − 4 x 2 + 9 x + 7 35.

8 = 8x −1 x

Not a monomial; when written in k

the form ax , the variable has a negative exponent. 15. − 2 x3 + 5 x 2 Not a monomial; the expression contains more than one term. This expression is a binomial.

6( x3 + x 2 − 3) − 4(2 x 3 − 3 x 2 ) = 6 x3 + 6 x 2 − 18 − 8 x3 + 12 x 2

11. 2x3 Monomial; Variable: x ; Coefficient: 2; Degree: 3 13.

Polynomial; Degree: 0

5 Not a polynomial; the variable in the x denominator results in an exponent that is not a nonnegative integer.

π

2

Polynomial; Degree: 2

25. 3x 2 −

The area enclosed by the circular pool is: 5002 ⎛ 500 ⎞ A = π r2 = π ⎜ = ≈ 79577.47 ft 2 ⎟ π π ⎝ ⎠ Thus, a circular pool will enclose the most area.

8x Not a monomial; the polynomial in x2 − 1 the denominator has a degree greater than 0. The expression cannot be written in the form ax k where k ≥ 0 is an integer.

= − 2 x3 + 18 x 2 − 18

37.

(

) (

9 y2 − 3y + 4 − 6 1 − y2 2

= 9 y − 27 y + 36 − 6 + 6 y

) 2

= 15 y 2 − 27 y + 30

39. x( x 2 + x − 4) = x3 + x 2 − 4 x 41. ( x + 2)( x + 4) = x 2 + 4 x + 2 x + 8 = x2 + 6 x + 8

1862

Appendix: Review 43. (2 x + 5)( x + 2) = 2 x 2 + 4 x + 5 x + 10 = 2 x 2 + 9 x + 10

45. ( x − 7)( x + 7) = x 2 − 7 2 = x 2 − 49 47. (2 x + 3)(2 x − 3) = (2 x) 2 − 32 = 4 x 2 − 9

Check:

( x 2 )(4 x − 3) + ( x + 1) = 4 x3 − 3x 2 + x + 1 The quotient is 4 x − 3 ; the remainder is x + 1 .

5 x 2 − 13 61. x 2 + 2 5 x 4 + 0 x3 − 3 x 2 + x + 1

49. ( x + 4) 2 = x 2 + 2 ⋅ x ⋅ 4 + 42 = x 2 + 8 x + 16

5x4

+ 10 x 2 − 13x 2 + x + 1

51. (2 x − 3) 2 = (2 x) 2 − 2(2 x)(3) + 32 = 4 x 2 − 12 x + 9

−13x 2

53. ( x − 2)3 = x3 − 3 ⋅ x 2 ⋅ 2 + 3 ⋅ x ⋅ 22 − 23 = x3 − 6 x 2 + 12 x − 8

55. (2 x + 1)3 = (2 x)3 + 3(2 x) 2 (1) + 3(2 x) ⋅12 + 13

Check:

(x

2

)(

− 26 x + 27

)

+ 2 5 x 2 − 13 + ( x + 27 )

= 8 x3 + 12 x 2 + 6 x + 1

= 5 x 4 + 10 x 2 − 13 x 2 − 26 + x + 27

4 x 2 − 11x + 23

= 5 x 4 − 3x 2 + x + 1 The quotient is 5 x 2 − 13 ; the remainder is x + 27 .

57. x + 2 4 x3 − 3 x 2 +

x+ 1

4 x3 + 8 x 2 − 11x 2 +

x

−11x 2 − 22 x 23 x + 1 23 x + 46 − 45

2 x2 63. 2 x − 1 4 x5 + 0 x 4 + 0 x3 − 3 x 2 + x + 1 3

4 x5

− x2 + x + 1 Check:

( 2 x − 1)( 2 x ) + ( − x 3

Check: 2

( x + 2)(4 x − 11x + 23) + (− 45) 3

2

2

= 4 x − 11x + 23 x + 8 x − 22 x + 46 − 45 = 4 x3 − 3x 2 + x + 1

The quotient is 4 x 2 − 11x + 23 ; the remainder is –45.

− 2 x2

2

2

)

+ x +1

= 4 x5 − 2 x 2 − x 2 + x + 1 = 4 x5 − 3 x 2 + x + 1 The quotient is 2x 2 ; the remainder is − x2 + x + 1 . x 2 − 2 x + 12 65. 2 x 2 + x + 1 2 x 4 − 3 x3 + 0 x 2 + x + 1

4x − 3 59. x 2 4 x3 − 3 x 2 + x + 1

4 x3

2 x 4 + x3 + x 2 − 4 x3 − x 2 + x

− 3x 2 + x + 1 −3x 2 x+ 1

−4 x3 − 2 x 2 − 2 x x 2 + 3x + 1 1 1 x2 + x + 2 2 5 1 x+ 2 2

1863

Appendix: Review

Check:

(

)(

)

2x + x +1 x − 2x + 1 + 5 x + 1 2 2 2 4 3 2 3 2 1 = 2x − 4x + x + x − 2x + x 2 2 5 1 1 + x − 2x + + x + 2 2 2 4 3 = 2 x − 3x + x + 1 2

2

The quotient is x 2 − 2 x + 12 ; the remainder is 5 x+ 1 . 2 2

− 4 x − 3x − 3 2

3

2

67. x − 1 − 4 x + x + 0 x − 4 − 4x + 4x

( x 2 + x + 1)( x 2 − x − 1) + 2 x + 2 = x 4 + x3 + x 2 − x3 − x 2 − x − x 2 − x −1 + 2x + 2 = x4 − x2 + 1 The quotient is x 2 − x − 1 ; the remainder is 2x + 2 . x 2 + ax + a 2

2

3

Check:

71. x − a x3 + 0 x 2 + 0 x − a3 x 3 − ax 2 ax 2

− 3x 2

ax 2 − a 2 x

2

−3 x + 3 x

a 2 x − a3

− 3x − 4 −3 x + 3

a 2 x − a3

0

−7 Check: Check: 2

( x − 1)(− 4 x − 3x − 3) + (− 7) 3

2

2

= − 4 x − 3x − 3x + 4 x + 3x + 3 − 7 3

2

= − 4x + x − 4

The quotient is − 4 x 2 − 3 x − 3 ; the remainder is –7.

( x − a)( x 2 + ax + a 2 ) + 0 = x3 + ax 2 + a 2 x − ax 2 − a 2 x − a3 = x3 − a3 The quotient is x 2 + ax + a 2 ; the remainder is 0.

73. x 2 − 36 = ( x − 6)( x + 6) 75. 2 − 8 x 2 = 2(1 − 4 x 2 ) = 2 (1 − 2 x )(1 + 2 x )

x2 − x − 1

69. x 2 + x + 1 x 4 + 0 x3 − x 2 + 0 x + 1 x 4 + x3 + x 2 − x3 − 2 x 2 − x3 − x 2 − x − x2 + x + 1 − x2 − x − 1 2x + 2

77. x 2 + 11x + 10 = ( x + 1)( x + 10) 79. x 2 − 10 x + 21 = ( x − 7 )( x − 3)

(

81. 4 x 2 − 8 x + 32 = 4 x 2 − 2 x + 8

)

83. x 2 + 4 x + 16 is prime over the reals because there are no factors of 16 whose sum is 4. 85. 15 + 2 x − x 2 = − ( x 2 − 2 x − 15) = −( x − 5)( x + 3) 87. 3x 2 − 12 x − 36 = 3( x 2 − 4 x − 12) = 3( x − 6)( x + 2)

1864

Appendix: Review 89. y 4 + 11y 3 + 30 y 2 = y 2 ( y 2 + 11y + 30) = y 2 ( y + 5)( y + 6)

(

= 3 ( x + 5) − 4 ( x + 5) = ( x + 5 ) ⎡⎣3 ( x + 5 ) − 4 ⎤⎦ = ( x + 5 )( 3 x + 15 − 4 )

)

2

93. 6 x + 8 x + 2 = 2 3x + 4 x + 1

= ( x + 5 )( 3 x + 11)

= 2 ( 3x + 1)( x + 1)

( )

95. x 4 − 81 = x 2

2

117. x3 + 2 x 2 − x − 2 = x 2 ( x + 2) − 1( x + 2 )

3

3

97. x − 2 x + 1 = ( x − 1)

119. x 4 − x3 + x − 1 = x3 ( x − 1) + 1( x − 1)

2

= ⎡⎣ ( x − 1)( x 2 + x + 1) ⎤⎦

= ( x − 1)( x3 + 1)

2

= ( x − 1)( x + 1)( x 2 − x + 1)

= ( x − 1) 2 ( x 2 + x + 1) 2

99. x 7 − x5 = x5 ( x 2 − 1) = x5 ( x − 1)( x + 1) 101. 16 x 2 + 24 x + 9 = ( 4 x + 3)

123. Since B is -6 then we need half of -6 squared to be the last term in our trinomial. Thus 1 (−6) = −3; (−3) 2 = 9 2 y 2 − 6 y + 9 = ( y − 3) 2

105. 4 y 2 − 16 y + 15 = (2 y − 5)(2 y − 3)

125. Since B is − 12 then we need half of − 12 squared

107. 1 − 8 x 2 − 9 x 4 = −(9 x 4 + 8 x 2 − 1) = −(9 x 2 − 1)( x 2 + 1) = −(3x − 1)(3x + 1)( x 2 + 1)

109. x( x + 3) − 6( x + 3) = ( x + 3)( x − 6)

127. 2 ( 3 x + 4 ) + ( 2 x + 3) ⋅ 2 ( 3 x + 4 ) ⋅ 3

= 2 ( 3x + 4 ) ( ( 3 x + 4 ) + ( 2 x + 3) ⋅ 3) = 2 ( 3x + 4 )( 9 x + 13)

( 3x − 2 )3 − 27 3 = ( 3 x − 2 ) − 33

129. 2 x ( 2 x + 5 ) + x 2 ⋅ 2 = 2 x ( ( 2 x + 5 ) + x )

2 = ⎣⎡( 3 x − 2 ) − 3⎦⎤ ⎡( 3 x − 2 ) + 3 ( 3x − 2 ) + 9 ⎤ ⎣ ⎦

= ( 3 x − 5 ) 9 x 2 − 12 x + 4 + 9 x − 6 + 9 − 3x + 7

x 2 − 12 x + 161 = ( x − 14 )2

= 2 ( 3x + 4 )( 3 x + 4 + 6 x + 9 )

= ( x + 2)( x − 3)

2

to be the last term in our trinomial. Thus 1 (− 12 ) = − 14 ; (− 14 ) 2 = 161 2

2

111. ( x + 2) 2 − 5( x + 2) = ( x + 2) [ ( x + 2) − 5]

( = ( 3x − 5) ( 9 x

121. Since B is 10 then we need half of 10 squared to be the last term in our trinomial. Thus 1 (10) = 5; (5) 2 = 25 2 x 2 + 10 x + 25 = ( x + 5) 2

2

103. 5 + 16 x − 16 x 2 = −(16 x 2 − 16 x − 5) = −(4 x − 5)(4 x + 1)

113.

= ( x + 2)( x 2 − 1) = ( x + 2)( x − 1)( x + 1)

− 92 = ( x 2 − 9)( x 2 + 9)

= ( x − 3)( x + 3)( x 2 + 9) 6

)

2

91. 4 x 2 + 12 x + 9 = (2 x + 3) 2 2

(

115. 3 x 2 + 10 x + 25 − 4 ( x + 5 )

)

)

= 2x ( 2x + 5 + x) = 2 x ( 3x + 5)

1865

Appendix: Review 131. 2 ( x + 3)( x − 2 ) + ( x + 3) ⋅ 3 ( x − 2 ) 3

2

2

= ( x + 3)( x − 2 ) ( 2 ( x − 2 ) + ( x + 3) ⋅ 3) 2

Section 4

= ( x + 3)( x − 2 ) ( 2 x − 4 + 3x + 9 ) 2

= ( x + 3)( x − 2 ) ( 5 x + 5 )

1. quotient; divisor; remainder

= 5 ( x + 3)( x − 2 )

3. True

2

133.

2

( x + 1)

( 4 x − 3)2 + x ⋅ 2 ( 4 x − 3) ⋅ 4 = ( 4 x − 3) ( ( 4 x − 3) + 8 x ) = ( 4 x − 3)( 4 x − 3 + 8 x ) = ( 4 x − 3)(12 x − 3) = 3 ( 4 x − 3)( 4 x − 1) = 6 ( 3 x − 5 )( 2 x + 1)

2

4

2

2

8

1 4 12

Quotient: x 2 + x + 4 Remainder: 12 2

2

( ( 2 x + 1) + ( 3x − 5) )

= 6 ( 3 x − 5 )( 2 x + 1) ( 2 x + 1 + 3 x − 5 ) 2

= 6 ( 3 x − 5 )( 2 x + 1) ( 5 x − 4 ) 2

137. Factors of 4: 1, 4 2, 2 –1, –4 –2, –2 Sum: 5 4 –5 –4 None of the sums of the factors is 0, so x 2 + 4 is prime.

Alternatively, the possibilities are ( x ± 1)( x ± 4 ) = x2 ± 5 x + 4 or

( x ± 2 )( x ± 2 ) = x 2 ± 4 x + 4 , none of which

equals x 2 + 4 . 139. When we multiply polynomials p1 ( x ) and

p2 ( x ) , each term of p1 ( x ) will be multiplied

by each term of p2 ( x ) . So when the highestpowered term of p1 ( x ) multiplies by the highest

powered term of p2 ( x ) , the exponents on the

variables in those terms will add according to the basic rules of exponents. Therefore, the highest powered term of the product polynomial will have degree equal to the sum of the degrees of p1 ( x ) and p2 ( x ) . 141. When we add two polynomials p1 ( x ) and

p2 ( x ) , where the degree of p1 ( x ) = the degree

of p2 ( x ) , the new polynomial will have degree ≤ the degree of p1 ( x ) and p2 ( x ) .

1866

2

1

135. 2 ( 3 x − 5 ) ⋅ 3 ( 2 x + 1) + ( 3x − 5 ) ⋅ 3 ( 2 x + 1) ⋅ 2 3

5. 2 1 − 1

7. 3 3

2 −1 9 33

3 96

3 11 32

99 2

Quotient: 3x + 11x + 32 Remainder: 99 9. −3 1

0 −4 0 1 0 −3 9 − 15 45 − 138

1 −3

5 − 15

46 − 138

Quotient: x 4 − 3x3 + 5 x 2 − 15 x + 46 Remainder: − 138

11. 1 4 4

0 −3 0 4 4 1

1 0 5 1 2 2

4

2

1

1

2

7

Quotient: 4 x5 + 4 x 4 + x3 + x 2 + 2 x + 2 Remainder: 7

13. −1.1 0.1

0

0.2

− 0.11 0.121

0 − 0.3531

0.1 − 0.11 0.321 − 0.3531

Quotient: 0.1x 2 − 0.11x + 0.321 Remainder: –0.3531 15. 1 1 0 0 0 0 − 1 1 1 1 1

1

1 1 1 1 1 4

0 3

Quotient: x + x + x 2 + x + 1 Remainder: 0

Appendix: Review

17.

2 4 −3 −8 4 8 10 4 4 5 2 8 Remainder = 8 ≠ 0. Therefore, x − 2 is not a factor of 4 x3 − 3 x 2 − 8 x + 4 .

19. 2 3 − 6 0 − 5 10 6 0 0 − 10 3 0 0 −5 0 Remainder = 0. Therefore, x − 2 is a factor of 3x 4 − 6 x3 − 5 x + 10 .

21. −3 3

(

2 2 x3 − 4 x 2 x x − 2 = x−2 x−2

3. True;

0 0 82 0 0 27 − 9 27 − 81 − 3 9 − 27

5.

3( x + 3) 3x + 9 3 = = 2 − + −3 x x x ( 3)( 3) x −9

7.

x 2 − 2 x x( x − 2) x = = 3 x − 6 3( x − 2) 3

9.

24 x 2 24 x 2 4x = = 2 − −1 6 (2 1) 2 x x x 12 x − 6 x

11.

( y + 5)( y − 5) y 2 − 25 = 2 2 y − 8 y − 10 2 y 2 − 4 y − 5

(

3 − 9 27 1 −3 9 0 Remainder = 0. Therefore, x + 3 is a factor of 3x 6 + 82 x3 + 27 .

23. − 4 4

0 − 64 0 1 − 16

0 − 15

64 0 0 − 4

16

13.

4 − 16 0 0 1 −4 1 Remainder = 1 ≠ 0. Therefore, x + 4 is not a factor of 4 x 6 − 64 x 4 + x 2 − 15 .

25.

1 2 −1 0 2 −1 2 1 0 0 1 2

0 0 2

15.

3

0

( y + 5)( y − 5 ) 2 ( y − 5 )( y + 1)

=

y+5 2 ( y + 1)

4 x 2 x3 − 64 ⋅ 2x x 2 − 16

=

8 − 22

1 − 4 11 − 17

Section 5

=

3( x + 2) x x 3x + 6 ⋅ 2 = ⋅ 2 2 − x x + 2) ( 2)( x −4 5x 5x 3 = 5 x( x − 2)

=

5

x3 − 2 x 2 + 3x + 5 −17 = x 2 − 4 x + 11 + x+2 x+2 a + b + c + d = 1 − 4 + 11 − 17 = −9

)

(

( x − 4 ) x 2 + 4 x + 16 4 x2 = ⋅ ( x − 4)( x + 4) 2x

Remainder = 0; therefore x − 12 is a factor of 2 x 4 − x3 + 2 x − 1 . 27. − 2 1 − 2 −2

)

17.

(

2 x ⋅ 2 x ( x − 4 ) x 2 + 4 x + 16

(

2 x ( x − 4 )( x + 4 )

2 x x 2 + 4 x + 16 x+4

)

)

)

8x x2 − 1 = 8x ⋅ x + 1 10 x x 2 − 1 10 x x +1 8x x +1 = ⋅ − + 1 1 ( x )( x ) 10 x =

4 5 ( x − 1)

1. lowest terms

1867

Appendix: Review

19.

4− x 2 4 + x = 4 − x ⋅ x − 16 4x 4+ x 4x 2 x − 16 4 − x ( x + 4 )( x − 4 ) = ⋅ 4+ x 4x ( 4 − x )( x − 4 ) = 4x =−

= =

4x

x2 4 x 2 − 4 ( x + 2 )( x − 2 ) − = = 2x − 3 2x − 3 2x − 3 2x − 3

23.

1 x2 + x2 − 4 x + = x2 − 4 x x x2 − 4

(

=

=

)

2

2x − 4

(

x x2 − 4

(

)

2 x2 − 2

)

x ( x − 2 )( x + 2 )

x x − 2 x − 7 x + 6 x − 2 x − 24 x x = − ( x − 6)( x − 1) ( x − 6)( x + 4) x( x + 4) x( x − 1) = − ( x − 6)( x − 1)( x + 4) ( x − 6)( x + 4)( x − 1)

=

4 x 2 + 10 x − 4 ( x − 2)( x + 2)( x + 3)

2(2 x 2 + 5 x − 2) = ( x − 2)( x + 2)( x + 3)

1868

+

2

( x − 1)( x + 1)

2

3 ( x + 1) + 2 ( x − 1)

( x − 1)2 ( x + 1)2 3x + 3 + 2 x − 2

( x − 1)2 ( x + 1)2 5x + 1

( x − 1)2 ( x + 1)2

x − 2 x −1 + + 2 x +1 x 33. x 2x − 3 − x +1 x ⎛ ( x − 2)( x + 1) ( x − 1)( x + 2) ⎞ ⎜ ( x + 2)( x + 1) + ( x + 1)( x + 2) ⎟ ⎠ =⎝ ⎛ (2 x − 3)( x + 1) ⎞ x2 ⎜ ( x + 1)( x) − x( x + 1) ⎟⎠ ⎝

⎛ x2 − x − 2 + x2 + x − 2 ⎞ ⎜ ⎟ ( x + 2)( x + 1) ⎝ ⎠ = 2 2 ⎛ x − (2 x − x − 3) ⎞ ⎜ ⎟ x( x + 1) ⎝ ⎠ ⎛ 2x2 − 4 ⎞ ⎜ ( x + 2)( x + 1) ⎟ ⎠ =⎝ 2 ⎛ −x + x + 3 ⎞ ⎜ x( x + 1) ⎟ ⎝ ⎠

4x 2 − x2 − 4 x2 + x − 6 4x 2 = − ( x − 2)( x + 2) ( x + 3)( x − 2) 4 x( x + 3) 2( x + 2) = − ( x − 2)( x + 2)( x + 3) ( x + 3)( x − 2)( x + 2) 4 x 2 + 12 x − 2 x − 4 ( x − 2)( x + 2)( x + 3)

( x + 1)

1 ⎛ x + 1 ⎞ ⎛ x +1⎞ ⎜ ⎟ ⎜ ⎟ x = ⎝ x x ⎠ = ⎝ x ⎠ = x +1 ⋅ x = x +1 31. 1 ⎛ x 1 ⎞ ⎛ x −1 ⎞ x x −1 x −1 1− − x ⎜⎝ x x ⎟⎠ ⎜⎝ x ⎟⎠

2

=

2

1+

5x x2 + 4 x − x2 + x = = ( x − 6)( x + 4)( x − 1) ( x − 6)( x + 4)( x − 1)

27.

3

( x − 1) =

( x − 4 )2

21.

25.

29.

35.

=

2( x 2 − 2) x( x + 1) ⋅ ( x + 2)( x + 1) −( x 2 − x − 3)

=

2 x( x 2 − 2) −2 x( x 2 − 2) = 2 −( x + 2)( x − x − 3) ( x + 2)( x 2 − x − 3)

( 2 x + 3) ⋅ 3 − ( 3x − 5) ⋅ 2 6 x + 9 − 6 x + 10 = ( 3 x − 5 )2 ( 3 x − 5 )2 =

19

( 3 x − 5 )2

Appendix: Review

37.

(

)

x ⋅ 2 x − x 2 + 1 ⋅1

(x

2

)

+1

2

=

=

=

39.

2 x2 − x2 − 1

(x

)

2

+1

2

x2 − 1

(x

)

2

+1

2

( x − 1)( x + 1)

(x

2

)

+1

1 x +1 = ⇒ a = 1, b = 1, c = 0 x x 1 1 x = 1+ = 1+ 1+ 1 x +1 + 1 x ⎛ ⎞ 1+ ⎜ x ⎟ x ⎝ ⎠ x + 1 + x 2x + 1 = = x +1 x +1 ⇒ a = 2, b = 1, c = 1

45. 1 +

2

( 3x + 1) ⋅ 2 x − x 2 ⋅ 3 6 x 2 + 2 x − 3x 2 = ( 3x + 1)2 ( 3x + 1)2 =

3x2 + 2 x

( 3x + 1)2 x ( 3x + 2 ) = ( 3x + 1)2 41.

(x

2

) (x

+ 1 ⋅ 3 − ( 3x + 4 ) ⋅ 2 x 2

)

+1

2

=

=

1+

(x

2

)

+1

1 x

1+

−3x 2 − 8 x + 3 2

2

2

2

=−

( 3x − 1)( x + 3)

(x

2

)

+1

⎛ 1 1 1 ⎞ = (n − 1) ⎜ + ⎟ f R R ⎝ 1 2 ⎠ ⎛ R + R1 ⎞ 1 = (n − 1) ⎜ 2 ⎟ f ⎝ R1 ⋅ R2 ⎠ R1 ⋅ R2 = (n − 1) ( R2 + R1 ) f f 1 = R1 ⋅ R2 (n − 1) ( R2 + R1 )

2

x +1 1 = 1+ 2 x +1 ⎛ 2x +1 ⎞ ⎜ x +1 ⎟ ⎝ ⎠

2 x + 1 + x + 1 3x + 2 = 2x + 1 2x +1 ⇒ a = 3, b = 2, c = 1 =

2

2

f =

1+

= 1+

1

3x 2 + 3 − 6 x 2 − 8 x

( x + 1) − ( 3 x + 8 x − 3) = ( x + 1)

43.

1

1+

1 1+

= 1+

1 1+

1 1+

1 x

1 2x + 1 = 1+ x+2 3 ⎛ 3x + 2 ⎞ ⎜ 2x +1 ⎟ ⎝ ⎠

3x + 2 + 2 x + 1 5 x + 3 = 3x + 2 3x + 2 ⇒ a = 5, b = 3, c = 2 If we continue this process, the values of a, b and c produce the following sequences: a :1, 2,3,5,8,13, 21,.... b :1,1, 2,3,5,8,13, 21,..... c : 0,1,1, 2,3,5,8,13, 21,..... In each case we have a Fibonacci Sequence, where the next value in the list is obtained from the sum of the previous 2 values in the list. =

R1 ⋅ R2 (n − 1) ( R2 + R1 )

0.1(0.2) (1.5 − 1)(0.2 + 0.1) 0.02 0.02 2 meters = = = 0.5(0.3) 0.15 15

f =

1869

Appendix: Review

Section 6

17.

2x − 3 = 5 2x − 3 + 3 = 5 + 3 2x = 8 2x 8 = 2 2 x=4 The solution set is {4}

19.

1 5 x= 3 12 1 5 3⋅ x = 3⋅ 3 12 5 x= 4

1. x 2 − 4 = ( x − 2 )( x + 2 ) x 2 − 3x + 2 = ( x − 1)( x − 2 )

The least common denominator is ( x − 1)( x − 2 )( x + 2 ) . 3.

( x − 3)( 3x + 5) = 0 x − 3 = 0 or 3x + 5 = 0 x=3 3 x = −5 5 x=− 3 ⎧ 5 ⎫ Solution set: ⎨− ,3⎬ ⎩ 3 ⎭

⎧5⎫ The solution set is ⎨ ⎬ ⎩4⎭

5. equivalent equations 7. False; the solution is 8 . 3 3x − 8 = 0 3x = 8 x=8 3 9. add;

25 4

11. False; a quadratic equation may also have 1 repeated real solution or no real solutions. 13. 3x = 21 3x 21 = 3 3 x=7 The solution set is {7} 15.

1870

5 x + 15 = 0 5 x + 15 − 15 = 0 − 15 5 x = −15 5 x −15 = 5 5 x = −3 The solution set is {−3}

21.

6 − x = 2x + 9 6 − x − 6 = 2x + 9 − 6 −x = 2x + 3 −x − 2x = 2x + 3 − 2x −3 x = 3 −3 x 3 = −3 −3 x = −1 The solution set is {−1} .

23. 2(3 + 2 x) = 3( x − 4) 6 + 4 x = 3 x − 12 6 + 4 x − 6 = 3 x − 12 − 6 4 x = 3 x − 18 4 x − 3x = 3 x − 18 − 3 x x = −18 The solution set is {−18} .

Appendix: Review 25. 8 x − ( 2 x + 1) = 3 x − 10 8 x − 2 x − 1 = 3 x − 10 6 x − 1 = 3 x − 10 6 x − 1 + 1 = 3 x − 10 + 1 6 x = 3x − 9 6 x − 3x = 3x − 9 − 3x 3 x = −9 3x −9 = 3 3 x = −3 The solution set is {−3} .

27.

1 3 x−4 = x 2 4 ⎛1 ⎞ ⎛3 ⎞ 4⎜ x − 4⎟ = 4⎜ x ⎟ ⎝2 ⎠ ⎝4 ⎠ 2 x − 16 = 3 x 2 x − 16 + 16 = 3x + 16 2 x = 3x + 16 2 x − 3 x = 3x + 16 − 3 x

31.

⎛6⎞ ⎜ y⎟ ⎝ ⎠

−1

= 3−1

y 1 = 6 3 y 1 6⋅ = 6⋅ 6 3 y=2

The solution set is {2} . 33.

( x + 7 ) ( x − 1) = ( x + 1)2 x2 − x + 7 x − 7 = x2 + 2 x + 1 x2 + 6 x − 7 = x2 + 2 x + 1 x2 + 6 x − 7 − x2 = x2 + 2 x + 1 − x2 6x − 7 = 2x +1 6x − 7 − 2x = 2x +1− 2x 4x − 7 = 1 4x − 7 + 7 = 1+ 7 4x = 8 4x 8 = 4 4 x=2 The solution set is {2} .

− x = 16 −1( − x ) = −1(16 ) x = −16 The solution set is {−16} .

29. 0.9t = 0.4 + 0.1t 0.9t − 0.1t = 0.4 + 0.1t − 0.1t 0.8 t = 0.4 0.8 t 0.4 = 0.8 0.8 t = 0.5 The solution set is {0.5} .

2 4 + =3 y y 6 =3 y

35.

(

)

z z 2 + 1 = 3 + z3 z3 + z = 3 + z3 z3 + z − z3 = 3 + z3 − z3 z =3 The solution set is {3} .

37.

x2 = 9 x x2 − 9 x = 0 x ( x − 9) = 0 x = 0 or x − 9 = 0 x=9 The solution set is {0,9} .

1871

Appendix: Review 49.

t 3 − 9t 2 = 0

39. t

2

2 x + 3 = 5 or 2 x + 3 = − 5 2 x = 2 or 2x = − 8 x = 1 or x = −4 The solution set is {–4, 1}.

(t − 9) = 0

t 2 = 0 or t − 9 = 0 t =0 t =9 The solution set is {0,9} .

41.

51.

3 2 = 2x − 3 x + 5 3 ( x + 5 ) = 2 ( 2 x − 3)

{ }

15 + 6 = x − 6 + 6 21 = x The solution set is {21} .

53.

55.

x2 = 4 x = ±2 The solution set is {−2, 2} .

57.

2 3 10 = + x − 2 x + 5 ( x + 5 )( x − 2 )

59.

LCD = ( x + 5 )( x − 2 )

( x + 5)( x − 2 ) ( x + 5)( x − 2 ) 2 ( x + 5 ) = 3 ( x − 2 ) + 10

+

2 x + 10 = 3x − 6 + 10 2 x + 10 = 3x + 4 10 = x + 4 6= x The solution set is {6} .

47.

2x = 6 2 x = 6 or 2 x = −6 x=3 x = −3 The solution set is {–3, 3}.

1872

−2 x = 4 x=2 The solution set is {2} .

3x 2 = 12

3( x − 2)

− 2x = − 8

2x = 4

3x 2 + 6 x − 6 x = 6 x + 12 − 6 x

=

or

x = − 4 or x=4 The solution set is {–4, 4}.

( x + 2 )( 3x ) = ( x + 2 )( 6 )

2 ( x + 5)

− 2x = 8 − 2x = 8

3x 2 + 6 x = 6 x + 12

45.

1 − 4t = 5

1 − 4t = 5 or 1 − 4t = −5 −4t = 4 or − 4t = −6 3 t = −1 or t= 2 3 The solution set is −1, . 2

3 x + 15 = 4 x − 6 3x + 15 − 3 x = 4 x − 6 − 3 x 15 = x − 6

43.

2x + 3 = 5

1 2 Since absolute values are never negative, this equation has no solution. x−2 = −

x2 − 4 = 0 x2 − 4 = 0

10 ( x + 5 )( x − 2 )

x2 = 4 x = ±2 The solution set is {−2, 2} .

61.

x2 − 2 x = 3 x 2 − 2 x = 3 or x 2 − 2 x = −3 x 2 − 2 x − 3 = 0 or x 2 − 2 x + 3 = 0

( x − 3)( x + 1) = 0

2 ± 4 − 12 2 2 ± −8 no real sol. = 2

or x =

x = 3 or x = −1 The solution set is {−1, 3} .

Appendix: Review

63.

73.

x2 + x − 1 = 1 x2 + x − 1 = 1

4 x 2 − 12 x + 9 = 0

or x 2 + x − 1 = −1

x2 + x − 2 = 0

( 2 x − 3)2 = 0

or x 2 + x = 0

( x − 1)( x + 2 ) = 0

4 x 2 + 9 = 12 x

2x − 3 = 0 2x = 3

or x ( x + 1) = 0

x = 1, x = −2 or x = 0, x = −1

The solution set is {−2, − 1, 0,1} .

x=

⎧3⎫ The solution set is ⎨ ⎬ . ⎩2⎭

x2 = 4 x

65.

x2 − 4 x = 0 x ( x − 4) = 0 x = 0 or x − 4 = 0

75.

z + 4 z − 12 = 0

6 x2 − 5x − 6 = 0

( 3x + 2 )( 2 x − 3) = 0

z+6 = 0

or z − 2 = 0 z=2 z = −6 The solution set is {−6, 2} .

3x + 2 = 0

or 2 x − 3 = 0 3 x = −2 2x = 3 2 3 x=− x= 3 2 ⎧ 2 3⎫ The solution set is ⎨− , ⎬ . ⎩ 3 2⎭

2 x2 − 5x − 3 = 0

( 2 x + 1)( x − 3) = 0 2x + 1 = 0 2 x = −1

or x − 3 = 0 x=3

1 x=− 2 ⎧ 1 ⎫ The solution set is ⎨− ,3⎬ . ⎩ 2 ⎭

71.

x ( x − 7 ) + 12 = 0 x 2 − 7 x + 12 = 0

( x − 4 )( x − 3) = 0 x − 4 = 0 or x − 3 = 0 x=4 x=3 The solution set is {3, 4} .

6 x

6x2 − 5x = 6

2

( z + 6 )( z − 2 ) = 0

69.

6x − 5 =

⎛6⎞ x ( 6 x − 5) = x ⎜ ⎟ ⎝ x⎠

x=4 The solution set is {0, 4} .

67.

3 2

77.

4 ( x − 2)

x −3

+

3 −3 = x x ( x − 3)

LCD = x ( x − 3) 4x ( x − 2)

x ( x − 3)

+

3 ( x − 3)

x ( x − 3)

=

−3 x ( x − 3)

4 x ( x − 2 ) + 3 ( x − 3 ) = −3

4 x 2 − 8 x + 3 x − 9 = −3 4 x 2 − 5 x − 9 = −3 4 x2 − 5x − 6 = 0

( 4 x + 3)( x − 2 ) = 0 or x − 2 = 0 4x + 3 = 0 4 x = −3 x=2 x=−

3 4

⎧ 3 ⎫ The solution set is ⎨− , 2 ⎬ . ⎩ 4 ⎭

1873

Appendix: Review

79.

x x+3 −3 − = x − 1 x2 − x x2 + x LCD = x ( x + 1)( x − 1)

87.

2

)

x 2 − x 2 + 4 x + 3 = −3 x + 3 x 2 − x 2 − 4 x − 3 = −3 x + 3

x = 0 or x + 5 = 0

or x − 4 = 0 x = −5 x=4 The solution set is {−5, 0, 4} . 89.

( x + 1) ( x 2 − 1) = 0 ( x + 1)( x − 1)( x + 1) = 0

−4 x = −3x + 6 −x = 6

x = −6 The solution set is {−6} .

2

)(

x +1 = 0

)

2 x3 − x 2 − 8 x + 4 = 0 x 2 ( 2 x − 1) − 4 ( 2 x − 1) = 0

x = ±2 or x = ±1 The solution set is {−2, −1,1, 2} .

( 2 x − 1) ( x 2 − 4 ) = 0 ( 2 x − 1)( x − 2 )( x + 2 ) = 0

( x + 2 )2 + 7 ( x + 2 ) + 12 = 0 2 let p = x + 2 → p 2 = ( x + 2 )

2 x − 1 = 0 or x − 2 = 0 or x + 2 = 0 2x = 1 1 x= 2

2

p + 7 p + 12 = 0

( p + 3)( p + 4 ) = 0 or p = −4 → x + 2 = −4 → x = −6

The solution set is {−6, −5} . 85. 2 ( s + 1) − 5 ( s + 1) = 3 2 p2 − 5 p = 3 2 p2 − 5 p − 3 = 0

( 2 p + 1)( p − 3) = 0 2 p + 1 = 0 or p − 3 = 0 1 1 3 → s +1 = − → s = − 2 2 2 or p = 3 → s + 1 = 3 → s = 2

p=−

{ }

3 The solution set is − , 2 . 2

1874

{

x = −2

}

93. x 2 = 25 ⇒ x = ± 25 ⇒ x = ±5 The solution set is {−5, 5} .

2

2

x=2

1 The solution set is −2, , 2 . 2

p + 3 = 0 or p + 4 = 0 p = −3 → x + 2 = −3 → x = −5

let p = s + 1 → p = ( s + 1)

2 x3 + 4 = x 2 + 8 x

91.

− 4 x2 − 1 = 0

2

or x − 1 = 0

x = −1 x =1 The solution set is {−1,1} .

x 2 − 4 = 0 or x 2 − 1 = 0

83.

x3 + x 2 − x − 1 = 0 x 2 ( x + 1) − 1( x + 1) = 0

−4 x − 3 = −3 x + 3

(x

)

x ( x + 5 )( x − 4 ) = 0

x 2 − ( x + 1)( x + 3) = −3 ( x − 1)

81. x 4 − 5 x 2 + 4 = 0

(

x x 2 + x − 20 = 0

x+3 −3 ⎞ ⎛ x − = x ( x + 1)( x − 1) ⎜ 2 ⎟ ⎝ x − 1 x2 − x x2 + x ⎠

(

x3 + x 2 − 20 x = 0

95.

( x − 1)2 = 4 x −1 = ± 4 x − 1 = ±2 x − 1 = 2 or x − 1 = −2 ⇒ x = 3 or x = −1 The solution set is {−1, 3} .

Appendix: Review

97.

( 2 x + 3)2 = 9

109.

2x + 3 = ± 9 2 x + 3 = ±3 2 x + 3 = 3 or 2x + 3 = −3 ⇒ x = 0 or x = −3 The solution set is {−3, 0} .

1 =0 2 1 1 x2 + x − = 0 3 6 1 1 x2 + x = 3 6 1 1 1 1 x2 + x + = + 3 36 6 36 3x 2 + x −

2

1⎞ 7 ⎛ ⎜ x + 6 ⎟ = 36 ⎝ ⎠

2

⎛8⎞ 99. ⎜ ⎟ = 42 = 16 ⎝2⎠ 2

1 ⎛ 1/ 2 ⎞ ⎛1⎞ 101. ⎜ ⎟ = ⎜ 4 ⎟ = 16 2 ⎝ ⎠ ⎝ ⎠ 2

⎛⎛ 2⎞⎞ 2 ⎜⎜− 3⎟ ⎟ ⎠ ⎟ = ⎛−1⎞ = 1 103. ⎜ ⎝ ⎜ ⎟ 9 ⎜ 2 ⎟ ⎝ 3⎠ ⎜ ⎟ ⎝ ⎠

105.

x+

2

x 2 + 4 x = 21

1 7 =± 6 36

1 7 =± 6 6 −1 ± 7 x= 6 ⎧ −1 − 7 −1 + 7 ⎫ The solution set is ⎨ , ⎬. 6 ⎭ ⎩ 6 x+

111. x 2 − 4 x + 2 = 0 a = 1, b = − 4, c = 2

x 2 + 4 x + 4 = 21 + 4

( x + 2 )2 = 25 x + 2 = ± 25 ⇒ x + 2 = ±5 x = −2 ± 5 ⇒ x = 3 or x = −7 The solution set is {−7,3} . 1 3 107. x 2 − x − = 0 2 16 1 3 x2 − x = 2 16 1 1 3 1 x2 − x + = + 2 16 16 16 2

1⎞ 1 ⎛ ⎜x− 4⎟ = 4 ⎝ ⎠

x=

4 ± 16 − 8 4 ± 8 = 2 2 4±2 2 = = 2± 2 2 =

{

1 1 1 =± =± 4 4 2 1 1 3 1 x= ± ⇒x= or x = − 4 2 4 4 1 3 The solution set is − , . 4 4

{ }

}

The solution set is 2 − 2, 2 + 2 .

113. x 2 − 5 x − 1 = 0 a = 1, b = −5, c = −1 x=

x−

−(− 4) ± (− 4) 2 − 4(1)(2) 2(1)

− ( −5 ) ±

( −5)2 − 4 (1)( −1) 2 (1)

5 ± 25 + 4 5 ± 29 = 2 2 ⎧⎪ 5 − 29 5 + 29 ⎫⎪ , The solution set is ⎨ ⎬. 2 ⎪⎭ ⎪⎩ 2 =

1875

Appendix: Review 115. 2 x 2 − 5 x + 3 = 0 a = 2, b = − 5, c = 3 x= =

−(− 5) ± (− 5) 2 − 4(2)(3) 2(2) 5 ± 25 − 24 5 ± 1 = 4 4

{ }

3 The solution set is 1, . 2

117. 4 y 2 − y + 2 = 0 a = 4, b = − 1, c = 2 −(−1) ± (− 1) 2 − 4(4)(2) y= 2(4) 1 ± 1 − 32 1 ± −31 = 8 8 No real solution.

=

4 x2 = 1 − 2 x

119.

4x2 + 2 x − 1 = 0 a = 4, b = 2, c = −1 x=

− 2 ± 22 − 4(4)(−1) 2(4)

− 2 ± 4 + 16 − 2 ± 20 = 8 8 − 2 ± 2 5 −1 ± 5 = = 8 4 ⎧ −1 − 5 −1 + 5 ⎫ The solution set is ⎨ , ⎬. 4 ⎭ ⎩ 4 =

121. x 2 + 3 x − 3 = 0 a = 1, b = 3, c = −3 x=

− 3±

( 3)

2

− 4 (1)( −3)

2 (1)

− 3 ± 3 + 12 − 3 ± 15 = 2 2 ⎧⎪ − 3 − 15 − 3 + 15 ⎫⎪ The solution set is ⎨ , ⎬. 2 2 ⎪⎩ ⎪⎭ =

123. x 2 − 5 x + 7 = 0 a = 1, b = −5, c = 7 b 2 − 4ac = (−5) 2 − 4(1) ( 7 ) = 25 − 28 = −3

1876

Since the discriminant < 0, we have no real solutions. 125. 9 x 2 − 30 x + 25 = 0 a = 9, b = −30, c = 25 b 2 − 4ac = (−30) 2 − 4(9) ( 25 ) = 900 − 900 = 0 Since the discriminant = 0, we have one repeated real solution.

127. 3x 2 + 5 x − 8 = 0 a = 3, b = 5, c = −8 b 2 − 4ac = (5) 2 − 4(3) ( −8 ) = 25 + 96 = 121 Since the discriminant > 0, we have two unequal real solutions.

129. Solving for R: 1 1 1 = + R R1 R2 ⎛ 1 1 ⎞ ⎛1⎞ RR1 R2 ⎜ ⎟ = RR1 R2 ⎜ + ⎟ R R R ⎝ ⎠ ⎝ 1 2 ⎠ R1 R2 = RR2 + RR1

R1 R2 = R ( R2 + R1 ) R1 R2 R ( R2 + R1 ) = R2 + R1 R2 + R1 R1 R2 RR = R or R = 1 2 R2 + R1 R1 + R2

131. Solving for R: mv 2 F= R ⎛ mv 2 ⎞ RF = R ⎜ ⎟ ⎝ R ⎠ RF = mv 2 RF mv 2 = F F mv 2 R= F

Appendix: Review 133. Solving for r: a S= 1− r

For ax 2 − bx + c = 0 : x1*

a

(1 − r ) ⋅ S = (1 − r ) ⋅ 1 − r =

S a 1− r = S a 1− r −1 = −1 S a −r = − 1 S a r = 1− S

x2* =

or r =

S −a S

−b − b 2 − 4ac −b + b 2 − 4ac + 2a 2a

−b − b 2 − 4ac − b + b 2 − 4ac 2a −2b = 2a b =− a =

137. In order to have one repeated solution, we need the discriminant to be 0. a = k , b = 1, c = k b 2 − 4ac = 0 12 − 4 ( k )( k ) = 0

− ( −b ) +

( −b )2 − 4ac

2a ⎛ −b − b 2 − 4ac ⎞ ⎟ = −⎜ ⎜ ⎟ 2a ⎝ ⎠ = − x1

141. a.

−b − b 2 − 4ac −b + b 2 − 4ac and x2 = 2a 2a

x1 + x2 =

2a

and

135. The roots of a quadratic equation are x1 =

( −b )2 − 4ac

⎛ −b + b 2 − 4ac ⎞ ⎟ = −⎜ ⎜ ⎟ 2a ⎝ ⎠ = − x2

(1 − r ) S = a (1 − r ) S a S

=

− ( −b ) −

x 2 = 9 and x = 3 are not equivalent because they do not have the same solution set. In the first equation we can also have x = −3 .

b.

x = 9 and x = 3 are equivalent because 9 =3.

c.

( x − 1)( x − 2 ) = ( x − 1)2

and x − 2 = x − 1 are

not equivalent because they do not have the same solution set. The first equation has the solution set {1} while the second equation has no solutions. 143. Answers will vary. 145. Answers will vary. Knowing the discriminant allows us to know how many real solutions the equation will have. 147. Answers will vary.

2

1 − 4k = 0 4k 2 = 1 1 k2 = 4 k =± k=

1 2

1 4 or k = −

1 2

139. For ax 2 + bx + c = 0 : x1 =

−b − b 2 − 4ac −b + b 2 − 4ac and x2 = 2a 2a

1877

Appendix: Review

Section 7

27.

1. True; if the complex number contains an imaginary part, its square can be negative. For

example, ( 2i ) = 4i 2 = 4 ( −1) = −4 . 2

3. False; in the complex number system, a quadratic equation has two solutions. 5.

2

⎛1 3 ⎞ 1 ⎛ 1 ⎞⎛ 3 ⎞ 3 2 i ⎟ = + 2⎜ ⎟⎜ i⎟+ i 29. ⎜ + 4 ⎝ 2 ⎠⎝ 2 ⎠ 4 ⎝2 2 ⎠

{−2i, 2i}

=

1 3 3 1 3 i + (−1) = − + i + 4 2 4 2 2

7. True; the set of real numbers is a subset of the set of complex numbers.

31. (1 + i ) 2 = 1 + 2i + i 2 = 1 + 2i + (−1) = 2i

9. (2 − 3i ) + (6 + 8i ) = (2 + 6) + (−3 + 8)i = 8 + 5i

33. i 23 = i 22 +1 = i 22 ⋅ i = i 2

11. (−3 + 2i ) − (4 − 4i ) = (−3 − 4) + (2 − (− 4))i = −7 + 6i

( )

15. 3(2 − 6i ) = 6 − 18 i 17. 2i (2 − 3i ) = 4i − 6i 2 = 4i − 6(−1) = 6 + 4i 19. (3 − 4i )(2 + i ) = 6 + 3i − 8i − 4i 2 = 6 − 5i − 4(−1) = 10 − 5i 21. (− 6 + i )(− 6 − i ) = 36 + 6i − 6i − i 2 = 36 − (−1) = 37 23.

25.

10 10 3 + 4i 30 + 40i = ⋅ = 3 − 4i 3 − 4i 3 + 4i 9 + 12i − 12i − 16i 2 30 + 40i 30 + 40i = = 9 − 16(−1) 25 30 40 = + i 25 25 6 8 = + i 5 5

2 + i 2 + i −i − 2i − i 2 = ⋅ = i i −i −i 2 − 2i − (−1) 1 − 2i = = = 1 − 2i −(−1) 1

11

⋅ i = (−1)11 i = −i

1 1 1 1 = = = i15 i14 +1 i14 ⋅ i (i 2 )7 ⋅ i 1 1 1 i i i = = = ⋅ = 2 = =i 7 − − − − i i i ( 1) −i (−1) i

35. i −15 =

13. (2 − 5i ) − (8 + 6i ) = (2 − 8) + (−5 − 6)i = −6 − 11i

1878

6 − i 6 − i 1 − i 6 − 6i − i + i 2 = ⋅ = 1 + i 1 + i 1 − i 1 − i + i − i2 6 − 7i + (−1) 5 − 7i 5 7 = = = − i 1 − (−1) 2 2 2

( )

37. i 6 − 5 = i 2

3

− 5 = (−1)3 − 5 = −1 − 5 = − 6

39. 6i 3 − 4i 5 = i 3 (6 − 4i 2 )

= i 2 ⋅ i (6 − 4(−1)) = −1 ⋅ i (10) = −10 i 41. (1 + i )3 = (1 + i )(1 + i )(1 + i ) = (1 + 2i + i 2 )(1 + i ) = (1 + 2i − 1)(1 + i ) = 2i (1 + i ) = 2i + 2i 2 = 2i + 2(−1) = − 2 + 2i

43. i 7 (1 + i 2 ) = i 7 (1 + (−1)) = i 7 (0) = 0

( ) + (i )

45. i 6 + i 4 + i 2 + 1 = i 2

3

2 2

+ i2 + 1

= (−1)3 + (−1) 2 + (−1) + 1 = −1 + 1 − 1 + 1 =0

47.

− 4 = 2i

49.

− 25 = 5i

Appendix: Review

51.

(3 + 4i )(4i − 3) = 12i − 9 + 16i 2 − 12i

5x2 − 2 x + 1 = 0 a = 5, b = −2, c = 1

= −9 + 16(−1) = − 25

b 2 − 4ac = ( −2 ) − 4(5)(1) = 4 − 20 = −16 2

= 5i

x=

53. x 2 + 4 = 0 x 2 = −4

}

1 2 1 2 − i, + i . 5 5 5 5

b 2 − 4ac = 12 − 4(1)(1) = 1 − 4 = −3 x=

x = −4 or x = 4

57. x 2 − 6 x + 13 = 0 a = 1, b = − 6, c = 13,

67. x3 − 8 = 0

(

)

( x − 2) x 2 + 2 x + 4 = 0

b 2 − 4ac = (− 6) 2 − 4(1)(13) = 36 − 52 = −16

x−2 = 0⇒ x = 2

− (− 6) ± −16 6 ± 4i = = 3 ± 2i 2(1) 2

or x 2 + 2 x + 4 = 0 a = 1, b = 2, c = 4

The solution set is {3 − 2i,3 + 2i} .

b 2 − 4ac = 22 − 4(1)(4) = 4 − 16 = −12

2

59. x − 6 x + 10 = 0 a = 1, b = − 6, c = 10

x=

− (− 6) ± − 4 6 ± 2i x= = = 3±i 2(1) 2

x 4 = 16

(x

61. 8 x 2 − 4 x + 1 = 0 a = 8, b = − 4, c = 1 b 2 − 4ac = (− 4) 2 − 4(8)(1) = 16 − 32 = −16

x 4 − 16 = 0

)( ( x − 2)( x + 2) ( x 2

}

) + 4) = 0

− 4 x2 + 4 = 0 2

x − 2 = 0 or x + 2 = 0 or x 2 + 4 = 0 x = 2 or

1 1 1 1 − i, + i . 4 4 4 4

}

The solution set is 2, −1 − 3i, −1 + 3i . 69.

The solution set is {3 − i, 3 + i} .

− (− 4) ± −16 4 ± 4i 1 1 = = ± i 2(8) 16 4 4

− 2 ± −12 − 2 ± 2 3 i = = −1 ± 3i 2(1) 2

{

b 2 − 4ac = (− 6) 2 − 4(1)(10) = 36 − 40 = − 4

{

−1 ± −3 −1 ± 3 i 1 3 = =− ± i 2(1) 2 2 2

⎧ 1 3 1 3 ⎫ i, − + i⎬ . The solution set is ⎨− − 2 2 2 2 ⎩ ⎭

The solution set is {−4, 4} .

The solution set is

{

65. x 2 + x + 1 = 0 a = 1, b = 1, c = 1,

55. x 2 − 16 = 0 ( x + 4 )( x − 4 ) = 0

x=

−(−2) ± −16 2 ± 4i 1 2 = = ± i 2(5) 10 5 5

The solution set is

x = ± −4 x = ±2i The solution set is {−2i, 2i} .

x=

5x2 + 1 = 2 x

63.

x = −2 or x 2 = −4

x = 2 or x = −2 or x = ± −4 = ±2i The solution set is {−2, 2, −2i, 2i} .

1879

Appendix: Review

71.

x 4 + 13x 2 + 36 = 0

(x

)(

2

)

2

+9 x +4 = 0

x2 + 9 = 0

or x 2 + 4 = 0

x 2 = −9

or

x 2 = −4

x = ± −9 or

87. z + z = (a + b i ) + (a + b i ) = a + bi + a − bi = 2a z − z = a + b i − (a + b i ) = a + b i − (a − b i)

x = ± −4

= a + bi − a + bi

or x = ±3i x = ±2i The solution set is {− 3i, 3i, −2i, 2i} .

73. 3x 2 − 3 x + 4 = 0 a = 3, b = − 3, c = 4 2

= 2b i

89. z + w = (a + b i ) + (c + d i ) = (a + c) + (b + d ) i = (a + c) − (b + d ) i = ( a − b i ) + (c − d i )

2

b − 4ac = (− 3) − 4(3)(4) = 9 − 48 = −39 The equation has two complex solutions that are conjugates of each other. 75.

2 x 2 + 3x = 4 2 x2 + 3x − 4 = 0 a = 2, b = 3, c = − 4 b 2 − 4ac = 32 − 4(2)(−4) = 9 + 32 = 41 The equation has two unequal real solutions.

77. 9 x 2 − 12 x + 4 = 0 a = 9, b = −12, c = 4 b 2 − 4ac = (− 12) 2 − 4(9)(4) = 144 − 144 = 0 The equation has a repeated real solution.

79. The other solution is 2 + 3i = 2 − 3 i. 81. z + z = 3 − 4i + 3 − 4i = 3 − 4i + 3 + 4i = 6 83. z ⋅ z = (3 − 4i )(3 − 4i ) = (3 − 4i )(3 + 4i ) = 9 + 12i − 12i − 16i 2 = 9 − 16(−1) = 25 V 18 + i 18 + i 3 + 4i = = ⋅ I 3 − 4i 3 − 4i 3 + 4i 54 + 72i + 3i + 4i 2 54 + 75i − 4 = = 9 + 16 9 + 12i − 12i − 16i 2 50 + 75i = = 2 + 3i 25 The impedance is 2 + 3i ohms.

85. Z =

1880

= a + bi + c + d i = z +w

91 – 93. Answers will vary.

Section 8 1. mathematical modeling 3. uniform motion 5. True; this is the uniform motion formula. 7. Let A represent the area of the circle and r the radius. The area of a circle is the product of π times the square of the radius: A = π r 2 9. Let A represent the area of the square and s the length of a side. The area of the square is the square of the length of a side: A = s 2 11. Let F represent the force, m the mass, and a the acceleration. Force equals the product of the mass times the acceleration: F = ma 13. Let W represent the work, F the force, and d the distance. Work equals force times distance: W = Fd 15. C = total variable cost in dollars, x = number of dishwashers manufactured: C = 150 x 17. Let x represent the amount of money invested in bonds. Then 50, 000 − x represents the amount of money invested in CD's. Since the total

Appendix: Review

interest is to be \$6,000, we have: 0.15 x + 0.07(50, 000 − x) = 6, 000

(100 )( 0.15 x + 0.07(50, 000 − x) ) = ( 6, 000 )(100 ) 15 x + 7(50, 000 − x) = 600, 000 15 x + 350, 000 − 7 x = 600, 000 8 x + 350, 000 = 600, 000 8 x = 250, 000 x = 31, 250 \$31,250 should be invested in bonds at 15% and \$18,750 should be invested in CD's at 7%.

19. Let x represent the amount of money loaned at 8%. Then 12, 000 − x represents the amount of money loaned at 18%. Since the total interest is to be \$1,000, we have: 0.08 x + 0.18(12, 000 − x) = 1, 000

(100 )( 0.08 x + 0.18(12, 000 − x) ) = (1, 000 )(100 ) 8 x + 18(12, 000 − x) = 100, 000 8 x + 216, 000 − 18 x = 100, 000 −10 x + 216, 000 = 100, 000 −10 x = −116, 000 x = 11, 600 \$11,600 is loaned at 8% and \$400 is at 18%.

21. Let x represent the number of pounds of Earl Grey tea. Then 100 − x represents the number of pounds of Orange Pekoe tea. 5 x + 3(100 − x) = 4.50(100) 5 x + 300 − 3 x = 450 2 x + 300 = 450 2 x = 150 x = 75 75 pounds of Earl Grey tea must be blended with 25 pounds of Orange Pekoe. 23. Let x represent the number of pounds of cashews. Then x + 60 represents the number of pounds in the mixture. 9 x + 3.50(60) = 7.50( x + 60) 9 x + 210 = 7.50 x + 450 1.5 x = 240 x = 160 160 pounds of cashews must be added to the 60 pounds of almonds.

25. Let r represent the speed of the current. Rate Time Distance 20 = 1 16 − r 60 3 1 Downstream 16 + r 15 = 60 4

Upstream

16− r 3 16+ r 4

Since the distance is the same in each direction: 16 − r 16 + r = 3 4 4(16 − r ) = 3(16 + r ) 64 − 4r = 48 + 3r 16 = 7 r 16 r= ≈ 2.286 7 The speed of the current is approximately 2.286 miles per hour. 27. Let r represent the speed of the current. Rate Time Distance 10 Upstream 15 − r 10 15 − r 10 Downstream 15 + r 10 15 + r Since the total time is 1.5 hours, we have: 10 10 + = 1.5 15 − r 15 + r 10(15 + r ) + 10(15 − r ) = 1.5(15 − r )(15 + r ) 150 + 10r + 150 − 10r = 1.5(225 − r 2 ) 300 = 1.5(225 − r 2 ) 200 = 225 − r 2 r 2 − 25 = 0 (r − 5)(r + 5) = 0 r = 5 or r = −5 Speed must be positive, so disregard r = −5 . The speed of the current is 5 miles per hour.

29. Let r represent Karen’s normal walking speed. Rate Time Distance 50 r + 2.5 With walkway 50 r + 2.5 50 Against walkway r − 2.5 50 r − 2.5 Since the total time is 40 seconds:

1881

Appendix: Review 50 50 + = 40 r + 2.5 r − 2.5 50(r − 2.5) + 50(r + 2.5) = 40(r − 2.5)(r + 2.5) 50r − 125 + 50r + 125 = 40(r 2 − 6.25)

35. l = length of the garden w = width of the garden a.

100r = 40r 2 − 250 0 = 40r 2 − 100r − 250 0 = 4r 2 − 10r − 25 r=

−(−10) ± (−10) 2 − 4(4)(−25) 2(4)

10 ± 500 10 ± 10 5 5 ± 5 5 = = 8 8 4 r ≈ 4.05 or r ≈ −1.55 Speed must be positive, so disregard r ≈ −1.55 . Karen’ normal walking speed is approximately 4.05 feet per second. =

31. Let w represent the width of a regulation doubles tennis court. Then 2w + 6 represents the length. The area is 2808 square feet: w(2 w + 6) = 2808 2w2 + 6w = 2808 2w2 + 6 w − 2808 = 0 w2 + 3w − 1404 = 0 ( w + 39)( w − 36) = 0 w + 39 = 0 or w − 36 = 0 w = −39 or w = 36 The width must be positive, so disregard w = −39 . The width of a regulation doubles tennis court is 36 feet and the length is 2(36) + 6 = 78 feet.

33. Let t represent the time it takes to do the job together. Time to do job Part of job done in one minute 1 Trent 30 30 1 Lois 20 20 1 Together t t 1 1 1 + = 30 20 t 2t + 3t = 60 5t = 60 t = 12 Working together, the job can be done in 12 minutes.

1882

The length of the garden is to be twice its width. Thus, l = 2w . The dimensions of the fence are l + 4 and w+4 . The perimeter is 46 feet, so: 2(l + 4) + 2( w + 4) = 46 2(2w + 4) + 2( w + 4) = 46 4w + 8 + 2 w + 8 = 46 6w + 16 = 46 6w = 30 w=5 The dimensions of the garden are 5 feet by 10 feet.

b. Area = l ⋅ w = 5 ⋅10 = 50 square feet c.

If the dimensions of the garden are the same, then the length and width of the fence are also the same (l + 4) . The perimeter is 46 feet, so: 2(l + 4) + 2(l + 4) = 46 2l + 8 + 2l + 8 = 46 4l + 16 = 46 4l = 30 l = 7.5 The dimensions of the garden are 7.5 feet by 7.5 feet.

d. Area = l ⋅ w = 7.5(7.5) = 56.25 square feet. 37. Let t represent the time it takes for the defensive back to catch the tight end. Time to run Time 100 yards

Rate

Distance

Tight End

12 sec

t

100 = 25 12 3

25 t 3

Def. Back

10 sec

t

100 = 10 10

10t

Since the defensive back has to run 5 yards farther, we have: 25 t + 5 = 10t 3 25t + 15 = 30t 15 = 5 t t =3 10t = 30 → The defensive back will catch the tight end at the 45 yard line (15 + 30 = 45).

Appendix: Review 39. Let x represent the number of gallons of pure water. Then x + 1 represents the number of gallons in the 60% solution. ( % )( gallons ) + ( % )( gallons ) = ( % )( gallons ) 0 ( x ) + 1(1) = 0.60( x + 1) 1 = 0.6 x + 0.6 0.4 = 0.6 x x=

4 2 = 6 3

2 gallon of pure water should be added. 3

41. Let x represent the number of ounces of water to be evaporated; the amount of salt remains the same. Therefore, we get 0.04(32) = 0.06(32 − x) 1.28 = 1.92 − 0.06 x 0.06 x = 0.64 0.64 64 32 = = = 10 23 x= 0.06 6 3 10 23 ≈ 10.67 ounces of water need to be

evaporated. 43. Let x represent the number of grams of pure gold. Then 60 − x represents the number of grams of 12 karat gold to be used. 1 2 x + (60 − x) = (60) 2 3 x + 30 − 0.5 x = 40 0.5 x = 10 x = 20 20 grams of pure gold should be mixed with 40 grams of 12 karat gold. 45. Let t represent the time it takes for Mike to catch up with Dan. Since the distances are the same, we have: 1 1 t = (t + 1) 6 9 3t = 2t + 2 t=2 Mike will pass Dan after 2 minutes, which is a 1 distance of mile. 3

3 t + =1 4 9 27 + 4t = 36 4t = 9 9 = 2.25 4 The auxiliary pump must run for 2.25 hours. It must be started at 9:45 a.m. for the tanker to be emptied by noon. t=

49. Let t represent the time for the tub to fill with the faucets on and the stopper removed. Since one tub is being filled, we have: t ⎛ t ⎞ + − =1 15 ⎜⎝ 20 ⎟⎠ 4t − 3t = 60 t = 60 60 minutes is required to fill the tub. 51. Let t represent the time spent running. Then 5 − t represents the time spent biking. Rate Time Distance Run 6 t 6t Bike 25 5 − t 25(5 − t )

The total distance is 87 miles: 6t + 25(5 − t ) = 87 6t + 125 − 25t = 87 −19t + 125 = 87 −19t = −38 t=2 The time spent running is 2 hours, so the distance of the run is 6(2) = 12 miles. The distance of the bicycle race is 25(5 − 2) = 75 miles. 100 meters/sec. In 9.99 seconds, 12 100 Burke will run (9.99) = 83.25 meters. 12 Lewis would win by 16.75 meters.

53. Burke's rate is

55. Let x = length of side of original sheet in feet. Length of box: x − 2 feet Width of box: x − 2 feet Height of box: 1 foot

47. Let t represent the time the auxiliary pump needs to run. Since the two pumps are emptying one tanker, we have:

1883

Appendix: Review

Section 9

V = l ⋅ w⋅h 4 = ( x − 2 )( x − 2 )(1)

1. x ≥ −2

4 = x2 − 4 x + 4

−2

0 = x2 − 4 x 0 = x ( x − 4) x = 0 or x = 4 Discard x = 0 since that is not a feasible length for the original sheet. Therefore, the original sheet should measure 4 feet on each side.

57. Let x be the original selling price of the shirt. Profit = Revenue − Cost 4 = x − 0.40 x − 20 → 24 = 0.60 x → x = 40 The original price should be \$40 to ensure a profit of \$4 after the sale. If the sale is 50% off, the profit is: 40 − 0.50(40) − 20 = 40 − 20 − 20 = 0 At 50% off there will be no profit. 59. It is impossible to mix two solutions with a lower concentration and end up with a new solution with a higher concentration.

Algebraic Solution: Let x = the number of liters of 25% solution. ( % )( liters ) + ( % )( liters ) = ( % )( liters ) 0.25 x + 0.48 ( 20 ) = 0.58 ( 20 + x ) 0.25 x + 9.6 = 10.6 + 0.58 x −0.33 x = 1 x ≈ −3.03 liters

3.

−2 = 2

5. negative 7. −5,5 9. True 11. Interval: [ 0, 2]

Inequality: 0 ≤ x ≤ 2 13. Interval: [ 2, ∞ )

Inequality: x ≥ 2 15. Interval: [ 0,3)

Inequality: 0 ≤ x < 3 17. a.

(

)

≈ 2.00424 hrs Thus, with no wind, the ground speed is 919 ≈ 458.53 . Therefore, the tail wind is 2.00424 550 − 458.53 = 91.47 knots .

30

b.

4 > −3 4 − 5 > −3 − 5 −1 > −8

c.

4 > −3 3 ( 4 ) > 3 ( −3) 12 > −9

1884

0

Appendix: Review d.

4 > −3

41. If x > − 4, then x + 4 > 0.

−2 ( 4 ) < −2 ( −3)

43. If x ≥ − 4, then 3 x ≥ −12.

−8 < 6

21. a.

45. If x > 6, then − 2x < −12.

2x +1 < 2 2x + 1 + 3 < 2 + 3 2x + 4 < 5

b.

47. If x ≥ 5, then − 4 x ≤ − 20. 49. If 2 x > 6, then x > 3.

2x +1 < 2 2x + 1 − 5 < 2 − 5 2 x − 4 < −3

c.

1 51. If − x ≤ 3, then x ≥ − 6. 2

2x + 1 < 2

53.

3 ( 2 x + 1) < 3 ( 2 ) 6x + 3 < 6

d.

2x + 1 < 2

x +1 < 5 x +1−1 < 5 −1 x −2 ( 2 )

x < 4} or (−∞, 4)

−4 x − 2 > −4

0

23. [0, 4]

55. 4

0

25. [4, 6) 4

0

27.

29.

1− 2x ≤ 3 − 2x ≤ 2 x ≥ −1 The solution set is { x x ≥ −1} or [−1, ∞) .

6

−1

[ 4, ∞ ) 0

4

0

57. 3x − 7 > 2 3x > 9 x>3

4

( −∞, −4 )

The solution set is { x x > 3} or (3, ∞) .

−4

0

31. 2 ≤ x ≤ 5 0

33. −3 < x < − 2 −3

−2

0

35. x ≥ 4

37. x < −3 −3

39. If x < 5, then x − 5 < 0.

0

2

61. − 2( x + 3) < 8 − 2x − 6 < 8 − 2 x < 14 x > −7

4

0

3

59. 3x − 1 ≥ 3 + x 2x ≥ 4 x≥2 The solution set is { x x ≥ 2} or [2, ∞) .

5

2

0

0

The solution set is { x x > − 7} or (− 7, ∞) . −7

0

1885

Appendix: Review 71. − 5 ≤ 4 − 3x ≤ 2 − 9 ≤ − 3x ≤ − 2 2 3≥ x≥ 3

63. 4 − 3(1 − x) ≤ 3 4 − 3 + 3x ≤ 3 3x + 1 ≤ 3 3x ≤ 2 2 x≤ 3

⎧ 2 ⎫ ⎡2 ⎤ The solution set is ⎨ x ≤ x ≤ 3⎬ or ⎢ , 3⎥ . ⎣3 ⎦ ⎩ 3 ⎭

⎧ The solution set is ⎨ x x ≤ ⎩

_2 1

0

65.

2⎫ 2⎤ ⎛ ⎬ or ⎜ −∞, ⎥ . 3⎭ 3⎦ ⎝ 73.

3

1 ( x − 4) > x + 8 2 1 x−2 > x +8 2 1 − x > 10 2 x < − 20

The solution set is { x x < − 20} or (−∞, − 20) . 0

−20

x x 67. ≥ 1− 2 4 2x ≥ 4 − x 3x ≥ 4 4 x≥ 3

1

3

The solution set is { x 3 ≤ x ≤ 5} or [3, 5] .

1886

2x −1 x2 − 1 − x − 6 > −1 −x > 5 x < −5 The solution set is { x x < − 5} or ( −∞, − 5 ) . −5

0

Appendix: Review 79. x(4 x + 3) ≤ (2 x + 1) 2 2

⎧ 10 ⎫ ⎛ 10 ⎞ The solution set is ⎨ x x > ⎬ or ⎜ , ∞ ⎟ . 3 ⎝ 3 ⎠ ⎩ ⎭

2

4 x + 3x ≤ 4 x + 4 x + 1 3x ≤ 4 x + 1 −x ≤ 1 x ≥ −1 The solution set is { x x ≥ −1} or [ −1, ∞ ) . −1

81.

87. 0 < ( 2 x − 4 )

5⎫ ⎡1 ⎬ or ⎢ , 4⎭ ⎣2

5⎞ . 4 ⎠⎟

( 4 x + 2 )−1 < 0

0

89.

1 2

2x < 8

0 − _1 −1

−4

2

2 3 < x 5 2 2 3 < 0< and x x 5 2 Since > 0 , this means that x > 0 . Therefore, x 2 3 < x 5 ⎛2⎞ ⎛3⎞ 5x ⎜ ⎟ < 5x ⎜ ⎟ ⎝ x⎠ ⎝5⎠ 10 < 3 x 10 3} or ( 3, ∞ ) .

1 0 . 2x − 4 Therefore, 1 1 < 2x − 4 2 1 1 < 2( x − 2) 2 1 ⎞ ⎛ ⎛1⎞ 2( x − 2) ⎜ ⎟ < 2( x − 2) ⎜ 2 ⎟ 2( 2) x − ⎝ ⎠ ⎝ ⎠ 1< x−2 3< x

5 4

1 2

−1

0
12 3x < −12 or 3 x > 12 x < − 4 or x > 4

{x

x < −4 or x > 4} or ( −∞, −4 ) ∪ ( 4, ∞ ) −4

93.

0

4

2x −1 ≤ 1 −1 ≤ 2 x − 1 ≤ 1 0 ≤ 2x ≤ 2 0 ≤ x ≤1 x { | 0 ≤ x ≤ 1} or [0,1] 0

1

2

1887

Appendix: Review 95.

1− 2x > 3 1 − 2 x < −3 or 1 − 2 x > 3 −2 x < − 4 or − 2 x > 2 x > 2 or x < −1 { x x < −1 or x > 2} or ( −∞, −1) ∪ ( 2, ∞ ) −1

97.

0

105.

3x + 6 We need 3x + 6 ≥ 0 3x ≥ −6 x ≥ −2 To the domain is { x x ≥ −2} or [ −2, ∞ ) .

107. 21 < young adult's age < 30

2

109. a.

−4 x + −5 ≤ 9 −4 x + 5 ≤ 9 −4 x ≤ 4 −4 ≤ 4 x ≤ 4 −1 ≤ x ≤ 1 { x | −1 ≤ x ≤ 1} or [ −1,1] −1

99.

0

b. Let x = age at death. x − 30 ≥ 51.03 x ≥ 81.03 Therefore, the average life expectancy for a 30-year-old female in 2005 will be greater than or equal to 81.03 years.

1

− 2 x ≥ −4

c.

2x ≥ 4 2 x ≤ − 4 or 2 x ≥ 4 x ≤ − 2 or x ≥ 2

{x

x ≤ − 2 or x ≥ 2} or ( −∞, −2] ∪ [ 2, ∞ ) −2

0

2

101. x differs from 2 by less than x−2
2 x + 3 < − 2 or x + 3 > 2 x < −5 or x > −1 { x | x < −5 or x > −1}

Let x = age at death. x − 30 ≥ 46.60 x ≥ 76.60 Therefore, the average life expectancy for a 30-year-old male in 2005 will be greater than or equal to 76.60 years.

By the given information, a female can expect to live 81.03 − 76.60 = 4.43 years longer.

111. Let P represent the selling price and C represent the commission. Calculating the commission: C = 45, 000 + 0.25( P − 900, 000) = 45, 000 + 0.25 P − 225, 000 = 0.25P − 180, 000

Calculate the commission range, given the price range: 900, 000 ≤ P ≤ 1,100, 000 0.25(900, 000) ≤ 0.25 P ≤ 0.25(1,100, 000) 225, 000 ≤ 0.25 P ≤ 275, 000 225, 000 − 180, 000 ≤ 0.25 P − 180, 000 ≤ 275, 000 − 180, 000

45, 000 ≤ C ≤ 95, 000

The agent's commission ranges from \$45,000 to \$95,000, inclusive. 45, 000 95, 000 = 0.05 = 5% to = 0.086 = 8.6%, 900, 000 1,100, 000

inclusive. As a percent of selling price, the commission ranges from 5% to 8.6%, inclusive. 113. Let W = weekly wages and T = tax withheld. Calculating the withholding tax range, given the range of weekly wages:

1888

Appendix: Review 700 ≤ W ≤ 900 700 − 693 ≤ W − 693 ≤ 900 − 693 7 ≤ W − 693 ≤ 207 0.25(7) ≤ 0.25 (W − 693) ≤ 0.25(207) 1.75 ≤ 0.25 (W − 693) ≤ 51.75

1.75 + 82.35 ≤ 0.25 (W − 693) + 82.35 ≤ 51.75 + 82.35

84.10 ≤ T ≤ 134.10 The amount withheld varies from \$84.10 to \$134.10, inclusive.

115. Let K represent the monthly usage in kilowatthours and let C represent the monthly customer bill. Calculating the bill: C = 0.0944 K + 12.55 Calculating the range of kilowatt-hours, given the range of bills: 76.27 ≤ C ≤ 248.55 76.27 ≤ 0.0944 K + 12.55 ≤ 248.55 63.72 ≤ 0.0944 K ≤ 236.00 675.00 ≤ K ≤ 2500.00 The usage varies from 675.00 kilowatt-hours to 2500.00 kilowatt-hours, inclusive. 117. Let C represent the dealer's cost and M represent the markup over dealer's cost. If the price is \$18,000, then 18, 000 = C + MC = C (1 + M ) 18, 000 1+ M Calculating the range of dealer costs, given the range of markups: 0.12 ≤ M ≤ 0.18

Solving for C yields: C =

1.12 ≤ 1 + M ≤ 1.18 1 1 1 ≥ ≥ 1.12 1 + M 1.18 18, 000 18, 000 18, 000 ≥ ≥ 1.12 1+ M 1.18 16, 071.43 ≥ C ≥ 15, 254.24 The dealer's cost varies from \$15,254.24 to \$16,071.43, inclusive.

119. a.

Let T represent the score on the last test and G represent the course grade. Calculating the course grade and solving for

the last test: 68 + 82 + 87 + 89 + T G= 5 326 + T G= 5 5G = 326 + T T = 5G − 326 Calculating the range of scores on the last test, given the grade range: 80 ≤ G < 90 400 ≤ 5G < 450 74 ≤ 5G − 326 < 124 74 ≤ T < 124 To get a grade of B, you need at least a 74 on the fifth test. b. Let T represent the score on the last test and G represent the course grade. Calculating the course grade and solving for the last test: 68 + 82 + 87 + 89 + 2T G= 6 326 + 2T G= 6 163 + T G= 3 T = 3G − 163 Calculating the range of scores on the last test, given the grade range: 80 ≤ G < 90 240 ≤ 3G < 270 77 ≤ 3G − 163 < 107 77 ≤ T < 107 To get a grade of B, you need at least a 77 on the fifth test. 121. Since a < b , a b < 2 2 a a a b + < + 2 2 2 2 a+b a< 2 a+b Thus, a < 2

and and and

a b < 2 2 a b b b + < + 2 2 2 2 a+b 0

and b > ab > 0

)

and b 2 >

ab

2

> a2

ab > a

and

(

ab

)

2

2ab − a (a + b) 2ab h−a = −a = a+b a+b 2ab − a 2 − ab ab − a 2 = = a+b a+b a (b − a) = >0 a+b Therefore, h > a . 2ab b(a + b) − 2ab b−h =b− = a+b a+b ab + b 2 − 2ab b 2 − ab = = a+b a+b b(b − a) = >0 a+b Therefore, h < b , and we have a < h < b .

127. Since 0 < a < b, then a − b < 0 and ab > 0. a−b < 0. So, Therefore, ab a b − 0 , so we have b 1 1 0< < . b a 129. Since x 2 ≥ 0 , we have x2 + 1 ≥ 0 + 1

1890

b > ab

Thus, a < ab < b .125. 1 1⎛1 1⎞ = + For 0 < a < b, h 2 ⎜⎝ a b ⎟⎠ 1 1⎛b+a⎞ ⋅h h⋅ = ⎜ h 2 ⎝ ab ⎟⎠ 1⎛b+a⎞ ⋅h 1= ⎜ 2 ⎝ ab ⎟⎠ 2ab h= a+b

x2 + 1 ≥ 1

Therefore, the expression x 2 + 1 can never be less than −5 .

2

Section 10 1. 9; −9 3. index 5. cube root 7.

3

27 = 3 33 = 3

9.

3

−8 =

11.

3

( −2 )3

= −2

8 = 4⋅2 = 2 2

13.

3

−8 x 4 = 3 −8 x3 ⋅ x = −2 x 3 x

15.

4

x12 y 8 =

17.

4

x9 y 7 4 8 4 = x y = x2 y 3 xy

4

(x ) ( y ) 3 4

2 4

= x3 y 2

19.

36 x = 6 x

21.

3 x 2 12 x = 36 x 2 ⋅ x = 6 x x

23.

(

539

) = ( 5) ( 9) 2

2

3

2

= 5 ⋅ 3 92 = 5 3 81 = 5 ⋅ 3 3 3 = 15 3 3

25.

(3 6 )( 2 2 ) = 6

12 = 6 4 ⋅ 3 = 12 3

27. 3 2 + 4 2 = ( 3 + 4 ) 2 = 7 2 29. − 18 + 2 8 = − 9 ⋅ 2 + 2 4 ⋅ 2 = −3 2 + 4 2 = ( −3 + 4 ) 2 = 2

Appendix: Review

31.

(

3+3

)(

) ( 3)

3 −1 =

2

+3 3 − 3 −3

49.

= 3+ 2 3 −3

2− 5 2− 5 2−3 5 = ⋅ 2+3 5 2+3 5 2−3 5

=2 3

33. 5 3 2 − 2 3 54 = 5 3 2 − 2 ⋅ 3 3 2 = 53 2 −63 2 = (5 − 6) 3 2

51.

=−3 2

35.

(

) ( )

x −1

2

x

=

2

4 − 2 5 − 6 5 + 15 4 − 45

=

19 − 8 5 8 5 − 19 = 41 −41

5 5 3 4 53 4 = ⋅ = 3 2 2 32 34 x+h − x = x+h + x

53.

− 2 x +1

=

= x − 2 x +1 37.

3

16 x 4 − 3 2 x = 3 8 x3 ⋅ 2 x − 3 2 x = 2x 2x − 2x 3

3

= ( 2 x − 1) 3 2 x 8 x3 − 3 50 x = 4 x 2 ⋅ 2 x − 3 25 ⋅ 2 x

39.

= 2 x 2 x − 15 2 x

55.

= ( 2 x − 15 ) 2 x

41.

3

4

16 x y − 3x 3 2 xy + 5 3 −2 xy

3

2t − 1

)

57.

5− 2

= = =

(

)

(

)

3 5+ 2 25 − 2 3 5+ 2 23

or

2 x + h − 2 x 2 + xh h

= 23

9 2

32

(

15 − 2x = x 15 − 2 x

)

2

= x2

x 2 + 2 x − 15 = 0 ( x + 5)( x − 3) = 0 x = −5 or x = 3

5+ 2 5− 2 5+ 2 3

=

15 − 2 x = x 2

− 3 − 3 5 − 15 = ⋅ = 5 5 5 5 3

x + h − 2 x 2 + xh + x x+h−x

{}

= ( − x − 5 y ) 3 2 xy or − ( x + 5 y ) 3 2 xy

47.

=

⎛9⎞ 3 3 ⎜ 2 ⎟ −1 = 9 −1 = 8 = 2 ⎝ ⎠ The solution set is 9 . 2

Check:

= ( 2 x − 3 x − 5 y ) 3 2 xy

45.

3

t=

= 2 x 3 2 xy − 3 x 3 2 xy − 5 y 3 2 xy

1 1 2 2 = ⋅ = 2 2 2 2

( x + h) − 2 x ( x + h) + x ( x + h) − x

2t − 1 = 8 2t = 9

4

= 3 8 x 3 ⋅ 2 xy − 3x 3 2 xy + 5 3 − y 3 ⋅ 2 xy

43.

=

2t − 1 = 2

3

(

x+h − x x+h − x ⋅ x+h + x x+h − x

15 − 2(−5) = 25 = 5 ≠ −5

Check –5:

5 3+ 6 23

Check 3: 15 − 2(3) = 9 = 3 = 3 Disregard x = −5 as extraneous. The solution set is {3}.

59. 82 / 3 =

( 8) 3

2

= 22 = 4

1891

Appendix: Review

61.

( −27 )1/ 3 = 3 −27 = −3 3/ 2

63. 16

=

65. 9−3/ 2 =

⎛9⎞ 67. ⎜ ⎟ ⎝8⎠

⎛8⎞ 69. ⎜ ⎟ ⎝9⎠

3/ 2

( 16 ) 1 9

3

2 −1/ 3 3/ 4

77.

3

= 4 = 64 1

=

( 9)

3

1 1 = 3 27 3

3

3

27 27 27 2 = = ⋅ 8 ⋅ 2 2 16 2 16 2 2

=

27 2 32 3

⎛ 9⎞ ⎛ 3 ⎞ =⎜ ⎟ =⎜ ⎟ ⎝ 8⎠ ⎝2 2⎠

3

2 =

3/ 2

3

3

( 2)

3

=

=

3

79.

(x y )

3 6 1/ 3

75.

1/ 3

81.

3

83.

2+ 3 ≈ 4.89 3− 5

85.

3 35− 2 ≈ 2.15 3

4 ≈ 1.59

27 27 = 8 ⋅ 2 2 16 2

( ) (y )

= x3

1/ 3

2 2/3

x2 / 3 y 2 / 3

6 1/ 3

(x ) =

= xy 2

2 1/ 3

=

x

( y )1/ 3 ( x )2 / 3 ( y 2 ) x2 / 3 y2 / 3

2 / 3 1/ 3 2 / 3 4 / 3

y x y x2 / 3 y2 / 3

= x 2 / 3 + 2 / 3− 2 / 3 y1/ 3 + 4 / 3− 2 / 3 = x 2 / 3 y1 = x 2 / 3 y

1892

2 ≈ 1.41

27 2 27 2 ⋅ = 32 16 2 2

( x y ) ( xy ) 2

8 x5 / 4 y 3/ 4

3

71. x3/ 4 x1/ 3 x −1/ 2 = x3 4 +1/ 3 −1/ 2 = x 7 /12 73.

3

2 1/ 4

3/ 2 −1/ 4

= 8 x5 / 4 y −3/ 4

3

=

=

1/ 4

−1/ 3 3/ 4

= 23 x3/ 2 −1/ 4 y −1/ 4 −1/ 2

( )

⎛9⎞ =⎜ ⎟ ⎝8⎠

3/ 4

x1/ 4 y1/ 2

⎛ 9⎞ ⎛ 3 ⎞ 3 =⎜ ⎟ =⎜ ⎟ = 3 8 ⎝ ⎠ ⎝2 2⎠ 2 2

−3/ 2

2 1/ 4

( ) (y ) = x (y ) ( 16 ) x y = 163/ 4 x 2

4

=

3/ 2

(16 x y ) ( xy )

2/3

Appendix: Review x + 2 (1 + x ) (1 + x ) x 1/ 2 + 2 (1 + x ) = 1/ 2 (1 + x) (1 + x)1/ 2 1/ 2

87.

1/ 2

x + 2 (1 + x )

=

(1 + x)1/ 2 x + 2 + 2x = (1 + x)1/ 2 3x + 2 = (1 + x)1/ 2

(

)

89. 2 x x 2 + 1

(

2

(

2

1/ 2

+ x2 ⋅

)

= 2x x +1

1/ 2

+

(

(

)

1 2 x +1 2 x3

−1/ 2

95.

)

x2 + 1

1/ 2

) ⋅ ( x + 1) + x ( x + 1) 2 x ( x + 1) 2x ( x +x = = ( x + 1) (x =

2x x +1

1/ 2

1/ 2

2 x3 + 2 x + x3

( x + 1) x ( 3x + 2 ) = ( x + 1)

1/ 2

2

3

=

2 2

) + 1)

1

+1 + x

3

1/ 2

3 x3 + 2 x

(x

2

)

+1

1/ 2

2

91.

4x + 3 ⋅

1 1 + x−5⋅ ,x >5 2 x−5 5 4x + 3

4x + 3 x −5 = + 2 x − 5 5 4x + 3 = = = =

4x + 3 ⋅ 5 ⋅ 4x + 3 + x − 5 ⋅ 2 ⋅ x − 5 10 x − 5 4 x + 3 5 ( 4 x + 3) + 2 ( x − 5 ) 10

( x − 5)( 4 x + 3)

20 x + 15 + 2 x − 10 10

( x − 5 )( 4 x + 3) 22 x + 5

10

( x − 5 )( 4 x + 3)

x+4

=

1/ 2

2

( x + 4 )1/ 2 − 2 x ( x + 4 )−1/ 2 ⎛ ⎞ 2x 1/ 2 ⎜ ( x + 4) − ⎟ 1/ 2 ⎜ ⎟ + x 4 ( ) ⎠ =⎝ x+4 1/ 2 ⎛ 2x 1/ 2 ( x + 4 ) ⎜ ( x + 4) ⋅ − 1/ 2 ⎜ ( x + 4) ( x + 4 )1/ 2 =⎝ x+4 ⎛ x + 4 − 2x ⎞ ⎜ ⎟ ⎜ ( x + 4 )1/ 2 ⎟ ⎝ ⎠ = −x + 4 ⋅ 1 = x+4 ( x + 4 )1/ 2 x + 4

3

1/ 2

1/ 2 +1/ 2

2

=

1/ 2

2

2

2

⋅ 2x

⎛ x ⎞ 1 ⎞ ⎛ ⎜ 1+ x − x ⋅ ⎟ ⎜ 1+ x − ⎟ 2 1+ x ⎠ ⎝ 2 1+ x ⎠ 93. ⎝ = 1+ x 1+ x ⎛ 2 1+ x 1+ x − x ⎞ ⎜ ⎟ 2 1+ x ⎠ =⎝ 1+ x 2(1 + x) − x 1 = ⋅ 2(1 + x)1/ 2 1 + x 2+ x = 2(1 + x)3/ 2

97.

−x + 4

( x + 4)

(x

x2 2

3/ 2

)

−1

1/ 2

=

4− x

( x + 4 )3/ 2

(

)

− x2 − 1

1/ 2

, x < −1 or x > 1

x2

(

) ( ( )

)

⎛ x 2 − x 2 − 1 1/ 2 ⋅ x 2 − 1 1/ 2 ⎜ 1/ 2 ⎜ ⎜ x2 − 1 =⎝ x2

(

) ⋅(x = ( x − 1) x − ( x − 1) 1 = ⋅ − x 1 ( ) x x2 − x2 − 1 2

2

1/ 2

2

)

−1

1/ 2

⎞ ⎟ ⎟ ⎟ ⎠

1/ 2

1 x2

2

1/2

2

=

⎞ ⎟ ⎟ ⎠

2

x2 − x2 + 1 1 1 ⋅ 2 = 1/ 2 1/ 2 x x2 − 1 x2 x2 − 1

(

)

(

)

1893

Appendix: Review 1 + x2 − 2x x 2 x 99. ,x > 0 2 1 + x2

(

)

(

3 1/ 2 x ,x > 0 2 3 3 = 1/ 2 + x1/ 2 2 x

109. 3x −1/ 2 +

)(

)

⎛ 1 + x2 − 2 x 2x x ⎞ ⎜ ⎟ ⎜ ⎟ 2 x ⎝ ⎠ =

(1 + x ) − ( 2 x )( 2 x x ) ⋅

=

2 2

=

=

1 + x2

2 x

1 + x2 − 4x2 1 ⋅ 2 x 1 + x2

(

)

2

1

(1 + x )

2 2

1 − 3x 2

=

)

(

2 x 1 + x2

( ( 3x

1/ 2

= 2x

2

−x−4

)

)

2

)

( ) + x ⋅ 4 ( x + 4) ⋅ 2x = ( x + 4 ) ⎡3 ( x + 4 ) + 8 x ⎤ ⎣ ⎦ = ( x + 4 ) ⎡⎣3 x + 12 + 8 x ⎤⎦ = ( x + 4 ) (11x + 12 ) 4/3

2

1/ 3

2

1/ 3

2

1/ 3

2

1/ 3

2

2

2

2

2

107. 4 ( 3 x + 5 )

( 2 x + 3)3/ 2 + 3 ( 3x + 5)4 / 3 ( 2 x + 3)1/ 2 1/ 3 1/ 2 = ( 3x + 5 ) ( 2 x + 3) ⎡⎣ 4 ( 2 x + 3) + 3 ( 3 x + 5 ) ⎤⎦ 1/ 3

= ( 3x + 5)

( 2 x + 3)1/ 2 (8 x + 12 + 9 x + 15) 1/ 3 1/ 2 = ( 3x + 5 ) ( 2 x + 3) (17 x + 27 ) 1/ 3

where x ≥ −

1894

3 . 2

113. T = 2π

2

96 − 0.608 ≈ 390.7 gallons 1

2

64 = 2π 2 ≈ 8.89 seconds 32

115. 8 inches = 8/12 = 2/3 feet ⎛2⎞ ⎜3⎟ ⎛ 1 ⎞ 1 = 2π ⎜ T = 2π ⎝ ⎠ = 2π ⎟ 32 48 ⎝4 3⎠ =

π 2 3

=

π 3 6

≈ 0.91 seconds

117. Answers may vary. One possibility follows: If a = −5 , then

= 2 x1/ 2 (3 x − 4)( x + 1)

105. 3 x 2 + 4

V = 40 (1)

b.

103. 6 x1/ 2 x 2 + x − 8 x3/ 2 − 8 x1/ 2 = 2 x1/ 2 3( x 2 + x) − 4 x − 4

96 − 0.608 12 ≈ 15, 660.4 gallons

V = 40 (12 )

111. a.

3 101. ( x + 1)3/2 + x ⋅ ( x + 1)1/ 2 2 3 ⎞ ⎛ = ( x + 1)1/ 2 ⎜ x + 1 + x ⎟ 2 ⎠ ⎝ ⎛5 ⎞ 1 = ( x + 1)1/ 2 ⎜ x + 1⎟ = ( x + 1)1/ 2 ( 5 x + 2 ) ⎝2 ⎠ 2

(

3 ⋅ 2 + 3 x1/ 2 ⋅ x1/ 2 6 + 3 x 3 ( x + 2 ) = 1/ 2 = 2 x1/ 2 2x 2 x1/ 2

a2 =

( −5 )2

= 25 = 5 ≠ a .

Since we use the principal square root, which is always non-negative, ⎧a if a ≥ 0 a2 = ⎨ ⎩− a if a < 0 which is the definition of a , so

a2 = a .

Appendix The Limit of a Sequence; Infinite Series 1. limit

and

3. converges 5.

2n ⎫ ⎬ ⎩ n + 1⎭ Dividing the numerator and denominator by n gives 2n 2n 2 = n = n +1 n +1 1+ 1 n n As n gets larger and larger, the value of the term 1 gets closer and closer to 0. We conclude that n 2n 2 lim = = 2 . Thus, the sequence n →∞ n + 1 1 converges.

11.

As n gets larger and larger, the values of 1

13.

5n

=

0 = 0 . Thus, the sequence 1

{sn } = {3n } The terms of the sequence are 3,9, 27,81, 243,... Each term is 3 times the value of the previous term. Therefore, the terms of the sequence are increasing without bound as n gets larger and larger. The sequence diverges.

15.

4 1 , , n n2

5 and n

get closer and closer to 0. We conclude that

n2 + 1 converges.

As n gets larger and larger, each term of the sequence is larger than the previous term. We conclude that the sequence diverges.

As n gets larger and larger, the values of

2 1 = . Thus, 4 2

Dividing the numerator and denominator by n 2 gives 5n 5 n2 = n n2 + 1 1 + 1 n2 n2

lim

Dividing the numerator and denominator by n 2 gives 2n 2 − 4n + 1 4 1 2− + 2 2 n n n = 5 4n 2 + 5 4+ 2 2 n n

=

5n ⎫ ⎬ 2 n ⎩ + 1⎭

n →∞

⎧ 2n 2 − 4n + 1 ⎪⎫ ⎬ 2 ⎪⎩ 4n + 5 ⎪⎭

2

{an } = ⎧⎨

n2

{bn } = ⎪⎨

2n 2 − 4n + 1

4n + 5 the sequence converges. n →∞

2

9.

all get closer and closer to 0. We

conclude that lim

{an } = ⎧⎨

⎧⎪ 5n ⎫⎪ 7. {bn } = ⎨ ⎬ ⎩⎪ 4n + 3 ⎭⎪ The terms of this sequence are 5 20 45 80 125 , , = 3, , ,... 7 11 15 19 23

5 n2

n 4 ⎞ ⎪⎫ ⎟ ⎬ ⎩⎪ ⎝ 5 ⎠ ⎭⎪ The terms of the sequence are 8 32 128 512 , , , ,... 5 25 125 625

{sn } = ⎪⎨2 ⎛⎜

As n gets larger and larger, the terms get closer

From Appendix B of Student’s Solutions Manual for Precalculus Enhanced with Graphing Utilities, Sixth Edition. Michael Sullivan, Michael Sullivan, III. Copyright © 2013 by Pearson Education, Inc. All rights reserved.

1895

Appendix: The Limit of a Sequence; Infinite Series

and closer to 0. Therefore, the sequence n

⎛4⎞ converges and we conclude lim 2 ⎜ ⎟ = 0 . n →∞ ⎝ 5 ⎠

17.

{bn } = ⎧⎨3 − ⎩

2⎫ ⎬ n⎭

We conclude that the sequence converges and

2 gets n closer and closer to 0. We conclude that 2⎞ ⎛ lim ⎜ 3 − ⎟ = 3 − 0 = 3 . The sequence n →∞ ⎝ n⎠ converges.

that lim

23.

⎧⎪ ( −1)n ⎫⎪ 19. {cn } = ⎨4 + ⎬ n ⎪ ⎩⎪ ⎭ The terms of the sequence are 9 11 17 19 3, , , , ,... 2 3 4 5

The terms of the sequence alternate between increasing and decreasing, but, as n gets larger, the terms get closer to 4. Thus, we conclude that the sequence converges and ⎛ ( −1)n ⎞⎟ =4. lim ⎜ 4 + n →∞ ⎜ n ⎟ ⎝ ⎠ gets closer to 0 as n

⎧ n + ( −1)n ⎫⎪ ⎬ n ⎪⎩ ⎪⎭

{an } = ⎪⎨

Note that

n + ( −1) n

n

= 1+

19 we note that the term

( −1)n n

( −1)

nπ ⎞ ⎫ ⎟⎬ ⎝ 2 ⎠⎭

{bn } = ⎨sin ⎛⎜ ⎩

⎛ 3π ⎞ sin ⎜ ⎟ = −1 , sin ( 2π ) = 0 , and ⎝ 2 ⎠ sin (θ + 2π ) = sin θ .

gets larger. 21.

⎧ 2n ⎫⎪ ⎬ ⎪⎩ n ⎭⎪ The terms of the sequence are 2 4 8 16 32 64 32 = 2, = 2, , = 4, , = ,... 1 2 3 4 5 6 3

{an } = ⎪⎨

⎛π ⎞ We know that sin ⎜ ⎟ = 1 , sin (π ) = 0 , ⎝2⎠

n

n

= 1+ 0 = 1

The terms of the sequence increase without bound so we conclude that the sequence diverges. 25.

( −1)

n

n

n →∞

As n gets larger and larger, the term

Note that the term

n + ( −1)

As n gets larger, the terms will oscillate between 0, 1, and −1 . Therefore we conclude that the sequence diverges. . From problem

n

gets closer to 0

n as n gets larger. Therefore, the terms of the sequence will get closer to 1 as n gets larger.

27.

{sn } = {cos ( nπ )} ⎧−1 if n is odd We know cos ( nπ ) = ⎨ . ⎩1 if n is even

Therefore, the terms will oscillate between −1

1896

Appendix: The Limit of a Sequence; Infinite Series

and 1 and we conclude that the sequence diverges. 29.

⎛1⎞ ∑ ⎜⎝ 3 ⎟⎠ k =1

k −1

0

1

2

3

4

⎛1⎞ ⎛1⎞ ⎛1⎞ ⎛1⎞ ⎛1⎞ = ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ⎜ ⎟ + ... ⎝ 3⎠ ⎝ 3⎠ ⎝ 3⎠ ⎝ 3⎠ ⎝ 3⎠ 1 1 1 1 = 1+ + + + + ... 3 9 27 81

S1 = 1

1 4 = 3 3 1 1 13 S3 = 1 + + = 3 9 9 1 1 1 40 S4 = 1 + + + = 3 9 27 27 1 1 1 1 121 S5 = 1 + + + + = 3 9 27 81 81 S2 = 1 +

31.

∑ k = 1 + 2 + 3 + 4 + 5 + ... k =1

S1 = 1 S2 = 1 + 2 = 3 S3 = 1 + 2 + 3 = 6 S4 = 1 + 2 + 3 + 4 = 10 S5 = 1 + 2 + 3 + 4 + 5 = 15

1897

Index Page references followed by "f" indicate illustrated figures or photographs; followed by "t" indicates a table.

A Abscissa, 2, 56 Absolute maximum or minimum, 1137 Absolute value, 5, 101, 103-104, 106, 128, 431, 654, 977-978, 981-982, 985, 1014, 1024, 1045, 1050-1051, 1053, 1055, 1076, 1857, 1859 defined, 104, 106, 128, 654, 977, 981, 1014, 1045, 1053, 1055 functions, 101, 103-104, 106, 128, 431, 1076 inequalities, 977, 1045, 1050, 1053 properties of, 101, 1045 real numbers, 101, 103, 106, 431, 982, 1045, 1055 Acceleration, 121, 162, 186, 248, 428-429, 512, 521, 529, 555, 608, 663, 700-701, 765, 771-772, 775, 1064, 1880 Accuracy, 230, 233, 248, 306, 575, 578, 584, 957, 993 Acute angles, 572, 574-575, 580, 584-585, 590, 616 Addition, 11-12, 42, 88, 133, 173, 192, 201, 575, 593, 598, 611, 639, 656, 662, 684, 706, 741, 761, 801, 829-830, 841, 843, 866, 877, 904, 927, 929, 944-945, 948, 967, 978, 980, 989, 1028-1029, 1047-1048, 1053, 1176, 1272, 1320, 1818, 1861 associative property for, 662 of algebraic expressions, 978 Addition Principle, 944-945, 948, 967 Addition rule, 1818 probability, 1818 Additive inverse, 999 real numbers, 999 Additive inverses, 793 Algebra, 1-2, 6, 64, 200-201, 230, 233-234, 266, 430, 522, 524, 557, 656, 670, 674, 710, 788, 803, 827-830, 836, 844, 918, 944, 973, 978, 982, 987, 993, 995, 1028, 1045, 1059 Algebraic expressions, 64, 973, 978, 995 Allocation, 871 Ambiguous case, 586-587, 622 Amortization, 893, 900-901, 932 formula, 893, 900-901, 932 Amortization formula, 901 Angle of depression, 576-577, 581-582, 619, 1556 Angle of elevation, 186, 512, 521, 576-578, 581-584, 589, 592-594, 617-618, 622, 765, 1450, 1458, 1524 Angle of rotation, 601 Angles, 7, 13, 30, 37, 400-408, 414, 417-421, 424, 426, 428, 431, 479-480, 482, 507, 514-515, 522, 530-531, 533, 535-538, 558-559, 572, 574-576, 580-582, 584-587, 590-591, 593, 595-598, 605, 616, 621-622, 637, 653, 672, 679, 681, 686-690, 696-697, 699-700, 924, 938, 988-991, 1444, 1474-1475, 1488, 1509, 1523, 1532, 1542, 1560, 1616, 1797, 1860-1861 acute, 572, 574-576, 580, 584-585, 590-591, 596, 598, 616, 681, 688 adjacent, 572, 574-576, 696, 700, 1523, 1616 complementary, 426, 522, 572, 574, 616 congruent, 7, 13, 421, 530, 988-990 corresponding, 7, 30, 37, 405, 408, 417, 426, 482, 653, 988-990, 1860-1861 obtuse, 584-585, 590, 596 right, 13, 37, 400, 402, 406, 420, 426, 533, 572, 574-575, 580, 582, 584, 587, 591, 596, 605, 616, 621, 681, 696, 989-991, 1616, 1860 sides of, 7, 406, 537, 572, 581, 597-598, 621, 988-991 straight, 402, 576, 582, 593, 621 vertex of, 591, 596 vertical, 30, 37, 479, 576, 581, 584, 653, 700

Annuities, 900 fixed, 900 Annuity, 893, 900-901, 932, 1803 Applied problems, 249, 255, 266, 304, 571-572, 575, 584, 589, 595, 597, 616, 712, 717, 721, 729, 732, 741, 774, 1035-1036 Approximately equal to, 957, 975 Approximation, 306, 406, 551, 615, 750, 931, 975, 1519, 1784-1785 Arc length, 405, 417, 621, 1435, 1553-1554 finding, 405 Arcs, 405 Area, 1, 6, 13, 15, 51, 56, 71, 76, 84, 97, 122-126, 131, 139, 141, 175, 179-181, 188, 248, 255, 257-258, 285, 291, 320, 334, 401, 408-413, 422-423, 478-479, 481, 506, 550-551, 571, 573-574, 601-607, 616-620, 623, 625, 679-680, 691, 693-697, 700, 705, 745, 827, 858, 860-861, 885, 905, 957, 979, 984-985, 987-988, 990-993, 1001, 1042-1044, 1064, 1069-1070, 1077, 1080-1081, 1109, 1129, 1139, 1158-1159, 1167, 1170, 1194-1196, 1212-1213, 1272-1273, 1299, 1434-1435, 1443-1444, 1540-1543, 1551, 1553, 1556-1557, 1593, 1601, 1612, 1616, 1708, 1737, 1739, 1835-1836, 1840-1841, 1847-1848, 1851, 1860-1862, 1880, 1882 of a circle, 51, 71, 126, 180, 285, 291, 401, 408, 410-412, 478-479, 601, 605, 617, 860, 979, 984, 990-991, 1042, 1077, 1542, 1851, 1880 of a parallelogram, 691, 694, 696-697, 1616 of a rectangle, 76, 123, 180, 255, 860, 984, 991-992 of a square, 71, 126, 606, 860-861, 1042 of a trapezoid, 1847 of a triangle, 13, 571, 601-605, 607, 616, 827, 984, 987, 991, 1080-1081, 1541, 1708, 1835-1836 Areas, 408, 573, 605, 618, 788, 860, 905, 943, 965, 988, 993, 1001, 1070, 1272, 1553 Argument, 5, 68, 70, 74, 111, 113, 128, 234, 285, 297, 324, 345, 416, 444, 449, 515-516, 525, 545-546, 596, 654-656, 658-660, 698, 704, 861, 1441-1442, 1499-1500, 1515 Arithmetic, 339, 342, 892, 906-911, 918-919, 921, 931-933, 936, 938-939, 982, 993, 1054, 1075, 1786, 1788-1789, 1792-1793, 1800-1801, 1803-1806, 1808 Arithmetic mean, 1054 Arithmetic sequence, 906-911, 918, 931-932, 1075, 1788-1789, 1803, 1806 Arithmetic sequences, 892, 906, 911 defined, 906 Arithmetic series, 1793 Array, 803, 814, 828, 877, 905 Associative property, 662, 830, 835, 840 Asymptotes, 238, 240-244, 247-250, 252, 254, 258, 266-269, 276-278, 281-282, 300, 305, 310, 312, 332, 367, 383, 385, 459, 461-465, 479, 732, 737-741, 743-745, 748, 757-758, 774, 776, 780-782, 785-786, 1075, 1250-1251, 1253, 1255-1261, 1266-1268, 1291-1294, 1296-1297, 1301-1302, 1413, 1639-1643, 1645-1646, 1663-1664, 1667, 1673-1674, 1726 oblique, 238, 242-244, 247-250, 258, 266-269, 276-278, 282, 738-739, 743, 745, 1250-1251, 1253, 1256-1257, 1268, 1294, 1301-1302 Augmented matrices, 837 row operations, 837 Average, 7-8, 16-17, 30, 75, 77-78, 81, 87, 93-94, 96-98, 127, 129-130, 132, 136, 144-148, 150, 157, 159, 181, 187, 191-193, 218, 264, 274, 305, 308-309, 316, 359-360, 379, 409, 457, 465, 470-471, 474-476, 482, 600, 618,

766, 801, 860, 883, 1036, 1039, 1042-1045, 1053-1054, 1138-1140, 1169, 1172, 1174-1175, 1178, 1181-1182, 1206-1207, 1214, 1231, 1284, 1323-1324, 1357, 1411, 1416, 1436, 1686, 1736, 1851, 1888 Average cost, 81, 97, 264, 359, 1139 Average rate of change, 30, 87, 93-94, 96-98, 129-130, 132, 136, 144-148, 150, 187, 192, 218, 274, 305, 308-309, 316, 457, 465, 1138-1140, 1169, 1172, 1174-1175, 1178, 1206-1207, 1214, 1231, 1323-1324, 1411, 1416, 1851 defined, 93, 98, 129, 136, 145, 305 finding, 30, 93-94, 187, 1174 Averages, 359-360, 731, 1044-1045 Axes, 2-3, 12, 14, 17, 23, 37, 43, 79, 120, 162, 172, 176, 180, 299, 303, 321, 551, 555, 682, 689, 710, 745-753, 774-776, 784, 1118 Axis, 2-4, 12, 14, 16, 19-26, 31, 36, 42, 46, 50-51, 54, 56-57, 60-61, 78, 80, 82-83, 86-89, 95, 97, 99-101, 110, 114-119, 121-123, 125, 129-131, 134, 137, 161-167, 169-171, 176, 180, 183-189, 191, 193, 195, 202-205, 207, 211-212, 215-216, 219, 229, 236, 239-240, 243, 246, 249, 251-252, 254, 258-261, 263, 268, 271, 277, 279, 282, 291, 309-312, 314, 316, 321, 325-326, 331, 350, 380, 401-402, 410, 412, 415-416, 423, 426, 430, 446-447, 451-454, 458, 468-469, 477, 486, 488, 506, 596, 608-609, 628, 630, 632, 634-636, 638, 642-646, 648-649, 651, 653-655, 659-660, 665, 668-669, 671-672, 681-683, 686-688, 699, 701-702, 704, 707-708, 711-737, 739-741, 743-745, 747-748, 750-751, 754-759, 762, 769, 775, 777-783, 785-786, 858, 894, 1079, 1086, 1088-1094, 1098, 1102, 1105, 1109, 1112-1113, 1118, 1120, 1131-1132, 1135-1136, 1139, 1148-1155, 1157, 1163, 1165-1166, 1168-1169, 1183-1186, 1188, 1191, 1197-1211, 1214-1215, 1221-1226, 1228-1230, 1242-1244, 1250-1252, 1274-1275, 1285-1287, 1296-1297, 1299-1300, 1302, 1322, 1324-1326, 1332, 1335, 1371, 1379, 1402, 1404-1410, 1414-1415, 1420-1421, 1426-1427, 1434, 1443, 1564, 1566, 1568-1581, 1590, 1604-1606, 1612, 1614, 1617, 1623, 1627-1646, 1648-1651, 1653-1656, 1660, 1662, 1664-1674, 1676-1679, 1726, 1743-1745, 1763-1764 ellipse, 711, 721-732, 737, 747, 750, 754-756, 759, 775, 779-780, 782-783, 785-786, 1630-1631, 1635-1637, 1644, 1646, 1648-1651, 1653-1655, 1664-1666, 1668-1671, 1673, 1679

B Base, 6, 13, 16, 59, 76, 84, 127, 131, 154-157, 175-176, 257-258, 305-307, 310, 312-315, 317, 322, 326-327, 329, 331, 335-336, 338-341, 343, 345-347, 350, 360, 381, 383, 413, 421, 429-430, 512, 551, 577, 581-584, 593, 600-602, 604, 694, 718, 720-721, 745, 778, 786, 860-861, 909, 914, 917, 933, 980, 982, 984, 987-988, 990-992, 1041, 1062, 1077, 1104, 1135, 1158, 1194-1195, 1326, 1333, 1338, 1341, 1378, 1449, 1489, 1524, 1538, 1627-1628, 1645, 1671, 1677, 1732, 1739 logarithmic, 305-307, 310, 312-315, 317, 322, 326-327, 329, 331, 335-336, 338-341, 343, 345-347, 350, 360, 381, 383, 1326, 1333, 1338, 1341, 1378 Bearing, 356, 579, 582, 589, 592-593, 599, 618, 672, 985, 1525, 1553 Binomial expansion, 928-929 Binomial theorem, 891-939, 1075, 1779-1808

1899

Binomials, 712, 722, 995-996, 1029, 1304 coefficient, 995, 1304 Bits, 560 Bonds, 359-360, 801-802, 813-814, 816-817, 868, 870-871, 876, 884, 1037, 1042, 1064, 1699-1701, 1745, 1750-1751, 1880-1881 government, 359, 1037 Brachistochrone, 769-770 Break-even point, 152, 1177

C Calculators, 3, 69-70, 101, 161, 173-174, 182, 203, 286, 306, 313, 339-340, 425, 509, 631-632, 883, 896, 982, 1197 Calculus, 63, 69, 73, 92, 98, 103, 180, 202, 207-209, 219, 229, 239-240, 248, 289, 306, 313, 338, 348, 365, 368, 404, 444, 493, 512, 518, 521, 541, 544, 546, 551, 560, 573, 650, 670, 764, 770, 788, 845, 905, 916, 918, 1002, 1013, 1041, 1059, 1061, 1067, 1821-1856 defined, 63, 73, 98, 202, 313, 764, 770, 905, 918, 1824, 1827, 1843, 1850 instantaneous rate of change, 1855 limits, 209, 1849-1850 power functions, 202 tangent line to, 1850 Capacity, 255, 366-368, 371, 375-376, 378, 843, 861, 881, 1044, 1361-1362, 1364, 1376, 1767-1768, 1853 Cardioid, 25, 643-644, 649, 702-703, 707, 1570, 1575, 1577-1578, 1606 Carrying, 366-368, 371, 375-376, 378, 466, 521, 1361-1362, 1364, 1376, 1853 Cartesian coordinate system, 3 Cartesian plane, 14, 17, 25, 44, 52, 55, 97-98, 145-146, 153, 157, 171, 174, 332, 350, 383, 520, 636, 798 Categories, 333 Census, 17, 181, 268, 333-334, 350-351, 378, 384, 883, 1852 Center, 16, 45-55, 59-61, 84, 134, 176-177, 180, 191, 271, 335, 387, 404, 407, 410, 412, 414, 425, 429, 484, 505, 560, 571, 577-578, 581, 583-584, 593, 599-601, 605, 607-608, 619, 621-622, 634, 638, 640-642, 649-650, 652-653, 658, 660, 677, 690, 701-702, 704, 706-707, 718, 720-737, 739-741, 743-745, 748, 750, 754-757, 759, 763-764, 774-777, 779-783, 785-786, 863, 883, 935, 956, 966, 970, 976, 986, 1077, 1105-1110, 1116-1117, 1120, 1389, 1430, 1526, 1531, 1537-1538, 1567-1569, 1576, 1581, 1597, 1604-1605, 1613, 1628, 1630-1646, 1648-1650, 1653-1655, 1663-1671, 1673-1676, 1678, 1726, 1739 Central angle, 404-405, 408-414, 477-478, 481, 483, 485, 506, 578, 583, 601, 605, 617, 1389, 1435, 1526-1527 defined, 409, 478 finding, 405, 408, 410, 583, 617 Circles, 1, 45, 49, 51, 53, 255, 400, 583, 637, 648-649, 711, 721, 724, 735, 852, 860, 862, 944, 961, 974, 1526, 1736 area of, 51, 255, 860 center, 45, 49, 51, 53, 583, 649, 721, 724, 735, 1526 completing the square, 45, 721 defined, 51 equation of, 45, 49, 51, 53, 721, 724, 735, 860 finding, 1, 49, 51, 255, 583, 860, 974 graphing, 1, 45, 49, 53, 255, 637, 648, 721, 724, 735, 852, 862 radius, 45, 49, 51, 53, 255, 649, 860, 1736 Circuits, 248, 457, 475, 1013, 1035 Circular cylinders, 1062 Circumference, 51, 126, 159, 180, 304, 401, 405, 410, 413, 415, 485, 731, 975, 979, 984, 987, 990-991, 993, 1042, 1077, 1159, 1389, 1861-1862 Circumference of a circle, 51, 180, 405, 979, 984, 1042 Clearing, 846, 1769 Closed interval, 91-92, 1045-1046, 1064 Coefficient, 36, 157, 216, 219, 223-225, 227-229, 232-234, 237, 252-253, 266, 268, 373, 449, 611, 748-749, 804, 820, 927-928, 930-931, 933, 994-995, 1000-1001, 1003-1004, 1006, 1020-1021, 1063, 1181, 1232-1233, 1246, 1299, 1301, 1304, 1725, 1770, 1798-1799,

1900

1803, 1808, 1862 algebraic expressions, 995 binomial, 927-928, 930-931, 933, 994, 1798-1799, 1803, 1808, 1862 correlation, 157, 373, 1181 leading, 216, 219, 224-225, 227-229, 232-234, 237, 252-253, 268, 995, 1001, 1004, 1006, 1021, 1232-1233, 1299, 1301 matrix, 804, 820, 1770 trinomial, 994, 1862 Coefficients, 212, 223-225, 227-229, 231-238, 245-246, 255, 266-267, 269, 271, 276, 659, 792-793, 796, 803-804, 820-821, 823-824, 840, 845-846, 848, 850, 927, 930, 952, 995, 999, 1004-1005, 1033-1034, 1249, 1301-1304, 1768-1769, 1772-1774 Combinations, 153, 801-802, 817, 942, 948, 951-953, 955-956, 964, 967-968, 1075, 1703, 1812, 1820 Combining like terms, 10, 1015, 1048 Commission, 1053, 1065, 1888 Common difference, 906-911, 919, 931-933, 1788, 1792, 1800, 1804-1806, 1808 Common factors, 232, 239, 249, 256, 845, 1007-1009, 1012, 1017, 1058, 1060, 1850 Common logarithm, 327, 340, 347 Common logarithms, 339-340, 342 Common ratio, 912-919, 931-933, 1071, 1789, 1792-1793, 1800, 1804-1806 Commutative property, 662, 675, 686, 830, 835 Complement, 426, 947, 957, 962, 967-968, 971, 974-975, 1818 Complementary angles, 426, 522, 574 Complementary events, 962 probability of, 962 Completing the square, 45, 48, 161, 163, 628, 712, 721, 732, 776, 1000-1001, 1014, 1018, 1020-1021, 1023, 1027, 1673 circles, 45, 721 quadratic equations, 161, 1014, 1018, 1020-1021, 1027 quadratic functions, 161, 163 Complex conjugates, 1249 Complex numbers, 65, 233-234, 236-237, 276, 653, 655-656, 659-660, 670, 674, 679, 697, 705, 841, 846, 930, 973, 1028-1032, 1034-1035, 1587, 1878 Complex plane, 627, 653-655, 659, 697-699 imaginary axis, 653-655, 659 real axis, 653-655 Compound interest, 352-353, 356-357, 360, 381, 701, 893, 900 continuous compounding, 360, 381 formula, 352-353, 356-357, 360, 381, 893, 900 Conditional equations, 523, 527 defined, 523 Cones, 429 Congruent triangles, 573, 985, 988-989 Conic sections, 711 circle, 711 degenerate conics, 711 ellipse, 711 hyperbola, 711 parabola, 711 Conjugate pairs theorem, 233-235, 266, 1302 Conjugates, 234-235, 1030-1031, 1033-1034, 1064, 1249, 1646, 1880 axis, 1646 complex, 234-235, 1030-1031, 1033-1034, 1064, 1249, 1880 defined, 1030 CONNECTED mode, 10, 104, 758 Constant, 17, 48, 86-87, 89-91, 95, 98, 101, 103, 106, 126, 128-130, 135, 144-148, 150, 180, 186-187, 189, 192, 196, 202, 224, 244, 265, 307-309, 330, 335, 343, 360-362, 364-365, 370-371, 409-410, 481, 593, 597, 608-609, 618, 668, 672-673, 678, 681, 690, 699-700, 722, 728, 730, 733, 741-743, 759, 763, 765, 768, 774, 793, 805, 808, 857, 860, 881, 883, 899, 907, 910, 913, 918, 936, 973, 975, 978, 989, 994-995, 1020, 1035, 1039-1040, 1042-1044, 1072, 1135-1136, 1175, 1178, 1195, 1207, 1213-1214, 1223, 1255, 1304, 1323-1324, 1341, 1645, 1786, 1789-1790, 1792, 1804-1805 Constant ratio, 975 Constant term, 224, 1304 Constraint, 872, 875

Constraints, 871-875, 878, 1745-1753, 1764, 1767 graphing, 871 Continuous compounding, 354-355, 360, 381 Continuous function, 92, 229, 266 Convergence, 916, 1071 Coordinate systems, 628 polar, 628 Coordinates, 1-2, 4-8, 13-18, 22, 25, 29, 44-45, 47, 52, 63, 78-79, 89, 104, 110-113, 154, 165, 184, 415, 417, 419, 440, 451-453, 518, 530, 571-572, 596, 608, 612-614, 619, 621, 627-708, 710, 712, 715, 719, 723, 730, 733, 743, 747-748, 754-756, 758-759, 764, 774, 862-864, 893, 976, 978, 982, 1082, 1176, 1274, 1321, 1404-1410, 1450, 1563-1619, 1623, 1680 Correlation, 157, 373, 1181 Correlation coefficient, 157, 373, 1181 Corresponding angles, 7, 426, 988-990, 1860-1861 Cosecant, 400, 416, 424, 431-434, 440-442, 459, 463, 491, 507, 509, 556, 574, 1076, 1446 defined, 416, 442, 459, 463, 509 Cosine, 321, 400, 415, 426, 431-435, 440-442, 444, 446-447, 449, 453-454, 458-459, 463, 469, 485, 491, 493-494, 499-501, 507, 514, 517-518, 524, 526-528, 530-533, 537, 544, 552, 556-557, 561, 574, 596, 598, 610, 637, 649-650, 653, 659, 675, 705, 746-747, 749, 752, 1076, 1390, 1410-1412, 1429, 1432, 1440-1441, 1444-1445, 1447, 1497, 1500, 1510, 1515, 1519, 1560, 1616-1617 inverse, 493-494, 499-501, 507, 530, 556-557, 749, 1441, 1500, 1616 Cosines, 517, 526, 552-555, 571, 585, 595-599, 601, 603, 605, 616, 618, 621-623, 674-675, 679, 687-689, 763, 1078, 1534, 1537-1539, 1555, 1557, 1559 defined, 674, 689, 763 law of, 571, 585, 595-599, 601, 603, 605, 616, 618, 621-623, 674-675, 679, 1078, 1537-1539, 1555, 1557, 1559 proof, 554, 585, 596, 603, 675, 688 theorem, 552-553, 585, 595-596, 598, 603, 616, 621, 675, 687-688, 1538 Costs, 42, 105, 108, 133, 135, 152-153, 192, 255, 258, 359, 801, 843, 876-877, 1135, 1172, 1176, 1273, 1284, 1687, 1765, 1889 average, 192, 359, 801, 1172, 1284 fixed, 42, 152-153, 192, 359, 877, 1176 total, 133, 255, 258, 359, 801, 843, 876, 1284, 1765 Cotangent, 400, 416, 424, 431-434, 440-443, 459, 462-463, 491, 507, 509, 511, 556, 574, 1076, 1446-1447, 1497 Counting, 3, 223, 362, 930, 941-971, 975, 1809-1820 combinations, 942, 948, 951-953, 955-956, 964, 967-968, 1812, 1820 permutations, 942, 948-954, 964, 967-968 subsets of a set, 943, 968 Counting numbers, 943, 975 Cramer, Gabriel, 788 Credit card interest, 359 Cross products, 691-692 Cubes, 260, 996, 999, 1056 perfect, 996, 1056 Cubic functions, 270 Cubic units, 706, 1601 Curve fitting, 400, 466, 484, 799, 801, 816, 880 Cycloid, 769-770, 773, 784

D Data, 49, 55-56, 63, 77, 97-98, 133, 143, 145-146, 153-161, 173, 177-178, 181-182, 187-191, 193, 195, 201, 214-215, 218, 266, 268, 270-271, 282, 284, 307, 309, 317, 368, 371-380, 382, 384-385, 387, 396, 466, 470-477, 480, 482, 484, 522, 560, 575, 596, 616, 721, 828, 843, 934-935, 944, 947-948, 966, 1177, 1182, 1196, 1213, 1216-1218, 1323-1324, 1363-1365, 1376-1377, 1436, 1459, 1807-1808, 1820, 1852 collection, 157 definition of, 307, 828 Data points, 474, 1177 Data sets, 190 Data values, 471, 1459 Days, 108, 152-153, 320, 334, 362, 367-370, 385, 395, 398, 402, 412, 458, 473, 484, 488, 920-921, 951, 955, 962-963, 967-968, 1053,

1358-1361, 1377, 1381, 1388, 1413, 1717, 1793-1794, 1812 Decay, 284, 361, 363-364, 366-370, 373, 381-382, 1358, 1380 exponential, 284, 361, 363-364, 366-370, 373, 381-382, 1358, 1380 radioactive, 363-364, 369-370 Decay rate, 366, 368-369, 1358 Decimals, 26, 401, 403, 406, 479, 883, 916, 975, 982 comparing, 916 Decreasing function, 90, 129, 207, 311-312, 367 Degenerate conics, 711 Degree, 84, 201-203, 206-208, 210-211, 213-217, 219-221, 223-225, 227, 230-231, 233-238, 243-246, 248-249, 251-253, 255, 259, 266-269, 271, 276, 280, 306, 402-404, 406-408, 414, 417, 477, 516, 544, 549, 574, 580, 583, 617, 746, 748, 788, 845, 852, 861, 882, 925, 945, 966, 994-995, 998-1003, 1005-1006, 1009-1010, 1016, 1018, 1020, 1027, 1063, 1220, 1222-1230, 1232-1233, 1242-1243, 1249, 1251-1253, 1272, 1285-1287, 1291, 1299, 1301, 1303, 1394, 1430, 1432, 1718, 1862, 1866 Degrees, 43, 84, 121, 182, 202, 271, 292, 304, 365, 401-408, 411, 413-414, 417, 419, 422, 424-425, 477-482, 572, 574-575, 579, 583, 627-628, 636, 658-660, 698, 858, 966, 983, 990, 994, 997, 1002, 1053, 1156, 1387, 1389, 1424, 1432, 1444, 1590, 1866 Denominator, 22, 70-71, 202, 238-239, 242-246, 248-256, 261, 280, 392, 524, 527-528, 535, 725, 755, 757, 820, 845-850, 954, 979, 995, 1007-1012, 1025, 1030, 1035, 1057, 1060, 1063, 1067-1068, 1124, 1161, 1167-1168, 1249-1253, 1272, 1285, 1291, 1301-1303, 1718, 1774, 1780, 1804, 1850, 1858, 1862, 1870, 1895 rationalizing, 1057 Denominators, 929, 1008-1012, 1017, 1054, 1057 common, 1008-1010, 1012, 1017 least common, 1009-1010, 1017 Dependent variable, 68, 70-72, 74, 128, 134, 144, 157, 159, 191-193, 214, 218, 270, 366, 374-375, 377-380, 387, 444, 484, 1129, 1177, 1181 Depreciation, 148, 152, 284, 319, 335, 351, 379, 384, 387, 396, 919, 1366 Descending order, 995, 1005 Determinants, 691-692, 788, 818, 820-821, 824, 826, 878, 880, 1759 defined, 691, 821 Diagrams, 154-155, 177, 187, 372, 470, 974-975 scatter, 154-155, 177, 187, 372, 470 Diameter of a circle, 54, 975 Difference, 5-6, 64, 69, 72-73, 76, 98, 128-130, 244, 260, 293, 321, 335, 337-339, 342-343, 381, 383, 385, 413-414, 493, 530-533, 535-536, 539, 541, 552-554, 556-557, 559-560, 584, 591, 627, 630, 637, 650, 653, 655, 658, 662, 665, 674, 725, 733, 736, 741-743, 745, 774-775, 792-793, 801, 827, 829, 845, 860, 878, 899, 906-911, 919, 931-933, 952, 956, 977, 994, 996, 999, 1006, 1029, 1038, 1042, 1072, 1078, 1218, 1434, 1474-1475, 1509, 1511, 1645, 1671, 1783, 1786, 1788, 1792, 1800, 1804-1806, 1808, 1859 function, 64, 69, 72-73, 76, 98, 128-130, 244, 260, 293, 321, 335, 339, 342-343, 381, 383, 385, 414, 530, 533, 535, 552, 556-557, 560, 584, 591, 650, 775, 801, 878, 908, 931, 1072, 1218, 1434, 1475, 1808 real numbers, 98, 129, 337, 339, 342-343, 662, 801, 845, 906, 996, 999 Difference quotient, 69, 76, 98, 128, 130, 321, 343, 541 Difference quotients, 560 Digits, 306, 575, 947, 955, 968-969, 973, 975, 982, 1810, 1817 Directed line segments, 661 Direction angle, 667-669, 672, 688, 698, 700, 1609 Discounts, 292 Discriminant, 165, 167, 169, 176, 184, 225, 518, 854, 861, 1022-1023, 1025, 1027, 1032-1033, 1063, 1183-1189, 1208-1210, 1214-1215, 1454, 1486, 1737-1738, 1876-1877 quadratic formula, 165, 861, 1022-1023, 1027, 1032-1033 Distance, 1-2, 4-7, 13-14, 16-17, 39, 42-43, 45, 49-50,

52-55, 79, 84, 86, 108, 118, 123-127, 135, 137, 153, 169, 171-172, 175-177, 179, 181, 186, 188, 215, 265, 304, 335, 407-410, 412-415, 428-430, 443, 466, 483-484, 507, 521, 530-531, 539, 545, 551, 576, 578-584, 589-590, 592-596, 598-601, 607-610, 613-614, 617, 619-622, 628, 632-633, 635-636, 654, 658, 661, 663, 665, 679, 681, 683, 689-690, 697-698, 710, 712-714, 716, 719, 721-723, 726-727, 729-734, 738, 740, 742-743, 745, 747, 753-757, 759-760, 765-766, 772-773, 775-777, 780, 783, 801, 857-858, 868, 905, 917, 920, 933, 936, 973, 976-978, 982-983, 992, 1024, 1036, 1039-1040, 1042, 1044-1045, 1050, 1064, 1077, 1085, 1105, 1110-1111, 1117, 1134-1135, 1158-1159, 1191, 1194-1195, 1321, 1389, 1396-1397, 1412, 1444, 1458, 1489, 1524-1525, 1527, 1530-1532, 1537, 1551-1554, 1561, 1567, 1597, 1617, 1627-1628, 1635-1636, 1645, 1656, 1662, 1671, 1673-1674, 1678, 1686, 1736, 1766, 1784, 1803, 1861, 1880-1883 angle of elevation, 186, 521, 576, 578, 581-584, 589, 592-594, 617, 622, 765, 1458, 1524 formula, 2, 4-7, 17, 39, 42, 45, 52-53, 55, 335, 408, 410, 429, 443, 530-531, 545, 551, 578, 584, 590, 594-596, 598, 601, 607, 635-636, 658, 679, 681, 683, 690, 710, 712, 719, 721-722, 732-733, 742, 747, 754, 772, 905, 917, 983, 1036, 1039, 1042, 1077, 1085, 1117, 1396-1397, 1784, 1880 infinite geometric series, 920, 933, 1803 minimizing, 171-172, 188 Distribution, 108, 333-334, 883, 969 Distributive properties, 1029 Distributive property, 300, 675, 681, 686, 835, 898, 976, 994, 999-1000, 1012, 1016-1017, 1029 Dividend, 221, 275, 920, 997-998, 1001-1002, 1004-1006, 1232, 1301 Division, 29, 70, 80, 220-226, 231, 236, 238, 243-246, 251, 288, 361, 366, 431, 656, 802, 845, 851, 973, 978-979, 993, 996-998, 1001, 1003-1007, 1167-1168, 1232, 1235-1241, 1244-1248, 1280, 1288-1290, 1300-1301, 1718, 1723, 1732, 1762, 1858 long, 221-225, 236, 243-246, 251, 802, 845, 993, 996-998, 1003, 1006, 1300-1301 of integers, 238, 993 of rational expressions, 1007 Divisor, 221, 231, 997-998, 1004-1006, 1063, 1245, 1866 Divisors, 221 Domain, 64-68, 70-75, 78-87, 91-93, 95, 98-104, 106-107, 109-112, 116-119, 122-132, 134-135, 137-141, 144, 147, 150-153, 160-162, 165-167, 170-171, 173, 178-179, 185, 188-189, 194, 196-197, 201, 203-204, 212, 214, 231, 238-240, 246-247, 249-254, 256-257, 262, 264-267, 269, 271-282, 285-288, 290-291, 293-297, 300-302, 304-305, 307, 309-314, 316-319, 322, 324-328, 331-333, 336, 338, 343-346, 367, 380-383, 385, 389-393, 396-398, 400, 426, 430-433, 440-442, 445-447, 452-453, 456, 461-462, 464-465, 478-481, 487-491, 494-495, 497-499, 501-505, 507, 511-512, 514, 528, 557, 561, 568, 760, 763, 767, 770, 893, 902, 931, 973, 979, 983, 1007, 1014, 1017, 1053, 1086, 1124, 1126-1128, 1131-1132, 1135-1136, 1144-1145, 1149-1152, 1157-1163, 1165-1168, 1176-1179, 1181, 1183-1186, 1188-1189, 1193, 1199, 1208-1210, 1215, 1223-1230, 1242-1244, 1249-1250, 1253-1264, 1266-1272, 1279-1280, 1282-1283, 1286-1287, 1291-1296, 1301-1303, 1305, 1307-1314, 1320-1322, 1324-1326, 1328-1329, 1332-1340, 1344-1345, 1367-1372, 1378-1379, 1402, 1405-1410, 1414-1416, 1426-1428, 1430, 1439, 1441-1442, 1444, 1475, 1499-1500, 1512, 1679, 1789, 1794, 1826, 1828-1829, 1843, 1858, 1888 algebraic expressions, 64, 973 defined, 64-65, 67-68, 70-74, 78, 91, 93, 98-99, 104, 106-107, 110, 128-129, 131, 160-161, 173, 285, 287, 290, 294-296,

300, 305, 307, 313, 322, 324, 327-328, 381, 442, 462, 478, 498, 501, 505, 511, 557, 561, 568, 760, 763, 767, 770, 893, 973, 979, 1007, 1014, 1053, 1168, 1266-1267, 1296, 1313, 1402, 1441, 1475, 1499, 1843 determining, 65-67, 91, 116, 147, 166-167, 170, 188, 249, 287, 294 exponential functions, 305, 307, 309-312, 314, 325, 381 rational expressions, 238, 262, 973, 1007, 1014, 1017 rational functions, 201, 203-204, 212, 214, 231, 238-240, 246-247, 249-254, 256-257, 262, 264-267, 269, 271-282, 1223-1230, 1242-1244, 1249-1250, 1253-1264, 1266-1272, 1279-1280, 1282-1283, 1286-1287, 1291-1296, 1301-1303 relations, 64-65, 67, 293 DOT mode, 104, 758 Dot product, 627, 674-675, 679-681, 685-686, 689-690, 693-694, 696-698, 705, 1593 Doubling time, 362

E Eccentricity, 732, 745, 754-756, 759, 782, 1646 ellipse, 732, 754-756, 759, 782, 1646 hyperbola, 732, 745, 754, 756, 759, 782, 1646 EDA, 970, 1811 Ellipse, 652, 710-711, 721-732, 737, 746-747, 750, 752-756, 759, 768, 771, 773-776, 779-780, 782-786, 1075, 1630-1631, 1635-1637, 1644, 1646, 1648-1651, 1653-1655, 1657, 1663-1666, 1668-1671, 1673, 1675, 1679 defined, 732, 771, 773-774 equation of, 721-730, 732, 737, 746-747, 750, 752-755, 768, 773-776, 779-780, 782, 786, 1636-1637, 1646, 1663-1665, 1671, 1673, 1675 graphing, 652, 721, 724-725, 728, 730-732, 737, 750, 753, 755-756, 771, 773-776, 1075, 1679 reflection property, 729 rotation of axes, 710, 746-747, 750, 752-753, 774-775 Ellipsis, 893, 975 Empty set, 186, 518, 943, 973, 1742 Endpoints, 7, 14-15, 50, 54-55, 233, 298, 450-451, 470, 1046, 1109, 1117, 1245, 1303, 1841 Equality, 39, 314, 338, 525, 547, 665, 684, 829, 939, 1028-1029, 1511, 1806 Equations, 1-3, 7, 9-10, 12, 18-19, 21, 26-27, 29, 31, 37-38, 42, 44, 49, 52-53, 63, 67, 77, 99, 104, 106, 144, 161, 220-221, 226, 230, 233, 266, 271, 284, 295, 305, 314, 322, 329, 338-339, 344, 346, 348, 361, 363, 372, 381-382, 493-494, 504, 513-514, 516-518, 523, 527, 530, 537-538, 546, 553, 555-557, 572-573, 584-585, 590-591, 595, 604, 627-628, 630, 634, 637, 639, 642-653, 655, 659, 670, 675, 688, 693, 697, 701, 710, 714, 716-717, 725-727, 736-737, 739-740, 746-747, 754, 756, 758, 760-777, 787-890, 905, 908, 973, 976, 999, 1008-1009, 1014-1015, 1017-1025, 1027-1028, 1031-1032, 1035, 1048, 1054, 1057-1058, 1063, 1077, 1110, 1187, 1211, 1531-1533, 1660-1662, 1672, 1676, 1680, 1681-1778, 1787-1788, 1792, 1801, 1870 exponential, 284, 295, 305, 314, 322, 329, 338-339, 344, 346, 348, 361, 363, 372, 381-382, 1732 logarithmic, 284, 295, 305, 314, 322, 329, 338-339, 344, 346, 348, 361, 363, 372, 381-382, 648 logistic, 284, 361, 372, 381-382 point-slope, 29, 38, 53, 1077, 1737 polynomial, 161, 220-221, 226, 230, 233, 266, 271, 322, 516, 717, 746, 845, 847, 850-851, 861, 999, 1008-1009, 1018, 1020, 1027, 1063 rational, 26, 220-221, 226, 230, 233, 266, 271, 305, 322, 844-846, 850-851, 880, 882, 973, 976, 1008-1009, 1014, 1017, 1028, 1054, 1058, 1718 slope-intercept, 29, 37-38, 42, 53, 144, 791, 1077 Equilateral triangle, 16, 126, 180, 911, 984, 1159 Equilibrium point, 149-150 Equivalent equations, 10, 792, 796, 1014-1015, 1048,

1901

1870 Equivalent inequalities, 1048 Eratosthenes, 413 Error, 17, 22, 239, 339, 498, 593, 597, 600, 618, 839, 883, 905, 970, 1051, 1519, 1778, 1808, 1819 Estimate, 1, 17, 121, 132, 191, 583, 616, 673, 970, 1085-1086, 1851 Estimator, 1 Euler Leonhard, 63, 321, 410, 964 Euler, Leonhard, 63, 321, 410, 964 Evaluating algebraic expressions, 64 Even functions, 95, 440 Events, 893, 957, 960-962, 967-968, 1813 certain, 957, 967 complement of, 962, 967 Excel, 133, 191, 270-271, 387, 484 Expectation, 963 Experiment, 191, 220, 297, 384, 484, 616, 639, 745, 762, 766, 773, 883, 957-960, 962, 964-965, 967, 970, 993, 1038, 1042 Experiments, 957, 1547 Exponential decay, 361 Exponential equations, 284, 305, 314, 344, 346, 348, 381-382 one-to-one property, 346 solving, 305, 344, 346, 348 two exponential expressions, 314 Exponential expressions, 314, 337 Exponential functions, 284, 305, 307, 309-312, 314, 325, 342, 381, 921 defined, 305, 307, 381 domain of, 307, 309, 311, 314, 325, 381 graphing, 305, 309, 311-312, 314 inverse of, 305, 381 Exponential growth, 284, 306, 310, 361, 366 Exponential growth model, 366 Exponential model, 372-374, 376-377, 382, 384, 1366 Exponential notation, 323 Exponential regression, 373, 1363, 1366, 1376 Exponents, 305-306, 314-315, 323-324, 335, 337, 380, 973, 979-982, 984, 1054, 1058-1061, 1326, 1331, 1866 irrational, 306 rules of, 1866 zero, 315, 324, 335, 982 Extrema, 1155

F Factor theorem, 222-226, 229, 231-232, 234-235, 266, 1244, 1302 Factorial notation, 950 Factorials, 896, 931 Factoring, 220, 225, 516, 524, 844, 854, 999-1000, 1002, 1007, 1014, 1018, 1021, 1023-1025, 1060, 1240, 1244, 1280, 1301, 1850 defined, 1007, 1014, 1025, 1850 polynomials, 220, 844, 999, 1002, 1007, 1014, 1018 Factoring polynomials, 220, 844, 999, 1014 completely, 999 Factors, 55, 121, 206, 223-225, 227, 232, 234, 236, 239, 249, 256, 338, 342, 361, 366, 379, 383, 385, 604, 844-848, 850, 878, 895, 950-951, 976, 980, 999, 1002, 1007-1010, 1012, 1017-1018, 1023, 1056, 1058, 1060, 1246-1247, 1299-1301, 1303, 1850, 1864, 1866 common factors, 232, 239, 249, 256, 845, 1007-1009, 1012, 1017, 1058, 1060, 1850 defined, 121, 895, 1007, 1850 Feet, 16-17, 25, 43, 51, 76, 84-86, 121, 127, 131-132, 134-135, 172, 175-177, 179-181, 186, 188, 257-258, 265, 285, 291, 304-305, 319, 334, 408-413, 428-430, 443, 458, 466, 476, 481, 483, 506, 512, 520-521, 545, 551, 576-578, 580-584, 592-594, 600-601, 605-606, 615, 617-620, 679-680, 690, 699-700, 718, 720-721, 729, 731, 741-742, 745, 765-766, 771-773, 775-776, 801, 860-861, 909, 911, 919, 933, 986, 991-993, 1043-1044, 1062, 1064, 1085, 1104, 1109, 1133-1134, 1167, 1170, 1194-1195, 1204, 1213, 1272, 1284, 1321, 1387-1389, 1396, 1417, 1422, 1430, 1434-1438, 1443, 1449, 1488-1489, 1524-1527, 1530-1533, 1538-1539, 1541-1542, 1551-1552, 1554, 1556-1557, 1560-1561, 1597, 1627-1628, 1635-1636,

1902

1645, 1660-1663, 1671-1672, 1677, 1679, 1686, 1737, 1739, 1746, 1789, 1793, 1803, 1833-1834, 1845-1846, 1861-1862, 1882-1884, 1894 Fibonacci numbers, 897, 1013 formula, 1013 Fibonacci sequence, 897, 904-906, 935, 1784, 1869 finishing time, 876, 1749 First quadrant, 512, 1743-1745, 1763 Fixed costs, 42, 152-153 Fixed points, 722, 730, 733, 743, 774 Focal length, 745, 1013, 1678 Focus of a parabola, 713 finding, 713 Formulas, 1-2, 52, 221, 356, 370, 381, 405-406, 477, 493, 530-533, 535-536, 539, 542-544, 546-549, 552-557, 559, 571, 596, 601-602, 604, 614, 616, 633-634, 636-637, 653, 678, 687, 746, 748-751, 753, 774, 860, 899-900, 908, 925-926, 979, 985, 987, 993, 995, 999, 1009, 1027, 1077-1078, 1304, 1413, 1477, 1496, 1675, 1782 defined, 381, 535, 557, 678, 774, 925, 979 Fourth quadrant, 52 Fractions, 406, 792, 809-810, 845-850, 1009, 1013, 1060, 1774 clearing, 846 dividing, 406 equivalent, 792 graphing, 809-810 improper, 845 like, 792, 846, 848, 850, 1774 multiplying, 846-847, 849, 1774 powers of, 846, 1774 proper, 845, 850 simplifying, 1013 Frequency, 257, 313, 555, 609-610, 614-616, 619, 1546 Function notation, 74, 77, 87, 128, 144, 159-160, 894, 1178 defined, 74, 128, 159-160 using, 87, 144, 159-160, 894, 1178 Functions, 63-141, 143-197, 199-282, 283-398, 399-492, 493-494, 497-498, 501, 504, 507-510, 513, 518, 523-524, 526-530, 536, 546, 556-557, 560, 571-625, 628, 637, 650, 653, 701, 713, 735, 760, 770, 844, 853-854, 893, 911, 921, 1076, 1078, 1123-1172, 1173-1218, 1219-1304, 1305-1383, 1385-1438, 1450, 1521-1561, 1808, 1828, 1830, 1849 algebraic, 64, 212, 221, 249-250, 259-260, 315-316, 344-348, 404, 507, 510, 536, 557, 853-854 average rate of change, 87, 93-94, 96-98, 129-130, 132, 136, 144-148, 150, 187, 192, 218, 274, 305, 308-309, 316, 457, 465, 1138-1140, 1169, 1172, 1174-1175, 1178, 1206-1207, 1214, 1231, 1323-1324, 1411, 1416 constant, 86-87, 89-91, 95, 98, 101, 103, 106, 126, 128-130, 135, 144-148, 150, 180, 186-187, 189, 192, 196, 202, 224, 244, 265, 307-309, 330, 335, 343, 360-362, 364-365, 370-371, 409-410, 481, 593, 597, 608-609, 618, 1135-1136, 1175, 1178, 1195, 1207, 1213-1214, 1223, 1255, 1304, 1323-1324, 1341 cube, 68, 77, 99-100, 102, 106, 128, 232, 264, 1076, 1143, 1244-1245 defined, 63-65, 67-68, 70-74, 76, 78, 90-91, 93, 98-99, 104, 106-107, 110, 121, 128-129, 131, 133, 136, 145, 149, 159-161, 164, 173, 202, 285, 287, 290, 292, 294-296, 298-300, 305, 307, 313, 321-322, 324, 327-328, 330, 334, 381, 403, 409, 416-419, 427, 442, 458-459, 462-463, 478, 498, 501, 509, 523, 557, 606, 760, 770, 893, 1143, 1168, 1266-1267, 1296, 1313, 1347-1348, 1351, 1402 difference, 64, 69, 72-73, 76, 98, 128-130, 244, 260, 293, 321, 335, 337-339, 342-343, 381, 383, 385, 413-414, 493, 530, 536, 556-557, 560, 584, 591, 637, 650, 653, 911, 1078, 1218, 1434, 1808 domain and range, 65-66, 74, 83, 85, 95, 110, 112, 130, 247, 275-276, 400, 445-446, 507, 557 evaluating, 64, 144, 164, 220, 254, 285-286, 316,

451, 509 even, 71, 87-88, 95-103, 109, 123, 129-130, 135-136, 138, 140, 152-153, 172, 195, 203, 207, 210-211, 215, 219, 226, 229, 231, 235, 239, 254-255, 265, 268, 282, 292, 321, 346, 362, 388, 404, 417, 430, 440-443, 447, 449, 479-480, 486, 501, 523, 615, 637, 650, 921, 1078, 1135-1136, 1139, 1142, 1145, 1152, 1164, 1177, 1179, 1192, 1222-1223, 1253, 1297, 1299, 1304, 1313, 1331, 1402 exponential, 283-398, 921, 1076, 1305-1383, 1808 family of, 120-121, 172, 585 function notation, 74, 77, 87, 128, 144, 159-160, 1178 graphs of, 77-79, 99, 110, 114, 150, 162, 171-172, 202, 209, 215, 259, 290, 297-299, 301, 310, 312-313, 317, 322, 325, 349-350, 380, 382, 400, 444, 448, 450, 454, 458-459, 463-464, 467, 494, 507, 509, 518, 613, 760, 854, 1134, 1148, 1190, 1546 greatest integer, 103-104, 128 identity, 101-102, 106, 128, 221, 430, 435, 437-439, 441, 443, 459, 518, 523-524, 526-529, 557, 560, 601, 844, 1076 inverse, 284, 292, 295-305, 322, 324, 326-328, 332, 336, 342, 380-383, 385, 493-494, 497-498, 501, 504, 507-509, 530, 536, 556-557, 628, 1313, 1315-1322, 1334-1337, 1368-1372, 1374, 1378-1379 linear, 143-197, 201-202, 218, 227, 234, 236, 242, 246, 257, 259, 268, 270, 299, 304-305, 307-309, 317, 322, 347-348, 372-373, 379-380, 391, 401, 409-410, 412-414, 478-479, 483, 513, 530, 557, 621, 844, 911, 1173-1218, 1301, 1321, 1323-1324, 1383, 1389, 1435 logarithmic, 283-398, 1305-1383, 1808 maximum value, 90-91, 95-98, 135, 140, 168, 170, 175-176, 188, 194, 196-197, 215, 220, 445, 447, 454, 472, 1167, 1187, 1211-1213, 1215 minimum value, 90-91, 93, 96-98, 140-141, 161, 168, 170-171, 173, 187-189, 194, 196, 215, 220, 445, 447, 1187, 1210 modeling with, 148 notation, 71, 74, 77, 80, 87, 103, 124, 128, 144, 159-160, 182-185, 203, 209, 215, 239, 259-262, 304, 310, 312-313, 323-328, 342, 348, 403, 410, 431, 436, 464, 893, 1168, 1178, 1197-1206, 1211, 1215, 1274-1281, 1283-1285, 1296-1298, 1303, 1332 odd, 87-89, 95-100, 102-103, 106, 109, 129-130, 135-136, 140, 153, 172, 195, 203-204, 207, 210-211, 219, 227, 232, 235, 237, 252, 254, 265, 268, 276, 292, 305, 321, 388, 430-432, 440, 442-443, 445, 449, 452, 460-463, 478-480, 486, 488-491, 501, 523, 637, 650, 653, 921, 1078, 1136-1137, 1139, 1142-1143, 1148, 1163-1164, 1179, 1192, 1222-1223, 1297, 1300, 1304, 1313, 1322, 1402, 1406, 1409, 1413-1416, 1427, 1430, 1433, 1828 one-to-one, 284, 292-296, 298-305, 309-312, 314, 316, 322, 331, 339, 344, 346, 380-382, 385, 388-390, 396, 398, 428, 494, 501, 504, 1313, 1320-1322, 1368, 1378 piecewise, 63, 99, 104, 107, 129, 133, 136, 305, 1143 polynomial, 161, 199-282, 322, 1219-1304, 1808, 1828 product, 64, 72-73, 129, 149, 160-161, 171, 174, 178-179, 183-184, 188, 190, 205, 227, 235-238, 292, 315, 337, 339, 345-346, 368, 387, 493, 556, 594, 596, 602, 604, 614, 844, 853, 1078, 1304 quadratic, 120, 143-197, 201-202, 214, 218, 220-221, 225-227, 230, 233-234, 236, 259, 268, 270, 304-305, 315, 322, 344-346, 348, 372, 379-380, 396, 513, 518, 557, 844, 854, 1173-1218, 1236, 1240, 1300-1301, 1365, 1808 quotient, 64, 69, 72-73, 76, 98, 128-130, 221, 223, 225, 231, 233, 245-246, 248, 251-253, 255, 261-262, 280, 321, 337, 339, 343, 435, 438, 443, 523, 526-527, 1232, 1240,

1248-1249, 1288-1290, 1301 rational, 109, 199-282, 292, 305-306, 322, 464, 844, 1148, 1219-1304, 1828, 1849 square, 69-71, 85, 97, 99, 101-102, 106, 112, 117, 120, 122, 124-128, 131, 141, 161-164, 169, 175, 180-181, 255, 257-258, 264, 285, 291, 297, 320, 329, 334, 385, 409, 411-412, 422-423, 483, 596, 600, 603, 605-606, 620, 625, 628, 713, 735, 844, 853, 1076, 1139, 1159, 1167, 1194, 1196, 1212-1213, 1272, 1298, 1322, 1354, 1381, 1430, 1541, 1556-1557, 1560 sum, 64, 72-73, 129, 201, 232, 234, 237-238, 243, 335, 337-339, 342, 351, 356-357, 381-383, 385, 484, 493, 530, 536, 556-557, 573-575, 580, 585, 594, 596-597, 605, 607, 612-616, 618, 653, 893, 911, 921, 1078, 1192, 1244, 1272, 1304, 1541, 1808 transcendental, 284, 321 trigonometric, 399-492, 493-494, 504, 507-510, 513, 518, 523-524, 528, 530, 536, 546, 556-557, 560, 571-625, 628, 637, 701, 1078, 1385-1438, 1521-1561, 1828 vertical line test, 82, 713, 1124, 1135, 1164, 1167 Fundamental identity, 529 Fundamental theorem of algebra, 233-234, 266, 844 Future value, 218, 351, 353, 373, 382, 900 of an annuity, 900

G Gallons, 877, 1044, 1053, 1062, 1883, 1894 Games, 159, 193, 776, 956, 968, 1181-1182, 1816 Geometric interpretation, 94 Geometric mean, 1054 Geometric sequences, 342, 892, 912, 918, 921 defined, 912, 918 Geometric series, 219, 892, 912, 915-916, 918-921, 931-933, 1071-1072, 1075, 1792-1793, 1803, 1805 infinite, 915-916, 918-921, 931-933, 1071-1072, 1793, 1803 Geometry, 1-2, 6-7, 42, 76, 126, 232, 405, 408, 414, 512, 541, 551, 580, 583-584, 601, 605-606, 617, 691, 709-786, 791, 795, 827, 858, 860, 924, 973, 985, 987, 992, 1042, 1077, 1621-1680 Golden ratio, 905 Grams, 65-66, 98, 145-146, 158-159, 193, 362, 369-371, 377, 611, 614, 619, 802, 816, 1044, 1054, 1181, 1358, 1361, 1364, 1883 Graphing calculator, 3, 13, 49, 70, 154, 270, 286, 289, 306, 387, 404, 425, 444, 447, 497-498, 657, 810, 894, 899-900, 915, 925-926, 951, 953, 982, 1029-1030, 1033, 1055, 1154, 1172, 1739 ZERO feature of, 13, 1739 Graphs, 1-61, 63-141, 150, 162, 171-172, 174, 183-184, 202-204, 208-210, 215-216, 219, 224, 228, 259, 263, 268, 290, 295, 297-299, 301, 310, 312-313, 315, 317, 322, 325, 349-350, 380, 382, 400, 434, 444, 448, 450, 454-456, 458-459, 463-465, 467, 471, 479, 494, 507, 509, 514, 518, 521, 525, 532, 607, 613, 615, 627, 637, 647-648, 651, 714, 716-717, 724, 727-728, 740, 756, 760, 762, 777, 791, 794-795, 852-855, 858, 860, 865, 867, 982, 1046, 1079-1121, 1123-1172, 1176, 1190, 1194-1195, 1352, 1382, 1546, 1560, 1678-1680 of intervals, 1046 of inverse functions, 494 Greater than, 18, 71, 90-91, 99, 104, 137, 149, 152, 221, 243, 245-246, 265, 322, 324, 335, 432, 460, 462, 544, 722, 963, 968, 976, 982-983, 995, 998, 1046, 1050, 1054, 1063, 1168, 1177, 1283, 1402, 1737, 1750, 1821, 1862, 1888 Growth, 284, 306-307, 309-310, 321, 350-351, 361-363, 366-369, 371, 373-376, 379, 381-382, 384, 391, 892, 904, 934, 1323, 1330, 1358, 1361, 1807, 1853 exponential, 284, 306-307, 309-310, 321, 350-351, 361-363, 366-369, 371, 373-376, 379, 381-382, 384, 391, 1323, 1330, 1358, 1361 limited, 366, 375 Growth models, 366, 375 logistic, 366, 375

uninhibited, 366 Growth rate, 350-351, 362, 366-369, 371, 374, 384, 934, 1330, 1358, 1361, 1807, 1853

H Half-life, 364-365, 369-370, 377, 384, 1359 Half-open interval, 1046 Harmonic mean, 1054 Hemisphere, 473 Heron of Alexandria, 603-604 Horizontal asymptotes, 269, 300, 305, 367 defined, 300, 305 graphing, 305, 367 Horizontal axis, 4, 78 Horizontal line, 2, 4, 34, 37, 40, 52-53, 80, 101, 128, 202, 240-241, 294-295, 301, 380, 396, 494, 499, 501, 638, 640, 648, 652, 702, 715-716, 721, 769-770, 779, 864, 1099-1100, 1111, 1118, 1135, 1175, 1207, 1313, 1322, 1368, 1378, 1568, 1581, 1622, 1624, 1629 graph of, 80, 101, 202, 240-241, 294, 301, 380, 494, 499, 501, 638, 640, 652, 715, 721, 779, 864, 1118, 1135, 1207, 1322, 1378, 1581 slope of, 34, 40, 53, 1100 Horizontal lines, 29, 33, 53, 721, 779, 1099, 1105, 1313, 1629 graphing, 33, 53, 721 test, 53 Hours, 16, 42-43, 75-76, 86, 97-98, 105, 107, 121, 124, 139, 145-146, 159, 193, 218, 257, 271, 291, 301-302, 320, 334, 363, 368-371, 375-376, 413, 473-474, 476-477, 484, 490, 505-506, 582, 597-598, 600, 618, 766-767, 801-802, 816-817, 843, 857, 870, 876-877, 881, 920, 1036, 1039-1041, 1043-1044, 1053, 1085, 1129, 1135, 1156, 1159-1160, 1172, 1181-1182, 1272, 1313, 1329, 1340, 1360-1363, 1389, 1397, 1422-1424, 1436, 1443, 1530, 1537-1538, 1552, 1591, 1662, 1690, 1749, 1751-1752, 1767, 1881, 1883-1884, 1889 Hyperbola, 652, 710-711, 732-748, 752-754, 756-757, 759, 773-776, 780, 782-783, 785-786, 854, 1075, 1637, 1639-1643, 1645-1646, 1648, 1651, 1653-1654, 1663-1667, 1669, 1671-1673, 1675-1676, 1726, 1740, 1743, 1764 defined, 732, 734, 736, 745, 773-774 eccentricity, 732, 745, 754, 756, 759, 782, 1646 equation of, 732-743, 745-748, 752-754, 757, 773-776, 780, 782, 786, 1637, 1645-1646, 1663-1665, 1671-1673, 1675-1676 finding, 734, 737, 740 graphing, 652, 732, 734-739, 741, 744, 753, 756-757, 773-776, 854, 1075 rotation of axes, 710, 746-748, 752-753, 774-775 writing, 732, 753, 773 Hypotenuse, 5, 13, 124, 420, 426, 572-574, 576, 579-580, 617, 985-986, 991, 1063, 1078, 1521-1522, 1524, 1526, 1547, 1553, 1555, 1859-1860

I Identity, 101-102, 106, 128, 221, 430, 435, 437-439, 441, 443, 459, 517-518, 523-529, 531-532, 534-535, 540, 542-544, 550, 554, 557, 559-560, 562, 601, 681, 763-764, 830, 835-841, 844, 846-850, 878, 1014, 1063, 1076, 1459, 1711-1712, 1714, 1716-1718, 1756, 1774 defined, 106, 128, 459, 523, 535, 557, 763-764, 835, 841, 1014 linear equations, 837, 839-840, 878, 1014 property, 517, 532, 681, 830, 835-836, 840, 844, 1014 Identity matrix, 835-841, 878, 1714 defined, 835, 841 using, 839-841, 878 Identity property, 836 Image, 19, 65, 67-68, 70, 74, 78, 128, 293-294, 551, 560 Imaginary axis, 653-655, 659-660, 704, 707-708, 1617 Imaginary part, 679, 1028-1029, 1878 Inches, 4, 43, 127, 159, 180, 232, 257-258, 302, 304, 384, 389, 391, 405, 411-413, 422-423, 481, 550, 573, 580, 593-594, 605, 609, 617-618,

680, 718, 720-721, 731, 860, 909, 911, 917-918, 920, 933, 991-992, 1044, 1062, 1103-1104, 1160, 1195, 1245, 1272-1273, 1321, 1375, 1387-1389, 1430, 1532, 1553-1554, 1627, 1636, 1671, 1677, 1736, 1803, 1861, 1894 Increasing function, 90, 129, 207, 295, 309-310, 348, 367 Independent variable, 68, 70-71, 74, 89, 128, 134, 144, 146, 154, 157-161, 177, 181-182, 188, 191-193, 214, 218, 270, 306, 373-380, 384, 387, 444, 484, 893, 1129, 1177, 1181 Independent variables, 76 Index of summation, 898-900, 1070 defined, 1070 Inequalities, 64, 71, 89, 143, 173, 182-183, 187, 200, 259-260, 262, 266, 322, 448, 787-890, 973, 976-977, 1045-1050, 1053-1054, 1681-1778 absolute value, 977, 1045, 1050, 1053 compound, 801, 883, 1687 defined, 64, 71, 173, 322, 803, 821, 828-829, 832-835, 841, 973, 977, 1045, 1053-1054 interval notation, 71, 182-183, 259-260, 262, 973, 1045-1046, 1048-1049, 1054 linear, 143, 173, 182-183, 187, 259, 322, 788-797, 799, 801-808, 810, 812-815, 817-818, 820, 826-828, 837, 839-841, 844-850, 852, 862-881, 1045, 1745, 1774-1775 nonlinear, 187, 788, 791, 852-858, 867, 879 polynomial, 200, 259-260, 262, 266, 322, 845, 847, 850-851, 861 properties of, 173, 200, 818, 824, 826, 830-831, 835, 877-878, 880, 976, 1045, 1047 quadratic, 143, 173, 182-183, 187, 259, 322, 798-799, 844-845, 848-850, 854, 856-857, 861, 878, 880, 973, 1737-1738, 1765, 1774 rational, 200, 259-260, 262, 266, 322, 844-846, 850-851, 880, 882, 973, 976, 1054, 1718 symbols for, 874 system of, 789-800, 802-815, 817-821, 823, 826-828, 839-844, 846, 848-849, 852-859, 861-862, 865-870, 872-874, 878-883, 1681, 1687, 1691, 1712-1714, 1719-1721, 1735-1736, 1745, 1764-1767, 1769, 1775-1776 Inequality symbols, 976, 1045, 1048 Infinite, 209, 513, 521, 749, 795, 877, 915-916, 918-921, 931-933, 943, 948, 1067-1073, 1716, 1771, 1793, 1803, 1895-1897 geometric series, 915-916, 918-921, 931-933, 1071-1072, 1793, 1803 sequences, 915-916, 918-921, 931-933, 1068, 1793, 1803 series, 915-916, 918-921, 931-933, 1067-1073, 1793, 1803, 1895-1897 Infinite geometric series, 915-916, 918-921, 931-933, 1793, 1803 Infinity, 209, 239, 366, 858, 1046 Initial point, 661-662, 664, 671, 675, 677, 683, 689, 696, 698 Inputs, 64, 66, 68, 293-294, 301, 309, 380, 893, 1313, 1368 Instantaneous rate of change, 1855 Integers, 4, 14, 103, 136, 172, 224, 228, 232, 238, 276, 874, 893, 902, 921-922, 925, 929, 931, 975, 980, 993-994, 996-997, 999, 1028, 1055, 1058, 1245, 1299-1301 dividing, 996 graphs of, 172 Intercepts, 1-2, 12-13, 15-16, 18, 20-26, 29, 36-38, 42-44, 47, 50, 52-55, 57-58, 61, 77, 79-83, 85, 87, 95, 98-100, 103-107, 109, 119-120, 130-132, 135-136, 138, 140, 144, 153, 161, 164-167, 169-172, 183-184, 186-189, 195, 197, 201, 205-207, 210-215, 217, 219-220, 224, 227, 229, 236, 239, 247-252, 254, 256, 258-260, 262, 265, 268-269, 271-276, 278-282, 309-312, 316, 319, 333, 367, 380, 393, 445, 447, 454-455, 461, 721, 723-725, 730-732, 734, 743-744, 1033, 1083-1084, 1086-1093, 1097, 1102-1103, 1106-1109, 1111-1112, 1116, 1118-1120, 1131-1132, 1135-1136, 1143-1145, 1149, 1154-1155, 1163, 1165-1166, 1168-1169, 1182-1188, 1191, 1193, 1197-1206, 1208-1211, 1214-1215, 1220, 1223-1225, 1227-1229, 1232, 1234-1244, 1250, 1253, 1255-1256, 1260-1263, 1265, 1269, 1271, 1274, 1280,

1903

1282, 1286-1287, 1293, 1296-1297, 1300, 1302, 1332, 1339, 1404, 1454-1455, 1630, 1632, 1637, 1676, 1826, 1843 slope-intercept form, 29, 36-38, 42, 53-55, 58, 95, 144 Interest, 25, 44, 52, 99, 122, 132, 141, 157, 218-219, 351-361, 374, 381-382, 384, 394, 458, 522, 701, 801-802, 843, 893, 900-901, 903-904, 973, 1035-1037, 1042, 1064, 1170, 1355-1357, 1375-1376, 1618, 1688, 1782-1783, 1803, 1881 compound, 352-353, 356-357, 360, 381, 701, 801, 893, 900, 1782 simple, 351-352, 355, 358-359, 394, 802, 1036-1037 Interest rate, 132, 141, 157, 352-356, 358, 360, 843, 901, 1036-1037, 1170, 1355-1356, 1375, 1618, 1782-1783, 1803 annual, 353-355, 358, 360, 1037 Intermediate value theorem, 220, 229, 232, 266-267, 269, 282, 1289, 1303 INTERSECT feature, 27, 1546 Intersecting lines, 541, 711, 745, 791 Interval notation, 71, 74, 124, 182-184, 259-262, 310, 312, 324-328, 464, 503, 973, 1045-1046, 1048-1049, 1052, 1054, 1168, 1197-1206, 1211, 1215, 1274-1281, 1283-1285, 1296-1298, 1303, 1332, 1441-1442, 1499-1500 defined, 71, 74, 324, 327-328, 973, 1045, 1054, 1168, 1296, 1441, 1499 Intervals, 64, 87, 89, 91, 93, 95, 103, 130, 141, 212, 214, 230, 259-262, 450, 624, 731, 900, 1045-1046, 1124, 1136, 1142, 1168, 1546 Inverse, 284, 292, 295-305, 322, 324, 326-328, 332, 336, 342, 380-383, 385, 493-505, 507-509, 511, 516, 530, 536, 556-557, 565, 628, 633, 749, 810, 827, 836-843, 878-879, 881, 886, 981, 999, 1029, 1313, 1315-1322, 1334-1337, 1368-1372, 1374, 1378-1379, 1441-1442, 1476, 1499-1500, 1512, 1616, 1709, 1711-1714, 1717-1718, 1756 functions, 284, 292, 295-305, 322, 324, 326-328, 332, 336, 342, 380-383, 385, 493-494, 497-498, 501, 504, 507-509, 530, 536, 556-557, 628, 1313, 1315-1322, 1334-1337, 1368-1372, 1374, 1378-1379 of matrix, 841 Inverse functions, 284, 292, 297, 301, 324, 326-327, 336, 494, 497, 501, 556 cosine function, 501 defined, 292, 324, 327, 501 finding, 324, 497 graphing, 297, 326-327, 497 one-to-one, 284, 292, 301, 494, 501 sine function, 494, 497, 556 trigonometric, 494, 556 Inverse matrices, 839 Irrational number, 232, 306, 313, 975 Irrational numbers, 975-976, 1028

L Law of Large Numbers, 970 Leading coefficient, 216, 219, 224-225, 227-229, 232-234, 237, 252-253, 268, 995, 1001, 1004, 1006, 1021, 1232-1233, 1299, 1301 test, 224-225, 268, 1299 Least common denominator, 1025, 1870 defined, 1025 equations, 1025, 1870 Least common multiple, 1006, 1009-1010, 1017, 1063 Length, 4-7, 15-17, 71, 76, 84, 121, 124, 126-127, 131-132, 175, 179-180, 188, 200, 230, 232-233, 257, 265, 270-271, 305, 400-401, 404-405, 409-411, 413-415, 417, 420-422, 426, 429, 433, 450-453, 459, 466, 468-470, 472-473, 476-477, 479, 481, 484, 506, 514, 521, 550-551, 572-573, 575-576, 579-580, 582-583, 585, 590-594, 596, 598, 600-601, 605, 618, 620-621, 623, 625, 661-663, 671, 675, 700, 713, 722, 729-732, 745, 765, 769, 777, 780, 801, 860, 883, 917, 919, 935, 984-988, 990-992, 1013, 1041-1044, 1062, 1064, 1069, 1077, 1109, 1129, 1159-1160, 1170, 1194-1195, 1244-1245, 1272, 1284, 1389, 1404-1410, 1417, 1434-1435, 1443, 1525-1527, 1532-1533, 1539, 1552-1554, 1556, 1631-1632, 1634-1636, 1671, 1678, 1686, 1735-1737, 1739, 1807, 1819, 1851,

1904

1860-1861, 1880, 1882-1884 Like terms, 10, 69, 211, 261, 748, 751, 792, 798, 848, 850, 994, 1015-1017, 1048, 1774 combining, 10, 1015, 1048 Limits, 209, 1072, 1849-1850 at infinity, 209 Line, 2, 4-7, 9-10, 13-17, 19, 29-45, 50-56, 59, 61, 78-80, 82-84, 87, 93-95, 97-99, 101, 103, 105, 128-129, 134, 136, 144, 146-148, 150, 152, 154, 156-160, 162-167, 169, 171, 177-178, 187-191, 202-203, 214, 240-244, 246, 249, 251-253, 260, 262, 294-295, 297-299, 301-302, 305, 313-314, 320, 325-328, 380, 396, 401, 413, 419-420, 426, 430, 460, 466, 494-495, 499, 501-502, 506, 576-577, 579-583, 589, 592-594, 600, 606, 613, 618, 621, 638-640, 642-646, 648-653, 661, 663-664, 674, 676-678, 696, 698-699, 702, 707-708, 711-717, 719-722, 727, 730, 733, 738, 740, 743-745, 754, 759-760, 769-770, 773, 775, 778-779, 791, 795, 798-799, 803, 827, 841, 852, 857-868, 872-873, 883, 908, 930-931, 935, 950, 966, 973, 976-978, 982-983, 988, 1024, 1043, 1052, 1064, 1067, 1077, 1079, 1098-1100, 1103-1105, 1109-1111, 1114, 1117-1118, 1120, 1124, 1131, 1133, 1135, 1138-1139, 1141-1142, 1164, 1167-1168, 1174-1175, 1179-1186, 1191, 1207-1210, 1213-1215, 1251, 1253-1255, 1257-1271, 1280-1282, 1291-1296, 1303, 1313, 1322, 1368, 1378, 1417, 1526, 1533, 1553, 1568-1577, 1579-1581, 1594, 1605-1606, 1613-1614, 1622-1624, 1629, 1660, 1663, 1665, 1671, 1673-1674, 1708, 1726, 1737-1738, 1740-1745, 1762-1764, 1774-1775, 1811, 1830-1831, 1844-1845, 1850, 1857, 1882 horizontal, 2, 4-5, 29, 32-34, 37, 40-41, 43, 52-53, 78, 80, 84, 101, 128-129, 147, 162, 188, 202, 240-241, 243-244, 246, 249, 252-253, 294-295, 301-302, 305, 313-314, 328, 380, 396, 430, 494, 499, 501, 576-577, 580, 592, 594, 600, 638, 640, 648, 652, 699, 702, 715-716, 721, 738, 740, 743, 745, 769-770, 773, 779, 864, 1099-1100, 1105, 1111, 1118, 1135, 1175, 1207, 1251, 1253-1255, 1258-1261, 1265-1267, 1270, 1281, 1291-1293, 1295-1296, 1303, 1313, 1322, 1368, 1378, 1568, 1581, 1622, 1624, 1629 of symmetry, 52, 162-167, 169, 171, 187-189, 297, 646, 712-717, 719-721, 775, 778-779, 1109, 1183-1186, 1191, 1208-1210, 1214-1215, 1623, 1629, 1673 point-slope equation of, 1077 regression, 157, 178, 191, 1180-1181, 1213-1214 secant, 94-95, 97-99, 1138-1139, 1141-1142 slope of, 29-32, 34-35, 38-40, 43-45, 53, 55-56, 87, 93-95, 97-99, 148, 150, 156-159, 191, 799, 908, 1100, 1110, 1117, 1138-1139, 1179-1180, 1850 slope-intercept equation of, 1077 tangent, 50-51, 99, 136, 426, 494, 501-502, 606, 621, 638, 649, 860-861, 1109-1110, 1141, 1526, 1553, 1737-1738, 1830-1831, 1844-1845, 1850 Line segments, 7, 16, 661, 664, 988, 1103 Linear data, 190 Linear equations, 18, 144, 322, 513, 788-797, 799, 803-808, 810, 812-815, 817-818, 820, 826-828, 837, 839-840, 852, 877-878, 881, 908, 1014-1015, 1018 one variable, 792, 796, 1014-1015 parallel lines, 791 slope, 144, 791, 794-795, 799, 908 standard form, 1018 system of, 789-797, 799, 803-808, 810, 812-815, 817-818, 820, 826-828, 839-840, 852, 878, 881, 908 two variables, 789, 791-792, 794-797, 803, 805, 818, 820, 827, 852, 878, 908 Linear functions, 143-145, 147, 154, 173, 187, 305, 317, 911 finding, 173, 187 graphing, 143-144, 154, 187, 305 Linear inequalities, 259, 322, 788, 862-874, 878, 881, 1745, 1775 defined, 322

graphing, 259, 862-869, 871 in two variables, 862-863, 865, 868, 872-873 properties of, 878 solving, 259, 788, 862, 867-868, 871, 873, 881 system of, 862, 865-870, 872-874, 878, 881, 1745, 1775 Linear regression, 157, 1180-1181, 1196, 1213, 1216 Linear relationship, 157, 1213 Linear systems, 788, 791 Linear velocity, 1389 Lines, 1-2, 17, 25, 29, 32-33, 35-40, 42-44, 49, 51-54, 59-60, 98, 144, 154, 180, 240, 242, 249, 413, 541, 607, 621, 648, 676, 682, 711, 718, 721, 738, 745-746, 779, 789, 791, 794-795, 799, 852, 862, 865-866, 868, 871, 873, 905, 966, 1004, 1099, 1102, 1104-1105, 1117, 1131, 1140-1141, 1253, 1255-1258, 1292-1294, 1313, 1526, 1629, 1674 defined, 29, 51, 98, 745, 905, 1105, 1313 graphing, 1-2, 17, 25, 32-33, 36, 42, 49, 53-54, 98, 144, 154, 240, 249, 648, 721, 738, 791, 852, 862, 865-866, 868, 871, 905, 1140-1141, 1255-1258, 1292-1294 parallel, 29, 37-38, 40, 42, 44, 52-54, 59, 676, 682, 711, 718, 779, 791, 794, 799, 868, 873, 1099, 1102, 1104, 1117, 1629, 1674 perpendicular, 29, 38-40, 42-44, 51-54, 59-60, 607, 676, 682, 711, 1102, 1117 slope of, 29, 32, 35, 38-40, 43-44, 53, 60, 98, 799, 1117 Liters, 1044-1045, 1884 LN key, 326 Location, 2, 17, 55-56, 200, 207, 224, 302, 389, 400, 404, 407, 412-413, 438, 473, 505-507, 636, 672, 682, 710, 720, 741-742, 764, 843, 873, 878, 976, 1302, 1313, 1645 median, 17 Logarithmic equation, 329, 344-347, 349 Logarithmic equations, 322, 329, 338-339, 344, 346, 381-382 single logarithm, 338, 381 solving, 329, 339, 344, 346 Logarithmic functions, 283-398, 1305-1383 common, 300, 304, 327, 339-340, 342, 347, 362 defined, 285, 287, 290, 292, 294-296, 298-300, 305, 307, 313, 321-322, 324, 327-328, 330, 334, 381, 1313, 1347-1348, 1351 evaluating, 285-286, 316 graphing, 283, 286, 289, 297-298, 305-306, 309, 311-312, 314-315, 318, 320-321, 326-327, 341, 343-349, 362, 364-367, 369-380, 382-385, 387, 1305, 1315, 1329, 1341 natural, 289, 326, 339-340, 342, 347, 361, 375, 381 Logarithms, 284, 323, 329, 335-342, 344, 346-347, 381, 383, 385, 598, 1075, 1347-1348, 1351, 1380 defined, 381, 1347-1348, 1351 Logistic growth model, 371, 375-376, 384 Logistic model, 366-367, 371-372, 375-376, 378-379, 381-382, 385 carrying capacity and, 367 Long division, 221-225, 236, 243-246, 251, 845, 993, 996-998, 1003, 1006, 1300-1301 Loops, 133 Lowest terms, 223, 239, 242, 249-254, 266, 277-282, 1006-1007, 1011-1012, 1058, 1063, 1251-1271, 1280, 1282, 1291-1296, 1301, 1850, 1867

M Magnitude, 32, 203-204, 335, 383-384, 461, 463, 654-655, 657-658, 660-669, 671-672, 678, 680-681, 684, 686, 688-689, 693, 695-699, 704, 707, 1594, 1617 Mandelbrot set, 660 Marginal cost, 171-172, 188, 1192, 1212-1213, 1856 Marginal revenue, 1856 Markups, 1889 Mass, 162, 265, 608-611, 613-614, 619, 624-625, 663, 671, 700, 1042, 1064, 1544, 1554, 1880 Mathematical expectation, 963 Mathematical induction, 657, 892, 899, 921-924, 930-933 defined, 932 proof by, 921 Mathematical models, 63, 122 Matrices, 672, 788, 792, 803-804, 806-808, 810, 812-813, 815, 817-818, 820-821, 827-835,

837, 839-841, 843-844, 878-879, 881, 883, 1715, 1776 additive identity, 830 augmented, 803-804, 806-808, 810, 812-813, 815, 817, 828, 837, 878-879, 881, 1715 coefficient, 804, 820 column, 803-804, 806-808, 810, 820-821, 827-829, 832-835, 883 defined, 803, 821, 828-829, 832-835, 841 dependent system, 810 equations, 788, 792, 803-804, 806-808, 810, 812-813, 815, 817-818, 820-821, 827-835, 837, 839-841, 843-844, 878-879, 881, 883, 1715, 1776 identity, 830, 835, 837, 839-841, 844, 878 inconsistent system, 812 multiplying, 806, 830-831, 833-834, 843 notation, 803-804, 806 row, 803-804, 806-808, 810, 812-813, 815, 817, 821, 827-829, 832-835, 837, 839-840, 878, 883, 1776 scalar multiplication, 831 square, 803, 828-830, 835, 841, 844, 878 subtraction of, 829 zero, 830 Matrix, 788, 803-815, 817-818, 820, 827-830, 832-833, 835-844, 877-879, 881-884, 1691, 1693-1702, 1711-1718, 1756-1758, 1770-1772, 1777 Maxima, 87, 90-92, 100, 120, 129, 201, 1136, 1168, 1230, 1617 absolute, 87, 91-92, 129 Maximum, 3, 43, 51, 84, 87, 90-93, 95-98, 124, 129-130, 132, 135, 138, 140-141, 146-147, 151-152, 161, 164, 168, 170-176, 179-181, 186-190, 194, 196-197, 211-213, 215-216, 220, 224-225, 231, 252, 268, 320-321, 369, 379, 384-385, 428-429, 445, 447, 454, 472-473, 488, 521, 545-546, 551, 555, 574, 606, 610, 614-615, 619, 639, 680, 701, 721, 731, 761, 765, 771-773, 775-776, 785, 871, 874-876, 882-883, 889-890, 955, 1104, 1109, 1134, 1136-1139, 1154-1155, 1160, 1163-1164, 1167-1168, 1187, 1192-1196, 1210-1213, 1215-1216, 1218, 1223-1229, 1233, 1242-1244, 1272, 1286-1287, 1299-1300, 1330, 1365, 1376-1377, 1397, 1413, 1458, 1488-1489, 1495, 1546, 1593, 1617, 1636, 1660-1662, 1671-1672, 1747-1753, 1764, 1776 Maximum profit, 172, 874-876, 1192, 1749-1750, 1752 Mean, 63, 67, 79, 195, 202, 209, 239, 265, 296, 360, 408, 412, 476, 581-582, 676, 713, 724, 732, 735, 739, 749, 777, 780, 905, 935-936, 943, 973, 977, 1045, 1054, 1056, 1068, 1191, 1205, 1636, 1784 defined, 63, 67, 202, 296, 732, 905, 973, 977, 1045, 1054 finding, 296, 408, 581-582, 676, 713, 905, 943, 1068, 1191 geometric, 63, 936, 1054 harmonic, 1054 quadratic, 195, 202, 973, 1191, 1205 Means, 1-3, 32, 37, 39, 47, 67-68, 72, 78, 112, 136, 148, 157, 195, 203, 206, 215, 227, 240, 245, 287, 298, 329-331, 340-341, 345, 348, 360, 372, 380-381, 393, 408, 410, 421, 466, 487, 495, 499, 501-502, 504, 509, 511, 514, 520, 556, 574-575, 579, 585, 653, 658, 722, 763, 769, 790, 795, 811, 813, 844-845, 952, 956, 958, 960, 976, 981, 988, 992, 1006, 1011, 1014, 1048, 1055, 1104, 1109, 1111, 1142, 1149, 1167, 1245, 1299-1300, 1321, 1330, 1345, 1363-1365, 1377, 1403, 1477-1480, 1542, 1593, 1616, 1678, 1768, 1777, 1818, 1849, 1887 Measures, 30, 143, 159, 191, 334-335, 385, 401-403, 405, 407-408, 479, 575, 581-582, 592, 606, 617, 621, 769, 992, 1170, 1389 Median, 16-17, 181, 788, 1218 Meters, 51, 76, 109, 126, 179-180, 186, 188, 248, 291, 304-305, 405, 411-412, 428-429, 483, 521-522, 529, 555, 571, 576, 580-582, 589, 593, 610, 614, 619, 678, 690, 700, 706, 731, 771-772, 775, 801, 860, 991, 1043-1044, 1130, 1159, 1194-1195, 1205, 1212, 1322, 1387, 1435, 1458, 1524-1525, 1543-1544, 1554, 1597, 1661, 1736, 1869, 1883 Midpoint, 1-2, 7, 14-17, 43, 52-53, 55-56, 171, 181,

232-233, 722, 726, 733, 737, 755-757, 1079, 1082, 1084, 1086, 1105, 1109-1111, 1117, 1218, 1245, 1645 Midpoint formula, 2, 7, 17, 52-53 Minima, 87, 90-92, 100, 120, 129, 135, 140, 201, 1164, 1168, 1225-1227, 1229-1230, 1243, 1287, 1300, 1617 absolute, 87, 91-92, 129, 135, 140 Minimum, 3, 17, 87, 90-93, 95-99, 101, 108-109, 123-124, 129-130, 132, 135, 138, 140-141, 161, 163-164, 168, 170-173, 187-189, 194, 196, 208, 212-213, 215, 219-220, 252, 255, 257-258, 265, 268, 413, 445, 447, 455, 488, 639, 700, 761, 772, 873-876, 889-890, 1104, 1136-1139, 1154-1155, 1163-1164, 1168, 1171-1172, 1187, 1191-1193, 1210, 1212-1213, 1223-1224, 1226-1229, 1242-1244, 1272-1273, 1284, 1286-1287, 1299, 1413, 1417, 1458, 1617, 1662, 1747-1748, 1750-1752, 1765, 1768 Minor of entry, 822 Minutes, 51, 63, 85-86, 107, 133, 135, 148, 152-153, 188, 257, 285, 315-316, 320, 322, 334, 363, 365-366, 370, 384, 392, 401, 403-404, 411, 474, 476, 479, 481, 521, 578, 582, 590, 598, 600, 817, 860, 881, 1041, 1043-1045, 1053, 1135, 1212, 1329, 1331, 1340, 1360-1361, 1376, 1382, 1388, 1403, 1430, 1443-1444, 1458, 1527, 1530, 1552, 1591, 1736, 1882-1884 Mixture problems, 1035, 1038 Mode, 10, 12, 104, 403, 424-425, 428, 497, 510, 514, 516, 574-575, 631-632, 639, 756-758, 761-762, 766-767, 894, 896, 901-902, 905, 1394, 1397, 1429-1430, 1432, 1782 Models, 63, 122, 143-144, 148, 154, 156, 159, 173, 177-178, 181, 187, 200-201, 214, 218, 266, 271, 284, 307, 309, 317, 335, 351, 361, 363, 366, 372-373, 375, 382, 396, 466, 470, 475-477, 480, 484, 575, 610, 870, 904, 942, 957, 959, 968, 1323-1324, 1366 defined, 63, 159, 173, 307, 904, 959 radioactive decay, 363 Modulus, 654, 704 Monomials, 201, 993-995 coefficient of, 994 degree of, 201, 994-995 Multiples, 406, 414, 419, 423-424, 431-433, 460-463, 479-480, 486, 488, 501, 512, 514, 535, 663, 666, 827, 830, 878, 1402, 1413, 1444, 1828 Multiplication, 201, 598, 656, 670, 831, 835, 840-841, 943, 945-946, 949-950, 953-954, 963, 967-968, 978, 1028-1029, 1047-1048, 1054, 1304, 1818 of algebraic expressions, 978 of integers, 1028 Multiplication Principle, 943, 945-946, 949-950, 953-954, 963, 967-968, 1818 Multiplicity, 201, 205-207, 211-212, 215-217, 219, 223, 226, 229, 237, 243, 252, 254-255, 258, 266, 268, 271-276, 280-281, 1018, 1022, 1222-1226, 1228-1230, 1235, 1237, 1239, 1242-1244, 1286-1288, 1290, 1297, 1299-1300 Mutually exclusive events, 961-962

N Napier, John, 342 Natural logarithms, 339-340, 342 Natural numbers, 921-924, 931, 933, 939, 975, 1789, 1794, 1806 Negative exponent, 980, 1220, 1862 Negative exponents, 980 Negative infinity, 209, 1046 Negative numbers, 209, 981, 1032 Newton, Isaac, 365 nonlinear, 144, 146, 150, 154-155, 157, 177, 187, 189, 192, 196, 788, 791, 852-858, 867, 879, 1179 solving, 788, 791, 852-856, 858, 867 Nonlinear inequalities, 867 graphing, 867 system of, 867 Notation, 71, 74, 77, 80, 87, 103, 124, 128, 144, 159-160, 182-185, 203, 209, 215, 239, 259-262, 304, 310, 312-313, 323-328, 342, 348, 403, 410, 431, 436, 464, 495, 503, 803-804, 806, 814, 819, 893-894, 897-899, 902-903, 916, 932-933, 943, 950-951, 964, 973-974, 979, 995, 1003, 1014, 1045-1046,

1048-1049, 1052, 1054, 1067-1068, 1168, 1178, 1197-1206, 1211, 1215, 1274-1281, 1283-1285, 1296-1298, 1303, 1332, 1441-1442, 1499-1500 exponential, 304, 310, 312-313, 323-328, 342, 348, 1332 interval, 71, 74, 103, 124, 182-184, 259-262, 304, 310, 312-313, 324-328, 464, 495, 503, 973, 1045-1046, 1048-1049, 1052, 1054, 1168, 1197-1206, 1211, 1215, 1274-1281, 1283-1285, 1296-1298, 1303, 1332, 1441-1442, 1499-1500 limit, 103, 209, 239, 313, 916, 1054, 1067-1068 set, 71, 74, 103, 128, 144, 159-160, 183-185, 203, 239, 259-262, 310, 312, 325-328, 348, 403, 431, 893-894, 902, 943, 964, 973-974, 979, 1014, 1048-1049, 1052, 1168, 1197-1206, 1211, 1215, 1274-1281, 1283, 1296-1298, 1303, 1332, 1442 set-builder, 973-974 sigma, 897, 1054 summation, 893, 897-899, 903, 932-933 nth partial sum, 1071-1072 nth power, 883, 980 nth root, 657-658, 660, 696, 1055 complex numbers, 660 defined, 1055 nth term, 894-896, 903-904, 906-910, 912-914, 918-919, 1070, 1787, 1806 Number line, 2, 13, 260, 262, 973, 976-978, 982-983, 1024, 1052, 1303, 1857 Numbers, 2, 36, 48, 54, 65, 68, 70-74, 77, 82, 89, 91, 96, 98-103, 106, 110-112, 129, 134-136, 140-141, 144, 161, 166-167, 185-186, 197, 201, 203-204, 209, 212, 214, 221, 224-229, 231, 233-240, 246, 260, 262, 264-267, 276, 285-287, 305-307, 309-312, 317, 321, 324-328, 335-337, 339-343, 346, 362, 367, 380, 388, 391-392, 396, 417-418, 431-432, 442-443, 445-447, 453-455, 459, 461-463, 465, 467, 486, 495-496, 507, 511, 520, 530, 621, 628, 653, 655-656, 659-662, 664, 670-671, 674, 679, 682, 687, 697, 705, 750, 799, 801, 803-804, 807, 814, 818, 820-821, 828, 830-831, 835, 841, 844-850, 860, 872, 874, 877, 884, 893, 897-898, 905-906, 912, 921-925, 927, 930-931, 933, 939, 943, 947, 955-956, 964, 966-971, 973, 975-983, 993, 996, 999, 1004, 1007-1008, 1013, 1015, 1018-1019, 1022, 1028-1035, 1045-1047, 1049-1052, 1055, 1058, 1069-1070, 1073, 1086, 1124, 1167, 1192, 1206, 1234-1241, 1249-1250, 1283, 1289, 1299, 1303, 1312, 1324-1326, 1334-1336, 1368-1369, 1378, 1398, 1402-1403, 1405-1410, 1442, 1512, 1587, 1692, 1698, 1735, 1789, 1794, 1806, 1810-1812, 1815-1819, 1828, 1843, 1858, 1862, 1878 composite, 285-287, 380, 388 irrational, 276, 306, 975-976, 1028, 1055 positive, 2, 71, 82, 98, 110, 134, 186, 203, 209, 227, 239-240, 260, 262, 264, 305-307, 309-310, 312, 324-325, 336-337, 339-340, 343, 362, 459, 462, 486, 507, 628, 656, 671, 682, 687, 893, 921-922, 925, 927, 931, 976-977, 979-981, 983, 993, 1032, 1046-1047, 1050, 1055, 1058, 1069, 1283, 1289, 1299, 1303, 1369, 1378, 1403, 1789 prime, 227, 321, 924, 969, 999, 1816, 1862 rational, 201, 203-204, 209, 212, 214, 221, 224-229, 231, 233-240, 246, 260, 262, 264-267, 276, 305-306, 844-846, 850, 973, 975-976, 1007-1008, 1013, 1028, 1055, 1058, 1234-1241, 1249-1250, 1283, 1289, 1299, 1303, 1828 real, 2, 36, 48, 65, 70-71, 74, 77, 82, 89, 91, 98-103, 106, 110-112, 129, 134-136, 140-141, 144, 161, 166-167, 185-186, 197, 201, 203-204, 212, 214, 221, 224-229, 231, 233-240, 246, 260, 262, 264-267, 276, 286-287, 305-307, 309-312, 317, 324-328, 336-337, 339-343, 346, 362, 367, 388, 391-392, 396, 417-418, 431-432, 443, 445-447, 453, 461-462, 467, 486, 495, 507, 511, 653, 655, 659-660, 662, 664, 670, 674, 679, 682, 705, 799, 801, 807, 818, 821, 828, 830-831, 835, 841, 844-845, 848,

1905

850, 872, 884, 898, 906, 912, 927, 956, 969, 973, 975-983, 993, 996, 999, 1015, 1018-1019, 1022, 1028-1035, 1045-1047, 1051-1052, 1055, 1058, 1086, 1124, 1167, 1206, 1234-1241, 1249-1250, 1283, 1289, 1299, 1303, 1312, 1324-1326, 1334-1336, 1368-1369, 1378, 1398, 1402-1403, 1405-1410, 1442, 1512, 1692, 1698, 1789, 1794, 1828, 1843, 1858, 1862, 1878 signed, 2, 628 Numerators, 1008

O Objective function, 871-875, 878, 889, 1747-1753, 1764-1765, 1767-1768, 1775 Oblique asymptote, 238, 242-251, 256, 258, 266, 268, 276-279, 282, 738, 743, 1250-1253, 1256-1257, 1262-1264, 1268-1271, 1282, 1294-1295, 1301-1302 Oblique asymptotes, 243-244, 247, 267, 269, 738-739, 745 defined, 745 graphing, 267, 738-739 Oblique triangles, 571, 585, 595 Odd functions, 87-88, 129, 292, 430, 440 Odds, 942 One-to-one functions, 284, 292-293, 305 defined, 292, 305 One-to-one property, 346 exponential equations, 346 Open interval, 89-90, 93, 95, 129, 1045-1046 Open intervals, 89 Optimal, 84, 189, 581, 701 Ordered pair, 2, 13, 17, 296, 628, 682, 742, 790, 792-793, 862, 1085, 1313, 1320, 1645, 1771 Ordered pairs, 17, 64, 66-67, 74, 128, 145-146, 153-154, 159-160, 292-293, 295-296, 301, 381, 795, 821, 1145, 1167, 1313, 1681-1684, 1692-1693, 1715 coordinates of, 17 Ordered triple, 682 Ordinate, 2, 56 Origin, 2, 13, 16, 19-25, 39, 44, 46, 50, 54, 57, 60, 83, 87-89, 95, 99-100, 123, 125, 130-131, 176, 188, 203-204, 219, 401, 410, 414-415, 417, 423, 425, 430, 437-438, 444-445, 454, 458, 461, 477, 486, 488, 512, 581, 596, 628, 630, 632-633, 635-636, 642, 650, 654, 658, 660, 664, 671, 682-683, 690, 696, 712, 715-718, 721-729, 732-737, 739-740, 743, 747, 769, 774, 782, 785, 863, 976, 1024, 1027, 1050, 1062, 1088-1094, 1109, 1112-1113, 1118, 1120, 1135-1136, 1139, 1158, 1163, 1195, 1213, 1302, 1402, 1413, 1430, 1450, 1567, 1597, 1630-1631, 1645-1646, 1648-1649, 1668, 1677 coordinate system, 2, 16, 401, 410, 414, 512, 596, 628, 632, 636, 664, 718, 721-722, 729, 733, 1645, 1677 symmetry, 19-25, 54, 57, 60, 83, 87-89, 95, 100, 123, 188, 219, 414, 423, 445, 461, 642, 650, 712, 715-718, 721, 723, 732, 1088-1094, 1109, 1112-1113, 1120, 1135, 1139, 1302, 1649, 1677 Orthogonal vectors, 674, 677, 697 Ounces, 109, 137, 870, 874, 1044, 1147, 1746, 1883 Outputs, 64, 66-68, 79, 110, 160, 293, 296, 307-309, 380, 415, 426, 495, 1179, 1323

P Parabola, 9, 102, 105, 120, 128, 162-164, 166-169, 174, 176-177, 180, 187, 193, 202-203, 428, 652, 710-721, 745-747, 751-754, 756-759, 762, 773-776, 778-779, 782-783, 785-786, 852-854, 858-861, 867, 1075, 1182, 1192-1193, 1195, 1201, 1213, 1218, 1258, 1265, 1614, 1622-1625, 1627-1629, 1644, 1646, 1649, 1651, 1653, 1663-1666, 1668-1669, 1671, 1673, 1675, 1677, 1679, 1740, 1742-1743, 1763 defined, 128, 164, 202, 745, 762, 773-774 equation of, 176-177, 180, 187, 712-719, 721, 745-747, 751-754, 757-758, 762, 773-776, 778-779, 782, 786, 858, 860-861, 1195, 1213, 1622-1625, 1627-1629, 1646, 1663-1665, 1671, 1673, 1675, 1677

1906

general form, 710, 746 graphing, 9, 120, 163-164, 166-169, 177, 187, 203, 652, 712-715, 717, 720-721, 753, 756-758, 762, 773-776, 852-854, 859, 867, 1075, 1258, 1265, 1679 intercepts, 105, 120, 164, 166-167, 169, 187, 721, 1182, 1193, 1201, 1265 vertex, 102, 162-164, 166-169, 174, 176, 187, 203, 711-721, 745, 751, 754, 756-757, 759, 762, 774-776, 778-779, 782-783, 785-786, 858, 1192-1193, 1201, 1213, 1614, 1622-1625, 1627-1629, 1644, 1649, 1651, 1653, 1663-1665, 1668-1669, 1671, 1673, 1677, 1679 Parallel lines, 29, 37-38, 44, 52-53, 791, 873 defined, 29 Parallelograms, 696 area of, 696 Parameters, 372 Parametric equations, 710, 760-777, 1662, 1680 defined, 760-764, 767, 770-771, 773-774 writing, 773 Partial fractions, 845 decomposition, 845 Pascal, Blaise, 770, 927, 963 Paths, 777 length of, 777 Patterns, 410, 616, 772, 957, 1013 Percent increase, 157 Percentages, 883 Percents, 1777 Perfect cube, 1056 Perfect square, 1000-1002, 1020, 1056 Perfect squares, 981, 996, 1000, 1020 Perimeter, 125-126, 175, 180, 605-606, 618, 623, 801, 860, 984, 987, 993, 1042, 1077, 1159, 1195, 1542, 1737, 1861, 1882 Periodic function, 443, 477 Periods, 353, 457, 475, 481-482, 551, 615, 1803, 1808 Permutations, 942, 948-954, 964, 967-968, 1075 Perpendicular lines, 29, 38, 40, 44, 52-53, 682 defined, 29 graphing, 53 vectors, 682 Phase shift, 400, 466-467, 469-470, 472, 475, 479, 481-483, 489-491, 610, 1417-1419, 1421-1423, 1428-1429, 1431, 1434, 1436, 1517 Piecewise-defined functions, 63, 99, 104, 129, 133 Pixels, 3 Plane, 2-4, 8, 14, 17, 25, 32, 37, 44-45, 52, 54-55, 78, 82, 97-98, 124, 129, 145-146, 153, 157, 171, 174, 319, 332, 350, 383, 410, 415, 429-430, 444, 505, 520-521, 551, 555, 579, 592, 621, 627-628, 636, 653-655, 659, 661, 664-665, 670, 672, 676, 679, 682-686, 688-689, 693, 695, 697-701, 706, 710-712, 719, 722, 730, 733, 743, 747-748, 751, 753-754, 759-760, 773-775, 783-784, 786, 795, 798, 863-864, 867-868, 1044-1045, 1062, 1458, 1526-1527, 1530, 1558-1559, 1590, 1594, 1616, 1676, 1686, 1775 Plots, 10, 188, 639, 762 scatter, 188 Plotting, 2-3, 7-10, 16, 18, 21, 24-25, 46, 53-55, 103, 110, 629, 632, 637, 643, 653, 655, 660, 697, 726, 738, 761, 763, 1174, 1672 Plotting points, 2-3, 7-10, 16, 18, 21, 24-25, 46, 53-55, 629, 637, 643, 697, 763 Point, 2-9, 12, 14-17, 19, 21, 23-25, 27-29, 32-34, 36, 38-41, 44-45, 48-55, 63, 78-83, 85-89, 92-95, 97-99, 105, 110, 113-115, 119, 122-125, 127, 129, 144, 147, 149-152, 156-157, 159, 161-169, 171, 174-175, 179-180, 186-187, 193, 200, 208, 212-213, 215, 217, 219, 229, 252, 270-271, 294, 298, 301, 310, 312, 315, 319-320, 332-333, 345-348, 350, 367-368, 380, 383, 385, 387, 398, 400-401, 414-419, 421-428, 430-440, 446, 458-459, 466, 470, 477, 480-484, 486-487, 493, 508, 512, 518, 521, 530, 533-534, 539, 543, 555, 572-573, 576, 579, 582, 584, 592-593, 600-601, 607-609, 612-613, 620, 628-637, 643-644, 650, 652-655, 661-665, 667, 671-673, 675, 677-678, 682-683, 689-690, 693, 696-699, 711-712, 715, 718-720, 722-723, 730-733, 741-742, 744-748, 754-755, 759, 761,

763-764, 766, 768-770, 772, 775-776, 782, 791, 794, 799, 827, 852, 855, 859-861, 863-866, 872-875, 878, 889-890, 911, 976, 986, 1024, 1043, 1070, 1077, 1079, 1084-1086, 1091, 1093, 1098-1102, 1109, 1114-1115, 1117-1118, 1131-1133, 1138-1139, 1141-1142, 1149, 1157-1158, 1174-1177, 1181-1182, 1186-1187, 1191-1193, 1195, 1206-1207, 1210-1215, 1220, 1224, 1228, 1252-1264, 1266-1268, 1270, 1272, 1280, 1282, 1291-1297, 1302, 1313, 1324, 1328, 1333, 1339, 1352-1353, 1368, 1374, 1378, 1389, 1391-1395, 1403-1404, 1430, 1433, 1443, 1482-1483, 1519, 1526, 1530, 1532, 1551, 1561, 1564-1567, 1576-1578, 1590, 1604-1605, 1612-1613, 1617, 1619, 1622, 1627-1629, 1635, 1645, 1663, 1671-1672, 1674, 1677, 1679-1680, 1708, 1727, 1729, 1737, 1740-1745, 1748-1750, 1752, 1762-1765, 1774-1776, 1828-1830, 1844, 1849-1850 equilibrium, 149-150, 152, 608, 673, 699, 1177, 1612 Points, 1-10, 12-25, 29-32, 34, 36-37, 40-41, 43, 45-46, 52-56, 59-60, 77-80, 82, 84, 93-95, 97-98, 100-104, 106, 110, 112-113, 119, 122-123, 129, 131, 135, 137, 146, 150, 154-159, 162, 169, 171, 176, 180, 184-185, 188, 193, 203-204, 207-208, 210-216, 219-220, 225, 235-236, 241-242, 249-250, 265, 268, 271, 279, 294, 297-298, 305, 309, 311-312, 317, 319, 325-326, 331, 350, 431, 435, 444, 446, 450-453, 456, 458-459, 462, 465, 468-470, 474-475, 480, 482, 517-518, 520-522, 581, 600, 609, 612-613, 621, 628-629, 631-632, 636-637, 639, 642-647, 649, 653, 658, 661-662, 672, 681-683, 689-690, 697, 706, 711-717, 719, 721-723, 725, 727, 730, 732-739, 741, 743, 746, 754, 757, 759-763, 767, 769-770, 773-775, 779, 791, 795, 798-799, 801, 816, 827, 854-855, 858, 862-870, 872-875, 878, 880-882, 887, 893, 963-964, 976, 978, 982, 1024, 1050, 1067, 1077, 1079, 1091-1092, 1094, 1098-1099, 1103-1105, 1111, 1114, 1117, 1133, 1135, 1148, 1168, 1174, 1177, 1179-1181, 1187, 1190-1191, 1211, 1220, 1223-1230, 1242-1244, 1253-1255, 1257-1269, 1271, 1281, 1286-1287, 1291-1294, 1296, 1299-1300, 1302, 1322, 1327-1328, 1398, 1404-1410, 1418-1419, 1434, 1458, 1519, 1526, 1553-1554, 1576-1578, 1594, 1619, 1622-1625, 1627-1629, 1645, 1671, 1673-1674, 1678, 1698, 1726-1728, 1734-1735, 1743-1753, 1763-1767, 1775, 1808 Point-slope equation, 1077, 1737 Point-slope form, 29, 33-34, 38-39, 53, 87, 94-95, 156, 1138-1139 Polar coordinate system, 628, 636 polar axis, 628, 636 pole, 628 Polar equations, 627, 637, 639, 642-643, 648-652, 697, 710, 754, 756, 773-774 graphing, 627, 637, 639, 643, 648, 651-652, 697, 756, 773-774 Polynomial, 161, 199-282, 322, 516, 544, 549, 717, 746, 845, 847, 850-851, 861, 995, 997-999, 1001-1003, 1005-1010, 1016, 1018, 1020, 1027, 1063, 1219-1304, 1808, 1828, 1848, 1862, 1866 Polynomial equations, 220-221, 226, 230, 233, 266, 717 conjugate pairs theorem, 233, 266 First-degree, 221 Polynomial functions, 200-202, 204, 212, 214, 216, 219-221, 233-234, 236, 238, 266, 268 factor theorem, 234, 266 graphs of, 202 intermediate value theorem, 220, 266 remainder theorem, 221 synthetic division, 220, 236 Polynomial inequalities, 259, 266 Polynomials, 201-202, 212, 220-221, 259-260, 271, 280, 544, 552, 844-845, 973, 993-999, 1002-1003, 1007, 1009-1010, 1014, 1018, 1232, 1303, 1808, 1866 defined, 202, 973, 1007, 1014 degree of, 201-202, 220-221, 845, 994-995, 998,

1002, 1303, 1866 dividing, 996, 998, 1007 factoring, 220, 844, 999, 1002, 1007, 1014, 1018 in one variable, 201, 994-995, 999, 1007, 1014 multiplying, 202, 1007 prime, 999, 1002, 1866 quadratic, 201-202, 220-221, 259, 844-845, 973, 1014, 1018, 1808 ratio of, 202, 845, 1303 special products, 993, 995, 999 Population, 76-77, 86, 98, 145-146, 177, 182, 195, 248, 293, 295-296, 320, 333-334, 350-351, 359, 362-363, 366-371, 375-378, 384, 883, 892, 904, 934-935, 948, 1354, 1359, 1361-1362, 1365, 1782, 1807, 1852-1853 census, 333-334, 350-351, 378, 384, 883, 1852 Population growth, 368-369, 892, 934 Position vector, 661, 664, 671, 674, 683-684, 689, 696-697, 700 Positive integers, 893, 902, 921, 931, 993, 1055, 1300 Positive numbers, 264, 1032 Pounds, 85, 265, 669-670, 672-673, 678-681, 699-700, 801, 843, 870, 877, 880-881, 934, 939, 1038, 1042, 1591-1592, 1594, 1612-1613, 1686, 1717, 1746, 1751, 1765-1766, 1806, 1881 Power, 25, 43, 63, 201-204, 209-210, 215-216, 243, 265, 268, 271, 289, 305, 307, 322, 324, 336-337, 339, 347, 359-360, 370, 373, 383, 458, 541, 656, 745, 883, 921, 980-981, 994-995, 1009-1010, 1032, 1055, 1057-1058, 1062, 1071, 1285, 1300, 1345, 1355, 1357, 1780, 1804 defined, 63, 202, 305, 307, 322, 324, 458, 745, 981, 1032, 1055 logarithms, 336-337, 339, 347, 383 Power functions, 202-204, 210, 215 Power Rule, 1355 Powers, 306, 338, 342, 383, 385, 544, 656, 846, 925, 1004, 1031-1032, 1774 Prediction, 935, 1231, 1807 Present value, 351, 353, 356, 358, 360, 381-382, 920 Price, 1, 8, 55-56, 66, 77-78, 85, 149-150, 152-153, 159, 161-162, 171, 173-174, 178-179, 182, 186, 188, 190-193, 265, 292, 319, 335, 359-360, 377, 379-380, 387, 605, 789, 794, 802, 882, 884, 920, 935, 945, 966, 1038, 1042, 1045, 1053, 1064-1065, 1177-1179, 1181, 1192-1194, 1197, 1204, 1212, 1284-1285, 1312, 1364, 1366, 1689, 1776, 1793, 1808, 1884, 1888-1889 sale, 55-56, 149, 173, 292, 1045, 1053, 1312, 1884 total, 77, 178, 188, 359, 379, 387, 789, 920, 1038, 1042, 1064, 1178, 1212, 1284, 1793 Prime numbers, 321 Prime polynomials, 999 Principal, 351-358, 394, 981, 1032, 1036-1037, 1042, 1055, 1322, 1355, 1894 Principal square root, 981, 1032, 1322, 1894 Principle of Superposition, 560 Probabilities, 882, 957-965, 967-968, 970, 1813, 1818, 1820 Probability, 315-316, 320, 334, 372, 395, 882-883, 930, 941-971, 1329, 1340, 1363, 1777, 1809-1820 addition rule, 1818 coin toss, 970 complementary events, 962 mutually exclusive events, 961-962 odds, 942 Probability of an event, 959, 964 Problem solving, 173, 973, 1035 Product, 18, 20, 38-39, 43, 52, 64, 72-73, 129, 149, 160-161, 171, 174, 178-179, 183-184, 188, 190, 205, 227, 235-238, 292, 315, 337, 339, 345-346, 368, 387, 493, 517, 535, 551-556, 559, 594, 596, 602, 604, 614, 627, 655, 663, 670, 674-675, 679-681, 684-686, 689-698, 705, 827, 831-835, 841-842, 844-845, 853, 855, 858, 860, 877-878, 886, 895, 976, 980, 982, 984, 994-995, 999-1002, 1006, 1009, 1014-1015, 1018, 1027, 1029-1032, 1040, 1042, 1047, 1054, 1072, 1078, 1102-1103, 1117, 1304, 1593, 1718, 1770, 1824, 1866, 1880 signs of, 895 Profit, 42, 77, 134, 152, 172, 843, 874-877, 889, 963, 1042, 1045, 1064, 1130, 1172, 1177, 1192, 1749-1752, 1855, 1884

average, 77, 1042, 1045, 1172 total, 77, 134, 843, 874, 876, 963, 1042, 1064, 1749-1752, 1855 Projectile, 162, 175-176, 179, 186, 195, 428-429, 512, 521, 545, 551, 555, 764-766, 771-773, 776, 1194-1195, 1218 Pyramid, 594, 905, 992, 1532-1533, 1861 surface area of, 905 Pythagorean identities, 436, 523, 529, 536, 1424-1426, 1433 Pythagorean theorem, 4-6, 13, 39, 43, 54, 60, 124, 134, 421, 426, 572-573, 575, 596, 598, 621, 683, 985-986, 1080-1081, 1085, 1117, 1538, 1553, 1556, 1636, 1662, 1861 Pythagorean triples, 993

Q Quadrantal angles, 414, 417-419, 480 defined, 417-419 Quadrants, 3, 31, 52, 56, 102, 516, 1157, 1398, 1432, 1515-1516 Quadratic, 18, 26, 51, 120, 143-197, 201-202, 214, 218, 220-221, 225-227, 230, 233-234, 236, 259, 268, 270, 304-305, 315, 322, 344-346, 348, 372, 379-380, 396, 513, 516-518, 557, 659, 712, 717, 746, 750, 765, 798-799, 844-845, 848-850, 854, 856-857, 861, 878, 880, 973, 1014, 1018-1025, 1027-1028, 1032-1034, 1057, 1077, 1173-1218, 1236, 1240, 1300-1301, 1365, 1454, 1486, 1660-1661, 1672, 1737-1738, 1765, 1774, 1808, 1870, 1877-1878 Quadratic equations, 18, 161, 233, 305, 344, 513, 973, 1014, 1018-1022, 1024-1025, 1027-1028, 1032 quadratic formula, 1014, 1018, 1021-1022, 1027, 1032 Quadratic formula, 165, 221, 516, 659, 750, 856, 861, 1014, 1018, 1021-1023, 1027, 1032-1033, 1077, 1236, 1240, 1301 defined, 1014, 1032 discriminant, 165, 861, 1022-1023, 1027, 1032-1033 using, 165, 221, 516, 750, 856, 861, 1014, 1018, 1021-1023, 1032-1033, 1236, 1240, 1301 Quadratic functions, 143-197, 259, 304-305, 1173-1218 completing the square, 161, 163 defined, 145, 149, 159-161, 164, 173, 305 graphing, 143-144, 154, 156-161, 163-164, 166-170, 177-179, 181-184, 187-189, 259, 305, 1173 quadratic inequalities, 259 Quadratic inequalities, 259, 322 defined, 322 Quadratic regression, 178, 1196, 1214, 1216, 1365 Quarterly compounding, 353 Quaternions, 670 Quotient, 64, 69, 72-73, 76, 98, 128-130, 221, 223, 225, 231, 233, 245-246, 248, 251-253, 255, 261-262, 280, 321, 337, 339, 343, 435, 438, 443, 523, 525-527, 541, 655, 851, 858, 975, 995, 997-998, 1002-1007, 1011, 1021, 1030, 1035, 1057, 1059, 1061, 1063, 1232, 1240, 1248-1249, 1288-1290, 1301, 1863-1864, 1866 functions, 64, 69, 72-73, 76, 98, 128-130, 221, 223, 225, 231, 233, 245-246, 248, 251-253, 255, 261-262, 280, 321, 337, 339, 343, 435, 438, 443, 523, 526-527, 1232, 1240, 1248-1249, 1288-1290, 1301 real numbers, 98, 129, 221, 231, 233, 246, 262, 337, 339, 343, 443, 975, 1249 Quotients, 225, 338, 526-527, 560, 653, 655-656, 697, 1009, 1011, 1021, 1040

R Radian measure, 406-407, 639 Radical sign, 981 Radicals, 1054-1058, 1061 defined, 1054-1055 like radicals, 1056 rationalizing the denominator, 1057 Radicand, 71, 1055-1056, 1161 Radioactive decay, 363, 369-370 Radius of a circle, 52, 410, 1389 Range, 65-68, 70-71, 74, 78-80, 82-83, 85-86, 95, 98-107, 109-112, 116-119, 122, 128,

130-132, 134-135, 137-141, 165-167, 170-172, 186, 188-189, 194, 196-197, 203-204, 212, 214, 247, 265, 271-276, 280-281, 285, 293-296, 301-302, 304, 309-314, 316-319, 324-328, 331-333, 367, 380, 382-383, 385, 389-393, 396-398, 400, 426, 428-432, 441-443, 445-447, 452-453, 456, 461-465, 478-481, 486-491, 494-495, 499, 502-505, 507, 509-512, 521, 545-546, 555, 557, 561, 568, 773, 1044, 1053-1054, 1086, 1124, 1131-1132, 1136, 1144-1145, 1149-1152, 1160, 1163, 1165-1168, 1183-1186, 1188-1189, 1193, 1208-1210, 1215, 1223-1230, 1242-1244, 1250, 1286-1287, 1313-1314, 1320-1322, 1324-1326, 1328-1329, 1334-1340, 1368-1372, 1378-1379, 1382, 1402-1403, 1405-1410, 1414-1416, 1426-1428, 1430, 1439, 1441-1442, 1444, 1499-1500, 1511-1512, 1514, 1843, 1888-1889 defined, 65, 67-68, 70-71, 74, 78, 98-99, 104, 106-107, 110, 128, 131, 285, 294-296, 313, 324, 327-328, 442, 462-463, 478, 505, 509, 511, 557, 561, 568, 773, 1053-1054, 1168, 1313, 1402, 1441, 1499, 1843 determining, 65-67, 116, 166-167, 170, 188, 294 Rates, 42-43, 109, 268, 284, 335, 351, 355, 359, 382, 801-802, 843, 892, 934, 1053, 1688, 1855 Ratio, 29, 31, 108, 180, 202, 242, 246, 307-309, 320, 333, 364, 405, 408, 416, 459-460, 462, 521, 754, 759, 845, 860, 877, 905, 912-919, 931-933, 975, 989, 1035, 1071, 1147, 1218, 1303, 1323-1324, 1527, 1735, 1752, 1789-1790, 1792-1793, 1800, 1804-1806 common, 845, 912-919, 931-933, 975, 1071, 1789, 1792-1793, 1800, 1804-1806 golden, 905 Rational equations, 26, 1014, 1017 Rational exponents, 306, 973, 1054, 1058-1060 defined, 973, 1054 radicals, 1054, 1058 roots, 973, 1054 Rational expression, 246, 261-262, 845-846, 850-851, 880, 882, 1006-1007, 1009-1013, 1017, 1058, 1303, 1718 Rational expressions, 238, 260, 262, 292, 844-846, 973, 1006-1011, 1013-1014, 1017 defined, 292, 973, 1007, 1014 dividing, 1006-1008, 1011 domain of, 238, 262, 973, 1007, 1014, 1017 multiplying, 846, 1007-1008, 1011, 1017 Rational functions, 199-282, 844, 1219-1304, 1849 defined, 202, 1266-1267, 1296 domain, 201, 203-204, 212, 214, 231, 238-240, 246-247, 249-254, 256-257, 262, 264-267, 269, 271-282, 1223-1230, 1242-1244, 1249-1250, 1253-1264, 1266-1272, 1279-1280, 1282-1283, 1286-1287, 1291-1296, 1301-1303 graphing, 199, 201, 203-207, 211-216, 218-219, 221-222, 224-225, 227, 229-230, 233, 237-240, 248-251, 254-255, 257-259, 261, 267-268, 270, 1219, 1223-1231, 1239, 1242-1244, 1254-1266, 1268-1273, 1281-1282, 1286-1287, 1291-1296 oblique asymptotes, 243-244, 247, 267, 269 Rational inequalities, 200, 259-260, 262, 266, 322 defined, 322 Rational numbers, 224-225, 238, 975-976, 1007-1008, 1028 Rational zero theorem, 1246 Rationalizing the denominator, 1057 of radicals, 1057 Ratios, 30, 37, 200, 224, 238, 410, 425, 572-573, 621, 1299, 1301 unit, 30, 410, 425 Ray, 401, 407, 412, 417, 505, 521-522, 579, 589-590, 608, 628, 632, 745, 747 bearing, 579, 589 defined, 417, 505, 745 Rays, 401, 404, 413, 477, 506, 637, 680, 705, 717-718, 720-721, 745, 775, 1389, 1443, 1593 Real axis, 653-655, 660, 704, 707-708, 1617 Real numbers, 2, 36, 48, 65, 70-71, 74, 77, 82, 89, 91, 98-103, 106, 110-112, 129, 134-136, 140-141, 144, 161, 166-167, 185-186, 197, 201, 203-204, 212, 214, 221, 226-227, 231,

1907

233-240, 246, 262, 264-267, 276, 286-287, 305-307, 309-312, 317, 324-328, 336-337, 339-343, 346, 362, 367, 388, 391-392, 396, 417-418, 431-432, 443, 445-447, 453, 461-462, 467, 486, 507, 511, 662, 664, 682, 799, 801, 807, 818, 821, 828, 830-831, 835, 844-845, 850, 872, 884, 898, 906, 912, 927, 975-976, 979-980, 982, 993, 996, 999, 1015, 1018-1019, 1022, 1028, 1033-1034, 1045-1047, 1052, 1055, 1058, 1086, 1124, 1206, 1249-1250, 1283, 1303, 1312, 1324-1326, 1334-1336, 1368-1369, 1378, 1398, 1402-1403, 1405-1410, 1442, 1512, 1692, 1698, 1789, 1794, 1828, 1843, 1858, 1862, 1878 absolute value, 101, 103, 106, 431, 982, 1045, 1055 complex, 65, 233-238, 266-267, 276, 844, 1028, 1033-1034, 1249, 1878 defined, 65, 70-71, 74, 91, 98-99, 106, 110, 129, 136, 161, 287, 305, 307, 324, 327-328, 417-418, 462, 511, 662, 821, 828, 835, 906, 912, 979, 1028, 1045, 1055, 1402, 1843 exponential expressions, 337 imaginary, 1028, 1878 in calculus, 98, 103, 239-240, 306 inequalities, 71, 89, 262, 266, 799, 801, 807, 818, 821, 828, 830-831, 835, 844-845, 850, 872, 884, 976, 1045-1047, 1692, 1698 integers, 103, 136, 238, 276, 975, 980, 993, 996, 999, 1028, 1055, 1058 irrational, 276, 306, 975-976, 1028, 1055 ordered pair, 2, 682 properties of, 99-101, 144, 201, 203-204, 238-239, 310, 312, 324-325, 336-337, 339, 341-342, 346, 367, 443, 445, 447, 461, 818, 830-831, 835, 898, 976, 1045, 1047 rational, 201, 203-204, 212, 214, 221, 226-227, 231, 233-240, 246, 262, 264-267, 276, 305-306, 844-845, 850, 975-976, 1028, 1055, 1058, 1249-1250, 1283, 1303, 1828 real, 2, 36, 48, 65, 70-71, 74, 77, 82, 89, 91, 98-103, 106, 110-112, 129, 134-136, 140-141, 144, 161, 166-167, 185-186, 197, 201, 203-204, 212, 214, 221, 226-227, 231, 233-240, 246, 262, 264-267, 276, 286-287, 305-307, 309-312, 317, 324-328, 336-337, 339-343, 346, 362, 367, 388, 391-392, 396, 417-418, 431-432, 443, 445-447, 453, 461-462, 467, 486, 507, 511, 662, 664, 682, 799, 801, 807, 818, 821, 828, 830-831, 835, 844-845, 850, 872, 884, 898, 906, 912, 927, 975-976, 979-980, 982, 993, 996, 999, 1015, 1018-1019, 1022, 1028, 1033-1034, 1045-1047, 1052, 1055, 1058, 1086, 1124, 1206, 1249-1250, 1283, 1303, 1312, 1324-1326, 1334-1336, 1368-1369, 1378, 1398, 1402-1403, 1405-1410, 1442, 1512, 1692, 1698, 1789, 1794, 1828, 1843, 1858, 1862, 1878 Real part, 679, 705, 1028-1029 Reciprocal identities, 435-436, 438-439, 443, 463, 523 Reciprocals, 39-40, 52, 463, 860, 1027, 1767 Rectangle, 3, 42, 59, 76, 86, 123-126, 131-132, 141, 175, 179-180, 188, 255, 368, 551, 573-574, 598, 602, 738-739, 757, 860, 974, 984, 987, 991-993, 1044, 1069, 1077, 1103, 1129, 1167, 1194-1195, 1272, 1541, 1686, 1735-1736, 1861-1862 Rectangles, 3, 905, 1069-1070 Fibonacci, 905 golden, 905 Rectangular coordinate system, 1, 3, 10, 16, 401, 410, 414, 596, 628, 631-632, 636, 664, 718, 721-722, 729, 731, 733, 1677 Rectangular equations, 637, 697, 760, 763-764, 767, 773-774 defined, 760, 763-764, 767, 773-774 Reduced row echelon form, 810, 812-813, 815, 817, 837-839, 1770-1771 Reference angle, 1440, 1498, 1512 Reflection, 19, 114-116, 118, 122, 137, 170, 203-204, 206, 267, 298, 311, 325, 454, 729, 1148, 1155, 1157 ellipse, 729

1908

graphing, 19, 114, 116, 170, 203-204, 206, 267, 298, 311 parabola, 203 Regression, 157, 178, 191, 215, 373-375, 474, 1180-1181, 1196, 1213-1214, 1216, 1363-1366, 1376-1377 exponential, 373-375, 1363-1366, 1376-1377 linear, 157, 178, 191, 373, 1180-1181, 1196, 1213-1214, 1216 Relations, 64-65, 67, 154-155, 187, 293, 374, 956 defined, 64-65, 67 Remainder, 220-223, 231-233, 246, 248, 265-267, 275, 444, 666, 713, 724, 739, 745, 749, 997-998, 1002-1006, 1063, 1232, 1235-1241, 1244, 1280, 1287, 1777, 1863-1864, 1866-1867 Remainder theorem, 221-222, 231, 265, 1244, 1287 Resultant, 668-669, 672-673, 677, 705 Revenue, 77, 108, 152, 160-162, 171-174, 178-179, 182, 186, 188-190, 196, 789, 793, 801-802, 832, 877, 1038, 1042, 1053, 1192-1193, 1197, 1204-1205, 1212-1213, 1216, 1688, 1752-1753, 1856, 1884 average, 77, 801, 1042, 1053 total, 77, 178, 188-189, 789, 801, 832, 1038, 1042, 1212, 1688, 1752 Ridge, 584 Right triangles, 37, 400, 571-575, 616, 621, 664, 723, 985, 991, 1526, 1558 Pythagorean theorem, 572-573, 575, 621, 985 significant digits, 575 solving, 575 trigonometric functions of, 572-573, 616 Rise, 29-30, 32-33, 37, 39, 43, 60, 246, 250, 254, 309-310, 312, 410, 450, 473, 606, 694, 731, 761, 1104, 1551, 1616 Roots, 71, 653, 656-660, 697-699, 860-861, 905, 930, 973, 981, 1014, 1021, 1027, 1032, 1054-1056, 1167, 1244-1245, 1585-1586, 1608, 1615, 1739, 1877 cube root, 658, 1055 nth root, 657-658, 660, 1055 of the equation, 653, 657, 1014, 1021, 1027 of unity, 660, 1586 radicals, 1054-1056 Roster method, 973-974 Rotations, 1812 Rounding, 1139, 1417, 1519, 1560 Row operations, 803, 805-810, 815, 837-839, 877-878, 1711-1712, 1714, 1717, 1756, 1770-1772 Run, 29-30, 32-33, 37, 39, 43, 59-60, 731, 771-772, 775, 883, 921, 970, 1043-1044, 1064, 1103-1104, 1636, 1882-1883

S Sale price, 55-56, 292, 1045, 1312 Sample, 159, 369, 377, 385, 877, 957-960, 962-965, 967, 1813-1815, 1818 Sample space, 957-960, 962-965, 967, 1813-1815, 1818 Savings, 351-352, 355, 358-360, 385, 901, 1036 Scalar multiplication, 831 matrices, 831 Scalar quantities, 662 defined, 662 Scalars, 662, 664, 683, 689 Scatter diagrams, 154-155, 177, 187, 372, 470 line of best fit, 154, 177, 187 Scheduling, 871, 876 Scientific notation, 203, 348 Scores, 108, 512, 1053-1054, 1889 Secant, 94-95, 97-99, 400, 416, 424, 431-434, 440-442, 459, 463, 481, 491, 507, 509, 511, 556, 574, 1076, 1138-1142, 1402, 1446, 1497, 1511 defined, 98-99, 416, 442, 459, 463, 509, 511, 1402 graphs of, 99, 400, 459, 463, 507, 509 Secant line, 94-95, 97-99, 1138-1139, 1141-1142 Second quadrant, 715 Seconds, 17, 76, 121, 186, 291, 304-305, 321, 334, 366, 391, 393, 401, 403-404, 410, 412-413, 429, 457, 466, 475, 479, 482, 512, 520, 609-611, 614-615, 619-620, 741, 744, 765, 768, 771, 773, 775, 1043-1044, 1062, 1130, 1204, 1322, 1376-1377, 1382, 1388, 1396, 1430, 1457, 1543, 1554, 1558, 1660-1663, 1672, 1833-1834, 1845-1846, 1881, 1883, 1894

Semicircle, 125-126, 180, 551, 605, 620, 650, 681, 764, 987-988, 992, 1195, 1556-1557, 1594, 1861 Sequences, 342, 484, 891-939, 956, 1068, 1777, 1779-1808, 1869 arithmetic sequences, 892, 906, 911 defined, 893, 895-897, 903-906, 912, 918, 925, 932 finite, 916 geometric, 342, 892, 912-921, 931-933, 936, 938-939, 1789-1790, 1792-1794, 1800-1801, 1803-1808 infinite, 915-916, 918-921, 931-933, 1068, 1793, 1803 nth term, 894-896, 903-904, 906-910, 912-914, 918-919, 1787, 1806 Series, 116, 191, 219, 248, 359, 892, 912, 915-921, 931-933, 936, 942, 956, 1014, 1048, 1067-1073, 1075, 1789, 1791-1793, 1801, 1803, 1805, 1895-1897 arithmetic, 892, 918-919, 921, 931-933, 936, 1075, 1789, 1792-1793, 1801, 1803, 1805 defined, 912, 918, 932, 1014, 1070 geometric, 219, 892, 912, 915-921, 931-933, 936, 1071-1072, 1075, 1789, 1792-1793, 1801, 1803, 1805 mean, 936, 1068 summation notation, 932-933 Set notation, 71, 979, 1014, 1048, 1052 Set-builder notation, 973-974 Sets, 11, 63-65, 72, 74, 128, 190, 295, 415, 660, 766, 801, 883, 902, 934, 939, 942-945, 947-948, 961, 966, 973-975, 1050, 1686-1687, 1806, 1810 empty, 943, 973 intersection, 72, 947-948, 974-975, 1810 solution, 11, 72, 74, 295, 766, 902, 943-945, 961, 974, 1050, 1686-1687 union, 947-948, 961, 974-975, 1050, 1810 Sides, 6-7, 10-11, 13, 16, 20-21, 25, 30, 36-37, 47-48, 59, 76, 81, 100, 127, 174, 183, 188, 234, 255, 257-258, 260-261, 265, 299-300, 339, 344, 356-357, 362-364, 368-369, 406, 410, 425-426, 504, 514, 528, 530-531, 537, 546, 551, 572-575, 579-581, 585-586, 591-592, 595-598, 601-604, 606-607, 620-621, 625, 634, 638, 641, 662, 664, 675, 679, 693-696, 700, 712, 717, 722, 724, 733, 742, 750-751, 792-794, 796-798, 837, 840, 856, 858, 860, 905, 917, 924, 927, 937-938, 977, 985-986, 988-991, 993, 1014-1015, 1017, 1020-1021, 1023, 1025, 1037, 1040, 1047-1049, 1062, 1072, 1096, 1103, 1117, 1119, 1191, 1272, 1326, 1345, 1350, 1430, 1527, 1552, 1557, 1594, 1616, 1639, 1641-1642, 1675, 1688, 1719-1725, 1737, 1739, 1755, 1760-1761, 1767, 1769, 1774, 1797, 1860-1861 of an angle, 425, 531, 575, 602 Signal, 701, 710, 970, 1617 Significant digits, 575 Signs, 430, 434, 480-481, 548, 803, 840, 895, 928, 997, 1245, 1303, 1646, 1804 Simple interest, 351-352, 355, 358-359, 394, 802, 1036-1037 Simple interest formula, 352, 1036 Simplification, 39, 527, 650, 1016, 1252 Simplify, 10-11, 20, 28, 38, 40, 47, 69, 76, 95, 98, 130, 132, 301, 344-346, 387, 493, 522, 524-525, 528, 557, 658, 712, 722, 733, 792-793, 797, 809, 841, 848, 855-856, 909, 983-984, 990, 1006-1013, 1015-1016, 1020, 1037, 1040, 1048-1049, 1051, 1054-1056, 1058-1061, 1636-1637, 1687-1688, 1751, 1765 complex number, 658, 841 defined, 76, 98, 557, 841, 1007, 1054-1055 radicals, 1054-1056, 1058, 1061 trigonometric identities, 493, 522 with rational exponents, 1054, 1058 Simulation, 191, 766, 970, 1820 Sine, 321, 400, 415, 426, 430-435, 440-442, 444-446, 449, 451-452, 454, 456, 458-459, 463, 467-468, 470-475, 484, 491, 493-498, 507, 509, 513, 515, 517-518, 524, 526-528, 530-533, 536-538, 544, 546, 553, 556-557, 574, 584, 591, 598, 602, 637, 649-650, 653, 659, 746-747, 752, 1076, 1402, 1406, 1409, 1411, 1429, 1432, 1434, 1439-1442, 1444-1445, 1447, 1496-1499, 1511-1512, 1515, 1519, 1617, 1858 inverse, 493-498, 507, 509, 530, 536, 556-557,

1441-1442, 1499, 1512 Sine wave, 1519 Sines, 342, 526, 552-555, 571, 584-591, 594-596, 598-599, 605, 616, 621-622, 763, 1078, 1530-1533, 1540, 1542, 1555-1556 defined, 763 law of, 571, 584-591, 594-596, 598-599, 605, 616, 621-622, 1078, 1530-1533, 1540, 1542, 1555-1556 Sinusoidal functions, 444, 448, 450, 466-467, 470, 480, 571, 613 applications of, 571, 613 Slope, 29-45, 52-56, 58-60, 87, 93-95, 97-99, 101, 128, 136, 144-145, 147-148, 150, 152-153, 156-160, 187, 189-193, 195, 197, 202-203, 309, 430, 512, 791, 794-795, 799, 908, 1077, 1097-1105, 1110-1111, 1114-1115, 1117-1118, 1120, 1135, 1138-1142, 1174-1175, 1177, 1179-1180, 1206-1207, 1214, 1216, 1253-1254, 1257, 1259-1262, 1264-1267, 1270, 1291-1293, 1296, 1324, 1605, 1638, 1641, 1645, 1666, 1737, 1830, 1850 undefined, 29-30, 32, 40-41, 52, 54, 58, 60, 1077, 1097, 1099, 1101, 1111, 1114, 1266, 1850 Slope of line, 31 Slope-intercept equation, 1077 Solution set, 74, 183-186, 259-263, 265, 268, 315, 329, 345-348, 504, 515-518, 537-538, 545, 824, 1014, 1016-1019, 1021-1022, 1024-1025, 1032-1034, 1049-1052, 1094-1097, 1113, 1168, 1176, 1190, 1197-1205, 1211, 1215, 1274-1281, 1283, 1288-1289, 1296-1298, 1301, 1303, 1326-1327, 1332, 1338-1339, 1345-1353, 1372-1374, 1380, 1442-1443, 1450-1456, 1474-1475, 1486, 1495, 1508-1511, 1515-1516, 1589, 1622, 1681, 1716, 1870-1877, 1879-1880, 1885-1887, 1891 Solutions, 18, 26-28, 54-55, 58, 81, 105, 131, 165, 167, 169, 179, 187, 189, 205, 215, 225-226, 230, 232-233, 237, 242, 260, 262, 268, 344, 346, 348-351, 383, 385, 482, 513-519, 521-522, 537-538, 545, 558-559, 588, 596, 619, 657, 659, 670, 699, 719, 723, 734, 749, 776, 790-791, 795-796, 798-799, 803, 810-812, 817, 821, 824, 852-854, 857-858, 861, 873-874, 877-878, 880-881, 894, 933, 969, 1014-1015, 1018-1019, 1022-1025, 1027, 1033-1034, 1044, 1048, 1057, 1064, 1077, 1079, 1095, 1105, 1108, 1119, 1123, 1173, 1191, 1219, 1222, 1232, 1236, 1239-1241, 1244, 1247-1249, 1269, 1271, 1288, 1303, 1305, 1385, 1439, 1452, 1454-1455, 1458, 1486, 1521, 1563, 1619, 1621, 1679, 1681, 1689, 1703-1706, 1708-1709, 1716, 1726-1733, 1738, 1761-1762, 1765, 1769, 1771, 1773-1775, 1779, 1807, 1809, 1821, 1857, 1870, 1876-1878, 1880, 1884, 1895 checking, 513 of an equation, 18, 26-27, 537, 1014-1015 Solving equations, 1, 26-27, 348, 788, 973, 1014, 1025 algebraically, 26, 1014 Special products, 225, 993, 995, 999 Speed, 16-17, 76, 85-86, 109, 126-127, 137, 172, 181-182, 257, 304, 384-385, 401, 409-410, 412-414, 428-429, 466, 478-479, 481-483, 521-522, 546, 581-582, 593, 597-598, 600, 608, 617-618, 661-662, 667-669, 672-673, 699-700, 764, 766, 771-773, 776, 801, 857, 860, 881, 883, 1036, 1039-1040, 1042-1045, 1064, 1134, 1147, 1192, 1321, 1377, 1389, 1435, 1524, 1551, 1590, 1612, 1616, 1686, 1736, 1766-1767, 1834, 1846, 1881-1882, 1884 Spheres, 429 Spiral of Archimedes, 652 Spirals, 648 logarithmic, 648 Spreadsheet, 122, 133, 191, 270, 387, 484 Square, 3-6, 16, 18, 22, 31, 40, 42, 45, 47-49, 51, 56, 69-71, 85, 97, 99, 101-102, 106, 112, 117, 120, 122, 124-128, 131, 141, 161-164, 169, 175, 180-181, 255, 257-258, 264, 285, 291, 297, 320, 329, 334, 385, 409, 411-412, 422-423, 483, 531, 537, 561, 596, 600, 603, 605-606, 620, 625, 628, 634-635, 638-641,

657, 659, 680, 690, 694, 712-713, 717, 719, 721-722, 724, 728-729, 732-733, 735, 741, 743, 751, 758, 762, 765, 769, 776, 803, 828-830, 835-836, 841, 844, 853, 855, 860-861, 878, 885-886, 905, 920, 973, 981, 984-987, 990, 992-993, 1000-1002, 1014, 1018-1021, 1023, 1025-1028, 1032, 1034, 1041-1044, 1047, 1054-1057, 1062-1063, 1069-1070, 1076, 1080-1081, 1109, 1139, 1159, 1167, 1194, 1196, 1212-1213, 1272, 1298, 1322, 1354, 1381, 1430, 1443-1444, 1541, 1556-1557, 1560, 1593, 1616, 1626-1627, 1633-1634, 1644, 1673, 1708-1709, 1734, 1739, 1851, 1860, 1862, 1878, 1880, 1882, 1894 matrix, 803, 828-830, 835-836, 841, 844, 878 Square roots, 71, 657, 659, 905, 973, 981, 1014, 1021, 1032, 1054, 1167 defined, 71, 905, 973, 981, 1014, 1032, 1054 functions, 71, 1167 negative numbers, 981, 1032 principal square root, 981, 1032 Square units, 6, 56, 97, 122, 124, 141, 561, 603, 620, 625, 694, 885, 1063, 1081, 1109, 1139, 1167, 1213, 1443-1444, 1556, 1616, 1708, 1851 Squares, 6, 31, 39, 48, 191, 596, 598, 689, 728, 741, 746-747, 752-753, 775, 858, 860, 920, 981, 986, 990, 996, 999-1000, 1020, 1069, 1642-1643, 1737 area of, 6, 858, 860, 1069 perfect, 981, 996, 1000, 1020 Squaring, 537, 547, 634, 981, 1015, 1191 Standard deviation, 1054 Standard form, 36, 45-46, 48-50, 53-54, 315, 345-346, 634, 654, 657, 660, 698, 856, 995, 997, 1001, 1018-1019, 1022-1023, 1028, 1030-1032, 1034-1035, 1040, 1626-1627, 1630-1634, 1639, 1641-1644, 1663-1664, 1670, 1691, 1759 complex numbers, 660, 1028, 1030-1032, 1034-1035 linear equations, 1018 Standard viewing window, 11 Statements, 37, 133, 206, 215, 222, 322-323, 381, 433, 584, 921, 932, 977, 993, 1014 and positive integers, 921 defined, 133, 322, 381, 932, 977, 1014 Statistics, 302, 315, 334, 360, 943, 969 population, 334 Subset, 65, 71, 285, 943, 947, 959, 970, 974-975, 1028, 1810, 1878 Substitution, 223, 230, 285, 767, 788-789, 791-792, 794, 803, 807, 852, 854, 867, 878-879, 881, 1023-1024, 1576-1578, 1686-1687, 1726-1729, 1737-1738, 1766, 1768, 1848-1849 Subtraction, 656, 684, 829, 841, 978, 1028 of integers, 1028 Sum, 5-6, 64, 72-73, 129, 201, 232, 234, 237-238, 243, 335, 337-339, 342, 351, 356-357, 381-383, 385, 484, 493, 530-533, 535-537, 539, 542-543, 552-557, 559, 573-575, 580, 585, 594, 596-597, 605, 607, 612-616, 618, 653, 655, 659, 662, 665, 668-671, 677, 689, 722-723, 730, 746-747, 752, 774, 792-793, 805, 813, 822, 827, 829, 835, 845, 851, 858, 860, 878, 883, 893, 897-900, 903, 905-906, 908-912, 914-922, 924, 931-933, 938, 957-959, 964-965, 967, 984, 986, 988, 990, 994-996, 1000-1002, 1005-1006, 1027, 1029, 1031, 1069-1072, 1078, 1192, 1244, 1272, 1304, 1475, 1488, 1496, 1541, 1591, 1612, 1768, 1777, 1788, 1791, 1793, 1797, 1803-1806, 1808, 1813-1814, 1818, 1864, 1866, 1869 Summation notation, 893, 897-899, 903, 932-933 defined, 893, 897, 903, 932 index of summation, 898-899 Sums, 48, 338, 526-527, 552-554, 557, 830, 899, 1011, 1070-1071, 1073, 1304, 1866 Surface area, 255, 257-258, 291, 605, 858, 905, 984-985, 987, 991, 1044, 1077, 1272-1273 of a cube, 985 of a sphere, 984, 991 Survey, 371, 828, 944, 947-948, 966, 969-970 Symbols, 68, 123, 304, 444, 803, 874, 927, 964, 975-976, 1036, 1042, 1045-1046, 1048, 1051, 1819

Symmetry, 1, 18-25, 52-55, 57, 60, 83, 87-89, 95, 100-101, 123, 140, 144, 161-167, 169-171, 187-189, 193, 195, 219, 258, 268, 297, 299, 414, 423-424, 440, 445, 447, 461, 506, 637, 642-648, 650-651, 653, 697-699, 708, 710, 712-721, 723, 732, 751, 756-757, 762, 775, 778-779, 786, 1088-1094, 1109, 1112-1113, 1120, 1131-1132, 1135, 1139, 1183-1186, 1191, 1208-1210, 1214-1215, 1302, 1570-1580, 1605-1606, 1614, 1623, 1628-1629, 1649, 1651, 1673, 1677 about y-axis, 1131-1132 line of, 187-188, 297 Synthetic division, 220, 222-226, 236, 973, 1003-1006, 1232, 1235-1241, 1244-1245, 1248, 1280, 1288-1290, 1301, 1723 defined, 973 solving, 226, 973, 1239-1240, 1244 System of inequalities, 862, 865, 868-869, 879-880, 882, 1764, 1766 graphing, 862, 865, 868-869, 879-880, 1766 linear, 862, 865, 868-869, 879-880 nonlinear, 879

T TABLE feature, 16, 21, 229-230, 240, 254, 894, 901, 1242, 1782 Tables, 2, 12, 53, 114-115, 243, 342, 773, 876, 889, 1690, 1750 Tangent, 50-51, 99, 136, 416, 426, 431-434, 440-443, 459, 461, 463, 486, 491, 494, 501-503, 507, 516, 535, 556, 565, 574-575, 606, 621, 628, 633, 638, 649, 860-861, 1076, 1109-1110, 1141, 1402-1403, 1439-1442, 1444-1445, 1447, 1475-1476, 1497-1499, 1510, 1512, 1515, 1526, 1553, 1737-1738, 1830-1831, 1844-1845, 1850 defined, 51, 99, 136, 416, 442, 459, 463, 501, 535, 565, 606, 1402, 1441, 1475, 1499, 1850 graphs of, 99, 459, 463, 494, 507 inverse of, 494, 502 Tangent lines, 621 Tautochrone, 769-770 Temperature, 25, 43, 86, 109, 121, 137, 139, 182, 218, 274, 292, 304, 320, 365-366, 370-371, 384, 387, 395, 471-472, 474-476, 482, 662, 911, 983, 985, 1053, 1147, 1156, 1231, 1360, 1362, 1376-1377, 1789 Terminal, 401-402, 417, 424-427, 477, 481, 483, 530, 572, 628-629, 631, 635, 661-662, 664-665, 667, 671, 677, 683, 689, 698, 1430, 1614 Test scores, 1054 Theoretical probability, 970 Third quadrant, 1680 Tons, 593, 673, 904 Total cost, 127, 133-134, 255, 258, 843, 1103, 1178, 1284, 1717, 1751-1752, 1767 Total profit, 1749, 1751, 1855 Total revenue, 178, 189, 832, 1038, 1042, 1688 Transcendental functions, 284, 321 defined, 321 Transformations, 63, 110, 116-117, 119, 122, 161-162, 169-170, 187-189, 201, 204, 216, 238, 246-247, 266-268, 305, 307, 309, 311, 318, 326-328, 338, 444-447, 450-452, 454, 461, 464, 468-469, 480, 552, 712, 716, 721, 727, 732, 740, 862, 1420-1421 graphing, 63, 110, 116-117, 119, 161, 169-170, 187-189, 201, 204, 216, 238, 267-268, 305, 309, 311, 318, 326-327, 444-447, 451-452, 461, 464, 468-469, 712, 721, 732, 862 horizontal, 110, 116-117, 122, 162, 188, 238, 246-247, 266-267, 305, 307, 309, 311, 318, 328, 446-447, 454, 461, 464, 468, 716, 721, 740 multiple, 480 reflection, 116, 122, 170, 204, 267, 311, 454 vertical, 110, 116-117, 122, 162, 169, 204, 238, 246-247, 266-268, 307, 326-328, 445, 447, 452, 461, 464, 468-469, 716, 721, 727, 740, 1421 Translations, 672 horizontal, 672 Transverse axis, 733-737, 739-741, 743-744, 748, 754, 756-757, 775, 780-782, 785, 1637-1643, 1646, 1648, 1664-1667, 1672-1674, 1726 Tree diagram, 946, 960

1909

Trees, 816 Triangles, 7, 13, 16, 30, 37, 126, 400, 410, 425-426, 530, 571-575, 578, 584-589, 591-592, 595-598, 601-606, 616, 618, 621-624, 664, 723, 905, 927, 985, 988-992, 1078, 1160, 1526-1527, 1529-1530, 1537, 1550, 1553, 1558-1559, 1860-1861 acute, 572-575, 584-585, 591, 596, 598, 606, 616, 1078 area of, 13, 410, 571, 573-574, 601-606, 616, 618, 623, 905, 985, 988, 991-992, 1553, 1860-1861 congruent, 7, 13, 530, 573, 664, 723, 985, 988-990 equilateral, 16, 126 isosceles, 16 law of sines, 571, 584-589, 591, 595-596, 598, 605, 616, 621-622, 1078, 1530 obtuse, 584-585, 596 right, 13, 16, 37, 126, 400, 426, 571-575, 584, 587, 589, 591, 596, 605, 616, 621, 664, 723, 985, 989-991, 1526, 1553, 1558, 1860 theorem, 7, 13, 37, 425-426, 530, 572-575, 585, 595-596, 598, 602-603, 616, 621, 664, 723, 905, 927, 985, 1553, 1861 Triangular numbers, 905 Trigonometric functions, 399-492, 493-494, 504, 507-508, 513, 518, 523, 530, 536, 546, 556-557, 560, 571-625, 628, 637, 701, 1078, 1385-1438, 1521-1561, 1828 cosecant, 400, 416, 424, 431-434, 440-442, 459, 463, 491, 507, 556, 574 cotangent, 400, 416, 424, 431-434, 440-443, 459, 462-463, 491, 507, 556, 574 domain and range, 400, 445-446, 507, 557 evaluating, 451 inverse functions, 494, 556 secant, 400, 416, 424, 431-434, 440-442, 459, 463, 481, 491, 507, 556, 574, 1402 sine and cosine, 400, 433, 435, 444, 449, 463, 518, 530, 557, 574, 637, 1432 tangent, 416, 426, 431-434, 440-443, 459, 461, 463, 486, 491, 494, 507, 556, 574-575, 606, 621, 628, 1402-1403, 1526, 1553 Trigonometric identities, 493, 522-523, 530, 560, 1078 double-angle, 493, 1078 fundamental, 522, 560, 1078 half-angle, 493, 1078 product-to-sum, 493, 1078 Pythagorean, 523 reciprocal, 523 sum-to-product, 493, 1078 Trigonometry, 400, 410, 425, 484, 493-569, 572-573, 575, 621-622, 1439-1519 angle of elevation, 512, 521, 622, 1450, 1458 conditional equations, 523, 527 functions, 400, 410, 425, 484, 493-494, 497-498, 501, 504, 507-510, 513, 518, 523-524, 526-530, 536, 546, 556-557, 560, 572-573, 575, 621-622, 1450 identities, 493, 513, 517-518, 522-525, 527, 529-532, 534, 536-537, 542-543, 545-546, 552, 557, 560 law of cosines, 621 law of sines, 621-622 right triangles, 400, 572-573, 575, 621 significant digits, 575 Trinomials, 1000 factoring, 1000 perfect square, 1000 Turning point, 208, 212-213, 252, 1224, 1228

U Uninhibited growth model, 384 Union of sets, 974 Unit circle, 46, 53, 67, 400, 414-426, 428, 430-436, 440, 443, 458, 477-478, 480-482, 486-487, 530, 606, 650, 1391-1393, 1397, 1403 defined, 67, 416-419, 458, 478, 606 Unit vectors, 665, 669, 681, 683, 696, 706, 1602 Universal set, 944, 947, 962, 974-975, 982

V Values of a function, 68-69 Variable costs, 42 Variables, 7-8, 10, 26, 29, 63, 68, 76-77, 123, 148, 155-157, 181, 189, 191, 214-215, 218, 268, 299-301, 304, 372, 374, 438, 538, 637, 651, 698, 717, 728, 746, 788-799, 803-805, 807,

1910

812, 815, 817-821, 823-824, 827, 837, 839-840, 846, 848, 852, 862-863, 865, 868, 871-874, 878, 881, 908, 978-979, 984, 993, 1007, 1035-1036, 1042, 1059-1061, 1063, 1213, 1285, 1681, 1684-1686, 1688-1689, 1754, 1766, 1769-1773, 1866 functions, 63, 68, 76-77, 123, 148, 155-157, 181, 189, 191, 214-215, 218, 268, 299-301, 304, 372, 374, 438, 637, 1213, 1285 Variation, 230 Variations, 523, 770 Vectors, 627-708, 1563-1619 addition, 639, 656, 662, 684, 706 defined, 654, 662, 674, 678, 680, 685-686, 689-691 direction angle, 667-669, 672, 688, 698, 700, 1609 direction of, 628, 661, 664, 666-669, 672-673, 678-680, 682, 685, 688-689, 699-700, 1590 dot product, 627, 674-675, 679-681, 685-686, 689-690, 693-694, 696-698, 705, 1593 equality, 665, 684 orthogonal, 674, 676-678, 680-681, 691, 693-700, 705-708, 1592-1594, 1600, 1602-1603, 1611, 1615 parallel, 662, 674, 676-678, 680, 682, 688-689, 693-695, 697, 699-700, 705, 707-708, 1592-1594, 1611, 1615 perpendicular, 676-677, 680, 682, 699, 705, 1594, 1611, 1613 position vector, 661, 664, 671, 674, 683-684, 689, 696-697, 700 scalar product, 674, 684 sum and difference, 653 unit, 650, 661, 663, 665-667, 669, 671, 681, 683-685, 689-690, 695-698, 700, 705-706, 1587, 1602 vertical and horizontal components, 700 zero, 661-662, 669-670, 676, 692-694, 1591, 1593, 1612 Velocity, 84, 86, 175, 179, 186, 191, 430, 512, 521, 545, 551, 555, 661, 667-669, 672, 765, 773, 1042, 1062, 1133, 1217, 1389, 1590-1591, 1612, 1662, 1830 angular, 1389 linear, 175, 179, 186, 191, 1217, 1389 Venn diagram, 948, 961, 975, 1810 Vertex, 6, 16, 102, 112, 125, 131, 161-171, 174-176, 183-185, 187-189, 195, 203, 401-404, 410, 477, 505, 590-591, 596, 607, 621, 628, 632, 675, 711-723, 727, 729-732, 734, 737, 740, 743-745, 751, 754, 756-757, 759, 762, 765, 774-776, 778-779, 782-783, 785-786, 858, 1167, 1183-1189, 1191-1193, 1197-1206, 1208-1215, 1332, 1526, 1614, 1622-1629, 1631-1632, 1634-1638, 1640-1641, 1644-1645, 1649, 1651, 1653, 1660-1661, 1663-1665, 1668-1669, 1671-1674, 1677, 1679, 1747-1753, 1764-1765, 1768 degree of, 203 even, 102, 195, 203, 404, 762, 1192 odd, 102, 195, 203, 1192 of angle, 591, 596 Vertical, 2, 4-5, 29-30, 32-33, 36-37, 40-41, 52-53, 56, 78-80, 82-83, 110, 112, 116-118, 122, 129, 134, 137, 162-164, 169, 176, 203-204, 238, 240-243, 246-254, 258, 261, 266-269, 276-279, 281-282, 300, 307, 310, 312, 325-328, 331, 380, 392-393, 397-398, 415, 445, 447-448, 452-453, 459-465, 468-473, 479, 503, 573, 576-577, 579, 581, 584, 620, 639-640, 648, 650, 652-653, 665, 668, 680, 700, 702, 705, 713, 716, 721, 727, 731, 740, 743, 745, 760, 766, 770, 773, 803-805, 807-808, 838-839, 931, 954, 968, 1079, 1099-1100, 1105, 1111, 1114, 1118, 1124, 1131, 1135, 1148, 1155, 1164, 1167, 1249-1261, 1263-1271, 1273, 1280, 1282, 1291-1297, 1301-1303, 1327, 1334-1336, 1372, 1379, 1413, 1421-1423, 1431, 1436, 1557, 1568, 1576, 1587, 1622-1624, 1646, 1665, 1673, 1770, 1816, 1829, 1843-1844, 1850 line test, 78, 82-83, 129, 380, 650, 653, 713, 1124, 1135, 1164, 1167 Vertical asymptotes, 238, 242, 247, 249-250, 252, 254, 258, 266-269, 276-278, 281-282, 310, 312, 459, 461, 463-465, 479, 743, 1250-1251, 1253, 1255-1261, 1268, 1291-1294, 1297, 1413

defined, 459, 463 graphing, 238, 249-250, 254, 258, 267-268, 312, 461, 464, 1255-1261, 1268, 1291-1294 identifying, 250 Vertical axis, 78 Vertical line, 2, 4, 29, 32-33, 36-37, 40, 52-53, 56, 78-79, 82, 129, 162, 164, 169, 240-241, 460, 579, 639-640, 648, 652, 702, 713, 721, 760, 770, 803, 931, 1079, 1099-1100, 1111, 1114, 1118, 1124, 1135, 1164, 1167, 1568, 1576, 1622-1623, 1665, 1673 graph of, 32, 36, 78-79, 82, 129, 162, 164, 169, 240-241, 460, 639-640, 652, 713, 721, 760, 1118, 1135, 1164 slope of, 29, 32, 40, 53, 56, 1100 Vertical lines, 33, 40, 721, 1131 Vertical-line test, 78, 83, 129, 650, 653 Viewing, 3-4, 11-12, 14, 16, 31, 54, 98, 203-204, 224, 227-229, 250, 289, 368, 444, 454, 506, 573, 639, 724, 761-762, 865, 896, 1234-1241 Viewing window, 3-4, 11-12, 14, 16, 31, 54, 98, 203-204, 224, 227-229, 250, 289, 639, 724, 761-762, 865, 896, 1234-1241 Volume, 126-127, 131-132, 232, 255, 257-258, 268, 291-292, 429, 604, 680, 696, 731, 860, 984-985, 987, 990-991, 1041-1042, 1053, 1062, 1077, 1160, 1170-1171, 1244, 1272, 1736, 1739, 1746, 1833, 1860 of a cone, 127, 292 of a cube, 232, 985 of a sphere, 984, 990-991, 1860

W Weight, 66, 85, 137, 158-159, 180, 186, 191, 193, 265, 304, 360, 377, 483, 593, 608, 669, 672-673, 680, 699-701, 877, 880, 934, 1042, 1053, 1181, 1217, 1273, 1321, 1590-1591, 1612, 1746, 1806

X x-axis, 2-4, 12, 16, 19-26, 42, 46, 50-51, 54, 56-57, 60, 80, 83, 86, 95, 97, 99, 110, 114-116, 118-119, 121-123, 125, 129-131, 137, 162, 165-167, 169, 176, 180, 183-186, 188, 202-205, 207, 211-212, 215-216, 219, 229, 236, 240, 243, 246, 249, 252, 254, 258-261, 268, 271, 282, 309-312, 314, 316, 321, 326, 380, 401-402, 410, 416, 423, 430, 446, 454, 458, 477, 506, 596, 608-609, 628, 630, 632, 635-636, 642, 644-646, 653, 659, 665, 668-669, 671-672, 681-683, 687-688, 708, 712-716, 719, 721-725, 727, 729-731, 733-736, 739-740, 743, 747, 762, 769, 775, 777-782, 894, 1079, 1086, 1088-1094, 1102, 1109, 1112-1113, 1118, 1120, 1135, 1139, 1148-1154, 1157, 1165-1166, 1168-1169, 1183, 1188, 1197-1204, 1206-1207, 1211, 1215, 1221-1226, 1228-1230, 1242-1244, 1250-1251, 1274-1275, 1285-1287, 1296-1297, 1299-1300, 1302, 1325-1326, 1332, 1335, 1371, 1414-1415, 1420-1421, 1426-1427, 1443, 1566, 1590, 1612, 1628-1643, 1645-1646, 1662, 1665-1666, 1670-1671, 1673, 1679, 1726, 1743-1745, 1763-1764 x-coordinate, 2-3, 12-13, 17, 27, 32, 45, 56, 78-79, 99, 102, 110, 112-114, 116, 122, 135, 154, 164, 166-167, 169, 171, 176, 195, 346-347, 415-416, 418, 426, 431, 452-453, 458, 468-469, 472, 518, 631, 769, 1079, 1086, 1168, 1183-1189, 1191, 1208-1210, 1214-1215 x-intercept, 12-13, 18, 20-21, 23, 25-28, 32-33, 36, 40, 43-44, 53, 57-59, 81, 99-101, 105, 165-167, 169, 172, 187, 206-207, 211-216, 220, 229, 236, 239, 248-250, 253, 260-261, 268, 271, 277-279, 281-282, 317, 325, 331, 348, 380, 513, 1083-1084, 1086-1087, 1091, 1093, 1099, 1101-1102, 1104-1105, 1114, 1132, 1142-1145, 1182, 1184, 1188-1189, 1193, 1206, 1220, 1225-1226, 1230, 1232, 1241, 1253-1254, 1256-1259, 1266-1268, 1270, 1274, 1291-1292, 1294-1296, 1300, 1726 defined, 99, 1105, 1143, 1266-1267, 1296 parabola, 105, 166-167, 169, 187, 1182, 1193, 1258 x-value, 91, 112, 1124-1125, 1155 xy-plane, 2-4, 8, 14, 17, 45, 54, 78, 82, 415, 444, 653,

661, 670, 682, 688, 698, 706, 747-748, 753, 760, 863-864, 867-868, 1594

ZERO feature, 13, 184, 1739

Y Yards, 175, 179, 412, 521, 599, 1043, 1194, 1388, 1537, 1561, 1672, 1882 y-axis, 2-4, 12, 16, 19-25, 42, 46, 50-51, 54, 56-57, 60-61, 80, 82-83, 87-89, 95, 99-101, 110, 114-119, 121-123, 125, 129-131, 134, 188, 203, 219, 239-240, 251, 291, 311-312, 321, 325, 331, 350, 380, 402, 416, 423, 426, 447, 451-453, 458, 468-469, 486, 488, 609, 628, 632, 642-643, 646, 653, 659, 665, 669, 682-683, 688, 712, 714-716, 718-719, 721, 723, 725-728, 730-731, 734, 736-737, 740-741, 743, 747, 777, 780-782, 1079, 1086, 1088-1094, 1098, 1109, 1112-1113, 1118, 1120, 1131-1132, 1135-1136, 1139, 1149, 1151, 1153-1155, 1163, 1165, 1299, 1302, 1322, 1324-1326, 1335, 1371, 1379, 1402, 1404-1410, 1434, 1604, 1628, 1630-1634, 1636, 1638-1644, 1646, 1660, 1662, 1664, 1671, 1673-1674, 1677-1679, 1743-1745, 1763-1764 symmetry, 19-25, 54, 57, 60, 83, 87-89, 95, 100-101, 123, 188, 219, 423, 447, 642-643, 646, 653, 712, 714-716, 718-719, 721, 723, 1088-1094, 1109, 1112-1113, 1120, 1131-1132, 1135, 1139, 1302, 1628, 1673, 1677 y-coordinate, 2-3, 5, 12-13, 17, 23, 27, 32, 56, 79-80, 102, 110-114, 116, 122, 135, 154, 164, 166-167, 169, 195, 348, 415-416, 418, 431, 458, 631, 769, 1079, 1084, 1086, 1149, 1157, 1168, 1183-1189, 1191, 1208-1210, 1214-1215, 1324 Years, 17, 43, 65, 76-77, 97, 121, 132, 134, 148-149, 152, 178, 181, 195, 233, 248, 293, 319, 335, 351-354, 356-360, 364-365, 368-371, 373-374, 377-380, 384, 387, 458, 470, 581-582, 593, 801, 858, 883, 892, 902, 904, 911, 918-920, 933-935, 939, 945, 948, 969, 1036-1037, 1042, 1053, 1065, 1129, 1170, 1177-1178, 1196, 1354, 1356-1359, 1375, 1381, 1617-1618, 1780, 1782-1783, 1789, 1806, 1810-1811, 1853-1854, 1888 y-intercept, 12-13, 18, 20, 22, 25, 29, 34-36, 40, 42, 44, 53-54, 57-60, 82-83, 99-101, 103-104, 109, 128, 135, 140, 144, 148, 150, 152, 157, 160, 164-170, 183-184, 187-189, 192, 195, 197, 202, 207, 210-211, 213, 217-219, 225, 236, 249-253, 258, 268, 271-282, 309-312, 316, 325, 331, 367, 380, 445, 447, 454-455, 461, 465, 794-795, 1077, 1083-1084, 1086-1087, 1091-1092, 1097, 1099-1105, 1111-1112, 1114-1115, 1120, 1131-1133, 1135-1136, 1143-1145, 1163, 1168, 1174-1175, 1177, 1179, 1182-1189, 1191, 1197-1211, 1214-1215, 1222-1232, 1242-1243, 1253-1264, 1266-1271, 1280, 1282, 1286-1287, 1291-1296, 1300, 1302, 1332, 1404, 1413, 1726, 1843 defined, 22, 29, 99, 104, 128, 160, 164, 202, 1105, 1143, 1168, 1266-1267, 1296, 1843 parabola, 128, 164, 166-169, 187, 202, 1182, 1201, 1258

Z z-axis, 682-683, 688 Zero, 13, 18, 20, 26-27, 56, 58, 71, 86, 147, 152, 162, 174, 176, 183-184, 186, 201-202, 205-207, 211, 213, 215-216, 219-221, 223-238, 242-243, 248-251, 259-262, 266-267, 269, 271, 275-276, 280, 282, 315, 319, 324-325, 333, 335, 345-346, 350, 356, 360, 366, 517, 537, 611-612, 661-662, 669-670, 676, 692-694, 746, 766, 768, 830, 836, 853, 855, 863, 976, 982, 995, 998, 1014-1015, 1018, 1040, 1067, 1094-1095, 1113, 1119, 1141, 1161, 1194, 1204, 1222, 1232-1242, 1245-1247, 1249, 1251-1254, 1256-1259, 1261, 1263-1271, 1280, 1282, 1288-1289, 1291-1292, 1294-1296, 1299-1303, 1328, 1333, 1339-1340, 1353, 1544, 1554, 1591, 1593, 1612, 1732, 1739, 1762, 1808 exponent, 206, 219-220, 995 matrix, 830, 836 rational zero theorem, 1246

1911