Essential Mathematics for the Australian Curriculum Year 9
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English Pages 794 Year 2015
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Table of contents :
Prelims......Page 1
Chapter 1......Page 18
Chapter 2......Page 92
Chapter 3......Page 158
Chapter 4......Page 228
Chapter 5......Page 304
Semester review 1......Page 364
Chapter 6......Page 372
Chapter 7......Page 432
Chapter 8......Page 502
Chapter 9......Page 560
Chapter 10......Page 640
Semester review 2......Page 702
Answers......Page 710
Acknowledgements......Page 792
Citation preview
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ESSENTIAL MATHEMATICS FOR THE AUSTRALIAN CURRICULUM SECOND EDITION
DAVID GREENWOOD | SARA WOOLLEY JENNY GOODMAN | JENNIFER VAUGHAN STUART PALMER Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
Introduction This second edition of Essential Mathematics for the Australian Curriculum has been developed into a complete resources pack comprising a revised and updated print textbook, a new interactive textbook with a host of cutting-edge features, and an online teaching suite. The successful elements of the first edition have been retained and improved upon. These include: • logical sequencing of chapters and development of topics • careful structuring of exercises according to the four Australian Curriculum proficiency strands plus enrichment • graduated difficulty of exercise questions within the overall exercise and within proficiency groups • Let’s Start and Key Ideas to help introduce concepts and key skills. Additions and revisions to the text include: • New topics reflecting updates to the Australian Curriculum and state syllabuses • Revision and extension topics are marked as ‘Consolidating’ or ‘Extending’ to help customise the course to each classroom’s needs • The working programs have been subtly embedded in each exercise to differentiate three student pathways: Foundation, Standard and Advanced • A ‘Progress quiz’ has been added about two-thirds of the way into each chapter, allowing students to check and consolidate their learning – in time to address misunderstandings or weaknesses prior to completing the chapter • Pre-tests have been revised and moved to the interactive textbook. Features of the all-new interactive textbook: • Seamlessly blended with Cambridge HOTmaths, allowing enhanced learning opportunities in blended classrooms, revision of previous years’ work, and access to Scorcher • Every worked example in the book is linked to a high-quality video demonstration, supporting both inclass learning and the ‘flipped classroom’ • A searchable dictionary of mathematical terms and pop-up definitions in the text • Hundreds of interactive widgets, walkthroughs and games • Automatically-marked quizzes and assessment tests, with saved scores • Printable worksheets (HOTsheets) suitable for homework or class group work. Features of the Online Teaching Suite, also powered by Cambridge HOTmaths: • A test generator, with ready-made tests • Printable worked solutions for all questions • A powerful learning management system with task-setting, progress-tracking and reporting functions. The chart on the next pages shows how the components of this resource are integrated.
x Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
Guide to the working programs in the exercises
FLUENCY
UNDERSTANDING
16
PROBLEM-SOLVING
Gradients within exercises and proficiency strands
—
ENRICHMEMENT
—
REASONING
The working programs that were previously available in separate supporting documents have been updated, refined and embedded in the exercises for this second edition of Essential Mathematics for the Australian Curriculum. The suggested worki ng programs provide three pathways through the course to allow differentiation for Foundation, Standard and Advanced students. As with the first edition, each exercise is structured in subsections that match the four Australian Curriculum proficiency strands (Understanding, Fluency, Problem-solving and Reasoning) as well as Enrichment (Challenge). The questions suggested for each pathway are listed in three columns at the top Foundation Standard Advanced of each subsection: • The left column (lightest shaded colour) is 1–2 2 — the Foundation pathway 3–5(½) 3–6(½) 3–6(½) • The middle column (medium shaded 7–8 8–10 8–10 colour) is the Standard pathway • The right column (darkest shaded colour) 11 11–13 13–15 is the Advanced pathway.
The working programs make use of the gradients that have been seamlessly integrated into the exercises. A gradient runs through the overall structure of each exercise – where there is an increasing level of mathematical sophistication required from Understanding through to Reasoning and Enrichment – but also within each proficiency strand; the first few questions in Fluency, for example, are easier than the last few, and the last Problem-solving question is more challenging than the first Problem-solving question.
The right mix of questions Questions in the working programs are selected to give the most appropriate mix of types of questions for each learning pathway. Students going through the Foundation pathway will likely need more practice at Understanding and Fluency, but should also attempt the easier Problem-solving and Reasoning questions. An Advanced student will likely be able to skip the Understanding questions, proceed through the Fluency questions (often half of each question), focus on the Problem-solving and Reasoning questions, and attempt the Enrichment question. A Standard student would do a mix of everything.
Choosing a pathway There are a variety of ways of determining the appropriate pathway for students through the course. Schools and individual teachers should follow the method that works for them. If required, the chapter pre-tests (now found online) can also be used as a diagnostic tool. The following are recommended guidelines: • A student who gets 40% or lower in the pre-test should complete the Foundation questions • A student who gets between 40% and 85% in the pre-test should complete the Standard questions • A student who gets 85% or higher in the pre-test should complete the Advanced questions. For schools that have classes grouped according to ability, teachers may wish to set one of the Foundation, Standard or Advanced pathways as their default setting for their entire class and then make individual alterations depending on student need. For schools that have mixed-ability classes, teachers may wish to set a number of pathways within the one class, depending on previous performance and other factors. * The nomenclature used to list questions is as follows: • 3, 4: complete all parts of questions 3 and 4 • 1-4: complete all parts of questions 1, 2, 3 and 4 • 10(½): complete half of the parts from question 10 (a, c, e, ... or b, d, f, ...)
Essential Mathematics for the Australian Curriculum Year 9 2ed
• 2-4(½): complete half of the parts of questions 2, 3 and 4 • 4(½), 5: complete half of the parts of question 4 and all parts of question 5 • — : do not complete any of the questions in this section.
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
An overview of the Essential Mathematics for the Australian Curriculum complete learning suite All non-core topics are marked Consolidating or Extending to assist with course planning
Examples with fully worked solutions and explanations show the thinking behind each step
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Let’s start activities get students thinking critically and creatively about the topic
‘Working with unfamiliar problems’ poster
Working programs subtly embedded in each Proficiency Strand to provide three learning pathways through the book
PDF TEXTBOOK
Downloadable Included with print textbook and interactive textbook
Note-taking Essential Mathematics for the Australian Curriculum Year 9 2ed
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Each topic in the print book comes with interactive HOTmaths widgets, walkthroughs and HOTsheets in the interactive textbook
For more detail, see the guide in the online Interactive Textbook
Search functions
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
Easy navigation within sections without scrolling
Walkthroughs, Scorcher and levelled question sets in every section
Interactive navigation and searching
Tasks sent by teacher
INTERACTIVE TEXTBOOK POWERED BY HOTmaths
Access to interactive resources at any time without leaving the page
Hundreds of interactive widgets
Online tests sent by teacher
Student reporting
Class reporting
Video demonstration for every worked example Access to all HOTmaths Australian Curriculum courses
ONLINE TEACHING SUITE POWERED BY HOTmaths
Teacher’s copy of interactive textbook
Test generator and readymade tests
Student results
Printable chapter tests and worksheets
Essential Mathematics for the Australian Curriculum Year 9 2ed
Plus: • worked solutions • teaching program • curriculum grids and more
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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For questions 1 and 2, start by working with smaller numbers and look for a pattern.
For questions 3 and 4, try making a list or table.
For questions 5 and 6, try drawing diagrams to help you visualise the problem.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
For question 7, use rounding to estimate the answer.
For question 8, try estimating by taking a sample.
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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For questions 9-13, try using algebra as a tool to work out the unknowns.
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For questions 14 and 15, try using concrete, everyday materials to help you understand the problem.
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For questions 16 and 17, try using a mathematical procedure to find a shortcut to the answer.
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
1
Reviewing number and financial mathematics
Chapter
What you will learn
Australian curriculum
1A Integer operations (Consolidating) 1B Decimals and significant figures 1C Rational numbers (Consolidating) 1D Operations with fractions (Consolidating) 1E Ratios, rates and best buys (Consolidating) 1F Percentages and money (Consolidating) 1G Percentage increase and decrease (Consolidating) 1H Profits and discounts 1I Income and taxation 1J Simple interest 1K Compound interest (Extending)
NUMBER AND ALGEBRA
Essential Mathematics for the Australian Curriculum Year 9 2ed
16x16 Money and financial mathematics Solve problems involving simple interest
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
32x32
Online resources • Chapter pre-test • Videos of all worked examples • Interactive widgets • Interactive walkthroughs • Downloadable HOTsheets • Access to HOTmaths Australian Curriculum courses
Global financial crisis The global financial crisis of 2008 and 2009 was one of the most serious financial situations since the Great Depression in the 1930s. Prior to the crisis, US interest rates were lowered to about 1%, which created access to easy credit and ‘subprime’ lending. House prices in the US rose about 125% in the 10 years prior to the crisis. When the housing bubble burst, house prices began to fall and lenders began foreclosing on mortgages if borrowers could not keep up with their repayments. At the beginning of the crisis, US household debt as a percentage of Essential Mathematics for the Australian Curriculum Year 9 2ed
personal income was about 130%. As house prices collapsed, financial institutions struggled to survive due to the increased number of bad debts. The crisis expanded to cause negative growth in the US general economy and in other countries. In Australia, our sharemarket All Ordinaries Index collapsed by 55% from 6874 in November 2007 to 3112 in March 2009.
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Up for a challenge? If you get stuck on a question, check out the 'Working with unfamiliar problems' poster at the end of the book to help you.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
Review of numbers 2 + (-3) = -1 For 8 and 12 - 4 - (-2) = -2 HCF is 4 15 ¸ (-5) = -3 LCM is 24 -7 ´ (-3) = 21 √36 = 6 3 (-2)2 = 4 √27 =3 43 = 64
Rates, ratios and best buy Rate: 180 km per 3 hrs = 60 km/h Ratio: 3 : 1 = 6 : 7 = 6 : 7 7 2 14 14 $200 divided into 7 : 3 10 parts is $200 1 part is $20 3 parts is $60 7 parts is $140 Best buy: 3 kg of carrots for $6.45 5 kg for $10.20 3 kg bag: 6.45 ¸ 3 = $2.15/kg 5 kg bag: 10.20 ¸ 5 = $2.04/kg ∴5 kg is best buy
Operations with fractions 1 5 - 7 = 11 - 7 6 12 6 12 = 22 - 7 12 12 15 = 12 = 11 4 1 11 ¸ 7 = 3 ´ 2 2 2 21 7 = 3 7
Reviewing number and financial mathematics
Rounding 1.284 to 1 d.p. is 1.3 2 sig. fig. is 1.3 472.543 to 1 d.p. is 472.5 1 sig. fig. is 500 0.0624 to 2 d.p. is 0.06 2 sig. fig. is 0.062
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Rational numbers (fractions) 11 = 1.375 13 = 8 8
mixed improper terminating number fraction decimal
1 = 0.166... = 0.16∙ 6
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2 and 4 are equivalent fractions 3 6 Irrational numbers cannot be expressed as a fraction
Compound interest (Ext) Percentages 3 = 3 ´ 100% = 75% 4 4 1 25 % = 25.5 ¸ 100% = 0.255
$3000 at 5% for 3 years Amount = $3000 ´ (1.05)3 = $3472.88
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20% of 60 = 0.2 ´ 60 = 12 5% of amount = 32 ∴amount = 32 ¸ 0.05 = 640
Simple interest $2000 at 5% for 4 years I = P ´ r ´ t = $2000 ´ 5 ´ 4 100 100 = $400
Applications of percentages Percentage profit or loss = Profit or loss ´ 100% Cost price Mark-up and discount Commission/tax
Essential Mathematics for the Australian Curriculum Year 9 2ed
Percentage increase and decrease Increase 20 by 6% 20 ´ 1.06 = 21.2
Decrease 20 by 5% 20 ´ 0.95 = 19 Percentage change = change ´ 100% original
Income and tax Employees can be paid: wage: hourly rate with overtime at time and a half = 1.5 or double time salary: annual amount commission: % of sales net income = gross income - tax
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
Chapter
2
Linear and simultaneous equations
What you will learn
Australian curriculum
2A Algebraic expressions (Consolidating) 2B Simplifying algebraic expressions (Consolidating) 2C Expanding algebraic expressions 2D Solving linear equations 2E Equations with brackets and pronumerals on both sides 2F Solving word problems 2G Inequalities 2H Using formulas 2I Simultaneous equations: substitution (Extending) 2J Simultaneous equations: elimination (Extending) 2K Applications of simultaneous equations (Extending)
NUMBER AND ALGEBRA
Essential Mathematics for the Australian Curriculum Year 9 2ed
Patterns and algebra Apply the distributive law to the expansion of algebraic expressions, 16x16 including binomials, and collect32x32 like terms where appropriate
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
Online resources • Chapter pre-test • Videos of all worked examples • Interactive widgets • Interactive walkthroughs • Downloadable HOTsheets • Access to HOTmaths Australian Curriculum courses
On a collision course Linear equations with two variables, such as 2x + 3y = 12, have an infinite number of solutions. If we plot them on a number plane we get a line. Each point on the line represents a possible solution. These equations can model many different situations or systems in real life. For instance the two variables might be total cost and quantity of an item, or the distance of an object from its starting point over time, when it moves at uniform speed in a straight line. Although there is no single solution to such a distance-time equation for one object, there is only one solution that Essential Mathematics for the Australian Curriculum Year 9 2ed
satisfies both equations for two objects moving at constant speed in the same plane (provided they are not moving on parallel tracks). In this situation the two equations are called simultaneous equations. The one solution that satisfies the simultaneous equations can be found using algebra, and this solution represents the position and time at which they meet or collide. Solving simultaneous equations can be applied to problems such as working out when and where one ship will intercept another, but navigation is only one of many areas where this aspect of algebra has applications.
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Up for a challenge? If you get stuck on a question, check out the 'Working with unfamiliar problems' poster at the end of the book to help you.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
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Monic quadratic trinomials (Ext) These are of the form: x 2 + bx + c Require two numbers that multiply to c and add to b e.g. x 2 − 7x −18 = (x − 9)(x + 2) since −9 × 2 = −18 and −9 + 2 = −7
Expansion
DOPS a2
The process of removing brackets.
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e.g. 2(x + 5) = 2x + 10
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2 (x + 3)(2x − 4) = x(2x − 4) + 3(2x − 4) = 2x 2 − 4x + 6x − 12 = 2x 2 + 2x − 12 Special cases
Factorisation The reverse process of expansion. Always remove the highest common factor first. e.g. 2ab + 8b HCF = 2b = 2b(a + 4) e.g. 3(x + 2) + y (x + 2) HCF = x + 2 = (x + 2)(3 + y)
Algebraic techniques
DOPS (difference of perfect squares) (a − b)(a + b) = a 2 − b 2 e.g. (x − 5)(x + 5) = x 2 + 5x − 5x − 25 = x 2 − 25 Perfect squares (a + b) 2 = a 2 + 2ab + b 2 (a − b) 2 = a 2 − 2ab + b 2 e.g. (2x + 3) 2 = (2x + 3)(2x + 3) = 4x2 + 6x + 6x + 9 = 4x2 + 12x + 9
Algebraic fractions Add/subtract Must find lowest common denominator (LCD) before applying operation. e.g. 3x + x −1 3 4 By grouping
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Trinomials of the form ax 2 + bx + c (Ext) Can be factorised using grouping also e.g. 3x 2 + 7x − 6, 3 × (−6) = −18 = 3x 2 + 9x − 2x − 6, Use 9 and –2 = 3x (x + 3) − 2(x + 3) since = (x + 3)(3x − 2) 9 × (−2) = −18 and 9 + (−2) = 7
Essential Mathematics for the Australian Curriculum Year 9 2ed
These involve algebraic expressions in the numerator and/or denominator.
Multiply/divide To divide, multiply by the reciprocal. Factorise all expressions, cancel and then multiply. 3 15 ÷ x 2 − 9 2x + 6 2(x + 3) 1 3 1 = × 15 5 (x − 3)(x + 3)1 2 = 5(x − 3) e.g.
Solving equations with algebraic fractions (Ext) Find lowest common denominator (LCD) and multiply every term by the LCD. e.g. LCD of 2x and 3x is 6x LCD of (x − 1) and (x + 3) is (x − 1)(x + 3) LCD of 2x and x 2 is 2x 2
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
Chapter
9
Probability and statistics
What you will learn
Australian curriculum
9A Probability review (Consolidating) 9B Venn diagrams and two-way tables 9C Using set notation 9D Using arrays for two-step experiments 9E Tree diagrams 9F Experimental probability 9G Summarising data: measures of centre 9H Stem-and-leaf plots 9I Grouped data 9J Measures of spread 9K Box plots (Extending)
S TAT I S T I C S A N D P R O B A B I L I T Y
Essential Mathematics for the Australian Curriculum Year 9 2ed
Chance List all outcomes for two-step chance experiments, both with and without replacement using tree diagrams or arrays Assign probabilities to outcomes and determine probabilities for events Calculate relative frequencies from given or collected data to estimate probabilities of events involving ‘and’ or ‘or’ Investigate reports of surveys in digital media and elsewhere for information on how data were obtained to estimate population means and medians Data representation and interpretation Identify everyday questions and issues involving at least one numerical and at least one categorical variable, and collect data directly from secondary sources Construct back-to-back stem-and-leaf plots and histograms and describe data, using terms including ‘skewed’, ‘symmetric’ and ‘bimodal’ Compare data displays using mean, median and range to describe and interpret numerical data sets in terms of location (centre) and spread Investigate techniques for collecting data, including census, sampling and observation Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
Online resources • Chapter pre-test • Videos of all worked examples • Interactive widgets • Interactive walkthroughs • Downloadable HOTsheets • Access to HOTmaths Australian Curriculum courses
Are lotteries worth it? The probability of winning first division in a lottery such as Tattslotto (also known as Saturday Lotto and Gold Lotto), based on the choice of six numbers chosen from 45, is 1 in 8 145 060. This means that you would expect to win first division once in every eight million, one hundred and forty-five thousand and sixty attempts. This is calculated by counting the number of possible ways of winning as well as the total number of ways that six numbers can be drawn from 45.
Essential Mathematics for the Australian Curriculum Year 9 2ed
If you played 1 game per week every week of the year, you would expect to win once every 1562 centuries. Many thousands of people take this chance every week. Looking at the statistics, adults in Australia gamble an average of about $1000 per year trying their luck in various national and state lotteries and other games of chance.
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
4UBUJTUJDT BOE 1SPCBCJMJUZ
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Frequency
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
0 10 20 30 40 50
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
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Two-step experiments These can be represented by tables or tree diagrams. They involve more than one component and can occur with or without replacement . e.g. a bag contains 2 blue counters and 1 green, 2 are selected at random. (I) Table with replacement. pick 1 b b g b (b, b) (b, b) (g, b) Sample pick 2 b (b, b) (b, b) (g, b) space g (b, g) (b, g) (g, g) of 9 outcomes Pr(2 of same colour) = 5 9
(II) Tree without replacement Pick 1 Pick 2 b bb b g bg b bb 6 b g bg outcomes b gb g b gb Pr(2 of same colour) = 2 = 6
Set notation
Venn diagrams and two-way tables
Within a sample space are a number of subsets. A’ = not A A A B A ∪ B means A or B; the union of A and B A ∩ B means A and B; A B the intersection of A and B A only is the elements in B A A but not in B n(A) is the number of elements in A Ø is the empty or null set e.g. A = {1, 2, 3, 4, 5} B = {2, 4, 6, 8} A ∪ B = {1, 2, 3, 4, 5, 6, 8} A ∩ B = {2, 4} n(B) = 4 B only = {6, 8} 3 ∈ A means 3 is an element of A
Probability and statistics
Stem-and-leaf plots These display all the data values using a stem and a leaf. An ordered back-to-back stem-and-leaf plot.
skewed
Leaf Stem Leaf 9872 1 03 7433 2 224 521 3 3 6 78 7 4 459 0 5 0 3⏐5 means 35 key
symmetrical
Experimental probability This is calculated from results of experiment or survey. Experimental = number of times event occurs total number of trials probability Expected number = probability × number of occurrences of trials
Summarising data: Measures of centre
Median is the middle value of data that is ordered. odd data set even data set 2 4 7 10 12 median
2 4 6 10 15 18 8 median
An outlier is a value that is not in the vicinity of the rest of the data.
Essential Mathematics for the Australian Curriculum Year 9 2ed
i.e. 6 in neither category, 7 in both categories Pr(A) =
12 21
Pr(B only)
4 7 3 = 21 = 17
=
Probability review The sample space is the list of all possible outcomes of an experiment. For all possible outcomes: Pr(event) = number of favourable outcomes total number of outcomes
1 3
Mode is the most common value (there can be more than one). Two modes means data is bimodal. Mean = average sum of all values = number of values
These organise data from two or more categories A A’ A B B 7 3 10 3 5 7 B’ 5 6 11 6 12 9 21 Venn diagram
Measures of spread Range = maximum value – minimum value Interquartile = upper quartile – lower quartile (Q3) range (IQR) (Q1) IQR is the range of the middle 50% of data. e.g. Finding Q1 and Q3. First, locate median (Q2) then find middle values of upper and lower half. e.g.
1 2 5 6 7 9 10 12 12 15
Q1 Q2 = 8 Q3 For odd number of values exclude median from each half.
e.g. roll a normal six-sided die Pr(>4) = 26 = 13 0 ≤ Pr(event) ≤ 1 Pr(not A) = 1 – Pr(A)
Grouped data Data values can be grouped into class intervals, e.g. 0–4, 5–9, etc. and recorded in a frequency table. The frequency or percentage frequency of each interval can be recorded in a histogram. e.g. Frequency
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Box plots (Ext) 25% Q Q Q3 min value 1 2 max value 25% 25% 25% Scale 25% of data is in each of the four sections.
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10
Chapter
Introduction to quadratic equations and graphs
What you will learn
Australian curriculum
10A Quadratic equations (Extending) 10B Solving ax 2 + bx = 0 and x 2 = d (Extending) 10C Solving x 2 + bx + c = 0 (Extending) 10D Applications of quadratic equations (Extending) 10E The parabola 10F Sketching y = ax 2 with dilations and reflections 10G Sketching y = x 2 with translations 10H Sketching parabolas using intercept form (Extending)
NUMBER AND ALGEBRA
Essential Mathematics for the Australian Curriculum Year 9 2ed
Linear and non-linear relationships Sketch simple non-linear relations with and without the use of digital technologies
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
Online resources • Chapter pre-test • Videos of all worked examples • Interactive widgets • Interactive walkthroughs • Downloadable HOTsheets • Access to HOTmaths Australian Curriculum courses
CSIRO Parkes Observatory The CSIRO Parkes Observatory in New South Wales is home to the Parkes Radio Telescope, which is famous for providing the first pictures to the world of the Apollo 11 moonwalk in 1969. The telescope’s gigantic dish has a diameter of 64 metres and receives radio and microwave signals from outer space. The shape of the dish is parabolic, meaning that its curvature can be described by a
Essential Mathematics for the Australian Curriculum Year 9 2ed
quadratic equation. The parabolic shape guarantees that all the radio and microwave signals bounce off the dish surface and are reflected to the receiver, which sits at a special point (called the focus) above the dish. The Parkes Radio Telescope is still in use today as part of the Australia Telescope National Facility.
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Sketching y = ax 2 + bx + c (Ext) Find x- and y-intercepts and turning point then sketch. e.g. y = x 2 − 6x + 5 y-int: x = 0, y = 02 − 6(0) + 5 =5 x-int: y = 0, 0 = x 2 − 6x + 5 0 = (x − 1)(x – 5) x − 1 = 0 or x − 5 = 0 x = 1 or x = 5 Turning point: halfway between the x-intercepts: y 1+5 5 x= 2 =3
Solving quadratic equations (Ext)
Applications (Ext)
substitute x = 3 into rule y = 32 − 6(3) + 5 = −4 (3, −4) minimum turning point
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Essential Mathematics for the Australian Curriculum Year 9 2ed
x
Define variable. Set up equation. Solve using NuII Factor Law. Check answer makes sense. Answer the question in words.
Rewrite equation in standard form, i.e. ax2 + bx + c = 0 Factorise Apply Null Factor Law: If a ×b = 0, a = 0 or b = 0, Solve remaining equation e.g. x2 + 5x = 0 x(x + 5) = 0 x = 0 or x + 5 = 0 x = 0 or x = −5 e.g. x 2 − 12 = 4x x2 − 4x − 12 = 0 (x − 6)(x + 2) = 0 x − 6 = 0 or x + 2 = 0 x = 6 or x = −2
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
4FNFTUFS SFWJFX
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
I − E
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Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
N
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
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C G − K
D − H
E − I
L −
M
B G
C
D
E
H
I
F
J −
B G
W −
K
C −
D
F
I −
E J
H
C G
D −
E −
K −
L −
M −
C
D
H
I −
E −
F
H −
B Y =
B Y =
JJJ
C Y =
JW
D &YQBOEJOH UIF CSBDLFUT JT GBTUFS CFDBVTF ÷ HJWFT B GSBDUJPO BOTXFS
C B=D−C E B = D(C + E)
F B = − DE C
G B= C D
I B = E(C − ) D
J B = DE − C
K B = C + DE
L B = CF + D E
H B = D( − E) C
B Y = C Y= D .FUIPE B EPOU IBWF UP EFBM XJUI OFHBUJWFT 'JOBM TUFQ JO NFUIPE B EJWJEF CPUI TJEFT CZ B QPTJUJWF OVNCFS 'JOBM TUFQ JO NFUIPE C EJWJEF CPUI TJEFT CZ B OFHBUJWF OVNCFS B Y = D Y=
E B−C
D B − C
F Y = −D
G Y=C
C Y=
B−C
PS B − C C − B H Y = E + CE + D B−C E Y=−
I Y = − BC + CD PS BC + CD B−C+ C−B−
M B = E − FG D
&YFSDJTF '
&YFSDJTF & B Y − C
G − Y
B Y + = D Y + = − B
F
J
JOUFHFS
D B = D − E C
F Y −
E −
C Y= D %JWJEJOH CPUI TJEFT CZ JT GBTUFS CFDBVTF ÷ JT BO
C XIFO UIF DPNNPO GBDUPS EJWJEFT FWFOMZ JOUP UIF 3)4
I
NBSCMFT F
WJ
B B = C + D
D
H
-FU X CF UIF XBHF (w − ) =
D OFFE UP ÷ − BT −Y = E TIPVME IBWF + CFGPSF × JJ −
F − J −
B TIPVME IBWF + CFGPSF ÷ C TIPVME IBWF × CFGPSF −
B J
B −
C −
C −
F −
G
J −
K
N
O
D Y −
E − Y
C + Y − = E Y − = D
E
H
I −
L
M
B Y − = Y − D (Y − ) =
C (Y + ) = E Y = Y +
F Y − = Y + B -FU F CF UIF OVNCFS PG HPBMT GPS &NNB C F+ D F + F + = E F = F &NNB TDPSFE HPBMT -FPOJF TDPSFE HPBMT B -FU X CF UIF XJEUI JO DFOUJNFUSFT D X + (X) = E X =
C MFOHUI = X
F MFOHUI = DN XJEUI = DN
P −
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
C
Y + Y + = LN LN Y + Y + =
2
1
2
3
4
5
−4 −3 −2 −1
0
x
−3 −2 −1
'JSTU MFH = LN 4FDPOE MFH = LN 5IJSE MFH = LN 'PVSUI MFH = LN
3FDUBOHMF - = N 8 = N 5SJBOHMF TJEF = N B
I
−1
E J Y Y + Y + B 5 = Y +
JJ C D 3 = Y
1
2
3
−2 −1
0
1
2
x x x
0
1
2
3
4
2
3
4
5
6
−3 −2 −1
0
1
K
x
2
3
−7 −6 −5 −4 −3 −2 B Y
E N − K B ≥
H Y ≥
I Z ≥ −
L Y
C O ≤ − D Y ≥ E B≥ F Y ≥ − G Y ≥ H Y < − I U ≤ − J N > −
B Y − D Z ≤
E B <
G C <
MFTT UIBO MFTT UIBO DN
&YFSDJTF (
B
B > E > −
C − < D − < −
1
C
D
E
Y = PS Y = PS Y = LN
B
0
0
M
1SPHSFTT RVJ[
B Y =
x
L
LN "OOB )FOSZ $IMPF UXJOT
Y=
x
2
J
E Y = F
B BC
1
−10 −9 −8 −7 −6 H
JJ JJ
0
G
IPVST QN
C J Y Y + Y + D J Y Y + Y +
x
F
* BN ZFBST PME &SJD JT ZST PME OPX
1
E
" # $ ýDUJPO OPOýDUJPO CPPLT
B
0
D
LN
x
−1
"OTXFST
Y + = EBZT
2
3
x
B J −. FUD
JJ OVNCFST NVTU CF MFTT UIBO
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
B B = 1
"OTXFST
C J −, − FUD
JJ OVNCFST NVTU CF HSFBUFS UIBO − D J Y −B
B Y < C Y F Y >
E Y ≤
G Y≥
B Y > C − D B D Y ≤ B(C + D)
E Y ≤ BD C
G Y ≥ C − DE
H Y< D−C B
I Y > C − DE B
J Y TP JT TUFFQFS
C −
C Z = Y −
E Z = −Y − G Z = −Y + D −
E −
F
C Z = −Y +
D Z = Y + B HSBEJFOU = ZJOUFSDFQU = −
C HSBEJFOU = − ZJOUFSDFQU = − D HSBEJFOU = − ZJOUFSDFQU = E HSBEJFOU = ZJOUFSDFQU = F HSBEJFOU = − ZJOUFSDFQU = G HSBEJFOU = ZJOUFSDFQU = H HSBEJFOU = ZJOUFSDFQU =
I HSBEJFOU = −. ZJOUFSDFQU = B Z = −Y + C Z = Y − D Z = −Y + F Y − Z =
t 0 1 2 3 4 J JJ E = U J LN JJ IPVST PS I NJO E = U C H = U D $ = .O 1 = U
H Y − Z =
E Z=Y+ G Y + Z = I Y − Z = −
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
B , −
G
y
y
(2, 4)
O
(1, −1)
−2 C , −
1
x
O
x 1
H −
y
y
(1, 1) x
O −1
x
O
(3, −4) D
y
I −
(2, 2)
y
( 3, 4 23 )
1 x
O
x
O − 13
E −
B −
y
y 4
2
(1, 3)
(2, 1) x
O
O
x 1
C , − F −
y
y
O
x
O
3
(1, 5)
−6
x 1 D −
y 3
O
(2, 2)
x
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
J , −
y
x
O
K −
y
O
y
x
3
(4, −1)
L
y
y
4
(5, 1)
O
H
x
3
y
(3, 73 )
4 3
x
O
M
y
(2, 1) x
O
x
O
J E
I −
x
O
−2 G −
x
4
−3
(2, −3)
−4 F , −
y
"OTXFST
E , −
y
JJ G
WJ B B
JJJ D
JW C
W F
y
2 O
3
x O
2
x
−5
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
C
H
y 6
y 2
x
O
D
O
B OP
y
C OP
3
x
D ZFT
E OP
D E G I B Z = Y + ZJOUFSDFQU JT
x
O
(1, −1)
E
y
C FYQBOE CSBDLFUT BOE TJNQMJGZ Z = L Z = − B Y + E , N = − B , D = E C C C C B Z = Y + C Z = −Y + D Z = Y + E Z = −Y −
&YFSDJTF (
5
(1, 4)
x
O
B Z = Y + D Z = −Y +
B G
C
B Z=Y+
F
y
G
2
x
y
O
E Z = Y +
F Z = Y
G Z = −Y −
B Z = Y +
C Z = −Y −
J Z = −Y + B J C J − D J − E J
H
y
Y =
8
E Z=Y− G Z = Y −
I Z = −Y + K Z = −Y −
JJ Z = Y + JJ Z = −Y +
JJ Z = −Y + JJ Z = Y −
,−
(1, 4) O
F
E Z = Y − G Z = −Y +
D Z = −Y +
H Z = −Y +
x
E
C Z = −Y +
D Z = −Y + F Z = −Y +
(4, 3)
D C Z=Y+
D Z = −Y − F Z = Y + B Z = Y +
O −1
C Z = Y − E Z = −Y −
x
7 = −U + 7 = - JOJUJBMMZ B Z = Y
C Z = .Y +
B
C
D −
E
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
C D=
B QPTJUJWF
C VOEFýOFE E OFHBUJWF − B N= C N = −
D OP ZJOUFSDFQU JT ýYFE GPS BOZ HJWFO MJOF Z = C Y B B Z = Y
C Z = −Y +
D Z = Y + F Z = Y +
E Z = −Y +
D [FSP
"OTXFST
B D =
B -IPVS C
6000
G Z = − Y −
B −
Volume (litres)
1SPHSFTT RVJ[ C
B OPU PO MJOF B
C PO MJOF
y
3000
3
O
0
x
2
D E B B
C
1
2 3 Time (hours) J JJ 7 = U J JJ IPVST Z = −Y + C Z − Y = − , −, y
4
y
(1, 1)
3
−2
x
−4
C −,
x
y
y 3 y=2
(3, 1) 2 O
y = 2x
x
(1, 2) 4
x
x=4 B Z = Y + D Z = Y −
C Z = −Y −
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
&YFSDJTF )
B
B Z = −Y + C
D
E −
B
C
D −.
E −.
B
C
D
B
F (−, )
C
D
G (−, −)
I
J
L (−, −.)
E
H
K (−., −.) M (., −.)
B
C
D
E
F J
G
H
I
D Z = −Y +
Z = Y − PS Z = Y + B J −
JJ −
B J −
JJ JJ −
B
C
Y + Y B Y = D .(, −.)
B Z = − Y +
D
C Z=
Z + Z
F OP
G ZFT
JJJ
JJJ (., −.)
#
D E B $ = O + B $ = O + C C
D
E
B ZFT
C OP
D OP
E ZFT
B Z = Y + D Z = −Y + F Z = Y −
C Z = Y + E Z = −Y − G Z = −Y +
B Z = −Y +
C Z = −Y +
D Z = Y −
E Z=Y+
F Z = Y −
G Z = −Y + JJ Z = − JJJ Z = JJ Y = − JJJ Y = JJ Y = − JJJ Y = JJ Z = − JJJ Z =
D
E /PUF 5IJT HSBQI TIPVME SFBMMZ CF B TFSJFT PG QPJOUT CVU B GVMM MJOF TFHNFOU JT VTFE BT B HVJEF
1200 800 400
E ZFT
C −
JJJ
W
D OP
B −
E J Z=
G Z = −Y +
O C ZFT
C J Y= D J Y=
F Z = Y −
C JJ (−, −.)
B ZFT
B J Z=
E Z = −Y −
B C B 8 = Y +
&YFSDJTF *
D Z = Y −
B J JJ C " = + O
B
E J (−, ) JJ (−, ) JW (−., .)
C Z = Y +
&YFSDJTF +
B J Y − Y JJ Z − Z & JJJ (Y − Y ) + (Z − Z ) C "OTXFST XJMM WBSZ
C D J
JW
JJJ
C −C B
#(, ) "(−, ) "(−, ) C (, −)
E Z=Y+
Z = Z = Y + Z = −Y +
C J
B (−, )
C Z = Y +
40
x
C
D /PUF 5IJT HSBQI TIPVME SFBMMZ CF B TFSJFT PG QPJOUT CVU B GVMM MJOF TFHNFOU JT VTFE BT B HVJEF
3000 2500 2000 1500 1000 500 O D
100
n
E B $ = U + JW Z = − JW Y = − JW Y = JW Z =
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
E B B D E B
D
C
1500 1000 500 300 t O 10 IPVST ' = U + C NJOVUFT D NJOVUFT 7 = − U C NJOVUFT PS IPVST NJOVUFT IPVST NJOVUFT PS NJOVUFT V
%7%T
C 12000 10000 8000 6000 4000 2000 0
1500
C D E B C B F B C B E H
t O 30 7 = − U UIF OVNCFS PG MJUSFT ESBJOFE QFS NJOVUF F NJOVUFT LNI SBUF DIBOHFT JF OFX HSBEJFOU = E = + U NT C − NT D N E TFD JODSFBTJOH BMUJUVEF BU SBUF PG NT J $ = .O + JJ J 1 = .O − JJ JJJ $ = Y + C D F G 1 = Y − 5 = Y − I Y
&YFSDJTF ,
C (−, )
D
B 5SVF B ZFT
B
C 'BMTF C ZFT
D 'BMTF D ZFT
E 5SVF
B ZFT F ZFT
C ZFT G ZFT
D OP H OP
E OP I OP
B
F
C
G
D (, −) H
E
I
K (−, )
L (−, −)
J
M
B
C/R 5000 4000 3000 2000 1000 O
C
R
(1200, 7200)
n 0 1000 2000 TFDPOET 1BSBMMFM MJOFT JF TBNF HSBEJFOU B
C
D OP JOUFSTFDUJPO E OP JOUFSTFDUJPO F
G
B C − B
C " = . VOJUT B VOJUT C VOJUT D VOJUT E VOJUT F VOJUT
1SPCMFNT BOE DIBMMFOHFT EBZT IPVST
EJBHPOBMT JOUFSTFDU BU UIFJS NJEQPJOU L = −. PS MFOHUI "# = MFOHUI "$ " = . TRVBSF VOJUT 1 = VOJUT
B #% : Z = −Y + "$ : Z = Y − C C = − Y = D VOJUT L = . PS −
.VMUJQMFDIPJDF RVFTUJPOT $ #
% #
" "
$ %
& %
4IPSUBOTXFS RVFTUJPOT
/PUF 5IJT HSBQI TIPVME SFBMMZ CF B TFSJFT PG QPJOUT CVU B GVMM MJOF TFHNFOU JT VTFE BT B HVJEF
C
/PUF 5IJT HSBQI TIPVME SFBMMZ CF B TFSJFT PG QPJOUT CVU B GVMM MJOF TFHNFOU JT VTFE BT B HVJEF
"OTXFST
C
B −,
C −
(750, 3750)
n 1000
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
B
F
"OTXFST
y
y
2 O
x
2
O
x
4
−4
C
G
y
y
9 1
22
x
O
−5 x
O
−3 D
H
y
y 3.5
5
x
O
O
1
x
22
−7 I
E
y
y
2
4
O
−4
4
O
x
x
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
B LN
B -IS C
10 000 9000 8000 7000 6000 5000 4000 3000 2000 1000
200 180
d = 175 − 70t
160 140 120 100 80 60
V = 2000t
O 1 2 3 4 D 7 = U E IPVST
40 20 t
0
V
B (SBEJFOU =
0.5 1 1.5 2 2.5 3
B Z= C Z = −
t
ZJOUFSDFQU =
y
D Y = −
8 7 6 5 4 3 2 1
y 5 4 3 2 1
c
a
O −5 −4 −3 −2 −1−1 −2 −3 −4 −5
1 2 3 4 5
x
G Z = −Y
b
C (SBEJFOU = −
O –3 –2 –1–1 –2 –3 –4 –5
(1, 5)
1 2 3 4 5
ZJOUFSDFQU =
8 7 6 5 4 3 2 1
d
e
−5 −4 −3 −2 −1−1O 1 2 3 4 5 6 7
C −
x
y
y 5 4 3 2 1
y = 2x + 3
O −5 −4 −3 −2 −1−1
E Y= F Z = Y
B
"OTXFST
C IPVST D d
y = −3x + 7
(1, 4)
1 2 3 4 5
x
x
f
D −
E
F
G −
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
D (SBEJFOU = − y
5 4 3 2 1
1 2 3 4 5 6 7
2y − 3x = 8
300 200
x
100 80 0 JW IPVST C J $ = + O JJ C
ZJOUFSDFQU = y
B Z = Y + D Z = Y −
1 2 3 4
350 300 200
C Z = −Y +
D
Z = Y −
E Z = −Y +
JJ JJ
B O = C O = B Z = Y + D Z = −Y − F Z = Y +
D O= C Z = −Y − E Z = Y +
G Y + Z = PS Z = − Y + B OP C ZFT C (−, )
(3, 200)
150 x
Z= Y−
D J . (, ) E J . (−, .)
C1 = 80 + 40n
250
B
JJ . JJ
C2 = 65 + 45n
400
C Z = −Y +
B J . (, ) C J . (., .)
n 8
450
7 (2, 7) 6 5 4 3 2 1
−5 −4 −3 −2 −1 −1 −2
JJJ
(8, 400)
400
(3, 1)
E (SBEJFOU =
B
B J IS JJ C
2x + 3y = 9
O −2 −1 −1 −2 −3 −4
&YUFOEFESFTQPOTF RVFTUJPOT
ZJOUFSDFQU =
100 50
D E B C D
0 1 2 3 4
"GUFS IPVST $ = W + J JJ QFS WBTF WBTFT
5
6
7
8
n
$IBQUFS &YFSDJTF "
B NN F N
C DN G LN
D DN
E N
DN DN B B = C =
C B = C =
D B = C = B N C DN
D NN
E LN
F N B DN
G DN C DN
D DN
E N
F DN B DN
G LN C N
D DN
B NN C N D DN F LN G LN
E NN
B Y= F Y=
E Y =
C Y = G Y =
D Y =
N
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
B DN
C N
D DN
B Y = DN
C Y = .
D Y=
C N D J
C 1 = Y D 1 = B + C F 1 = (B + C)
E N
G 1 = Y "MM WFSUJDBM TJEFT BEE UP DN BOE BMM IPSJ[POUBM TJEFT BEE UP C DN DN
D DN DN E DN DN DN DN B J DN JJ DN JJJ DN C 1 = ( + Y) ∴ 1 = Y + D J DN JJ DN E J Y = . JJ Y = . F /P BT XJUI OP GSBNF UIF QJDUVSF IBT B QFSJNFUFS PG DN
&YFSDJTF #
B DN
B è
C NN C è
D .è
F + è G + è H + è
E + è I + è
J + è
B
C
D
E
C DN D NN E LN
B DN F N
C N D DN E DN G LN
B N B è
C DN C è
D NN D è
F è B + è
G è C + è
D è +
G
E + è F è + N
E N E è
G è +
C G
D
E
B SFDUBOHMF C DJSDMF E TFDUPS PG DJSDMF
D SIPNCVTLJUF F USJBOHMF G USBQF[JVN
H QBSBMMFMPHSBN B C
I TRVBSF D
J TFNJDJSDMF E
F G B NN C DN D DN E N F N G LN B N F N
C DN D LN G N
E N
B NN C DN D DN F N G LN NN
C N DN
E F G B N C LN N
E N D DN
DN D DN
DN
B C D DN E N F N G N B . × DN D N
C NN
N LN N
%
" è C J DN
B S =
JJ N
JJJ LN
B J ◦ JJ ◦ C BT BOHMF XPVME CF HSFBUFS UIBO ◦ XIJDI JT OPU QPTTJCMF
N DN B DN C J DN
JJ DN
N JT UIF MBSHFTU BSFB QPTTJCMF JF GVMM DJSDMF
B J N JJ N C N D ZFT
D B .è N D + è F + è
C . + .è E + .è G + .è
B S = $ è C J DN
B F
B DN
B N F
&YFSDJTF $
DN B DN DN
JJ
"OTXFST
B 1 = B + C E 1 = Y + Z
B N
&YFSDJTF %
JJ N
D E=$ è
B TFNJDJSDMF BOE SFDUBOHMF C USJBOHMF BOE TFNJDJSDMF D SIPNCVT BOE QBSBMMFMPHSBN
E DN
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
B 1 = × + + × èS = + + .è
C
Rectangular prism
D
Triangular prism
" = CI + èS = × + × è × .
= + .è
= + .è
= . N
= . N
C 1 = + + + + = DN
" = MX − CI = ×− ×× = −
B N D N N
= DN C N N E N N
F N N B DN C DN
G NN NN D DN
N
E N B N
F DN G DN C DN
E D N
Square pyramid
E N F DN G N B N N C NN NN D DN DN F DN DN B DN F N
E N N G N N
C N D N G DN
I DN N
E DN H NN
J NN
DN B DN
C N
D N
C
D − è
F
Tetrahedron (triangular pyramid)
G
Pentagonal pyramid
DN B + è E è
F . + .è
G + è
DN B IZQPUFOVTF EJBNFUFS XPVME FRVBM OPU C IZQPUFOVTF TMPQFE FEHF TIPVME CF DN OPU DN D IZQPUFOVTF EJBNFUFS TIPVME CF OPU N DN B DN DN D DN
&YFSDJTF &
B
Cube
C DN
B 54" = × × + × × + × × = + + = N C 54" = × × × + × + × + × = + + + = DN B DN C DN D N
E DN
F N G N
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
C DN D N
G LN
H N
B DN D N
C DN E N
I N DN
F DN N
G DN
DN B DN
N DN
C J N D
B N C B DN C DN B 10 cm2
D
J NN
JJ N
B è N C è DN D è N )BMG DZMJOEFS JT NPSF UIBO IBMG TVSGBDF BSFB BT JU JODMVEFT OFX C 16 cm2
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
DN
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
&YUFOEFESFTQPOTF RVFTUJPO B J JJ JJJ ZFBST C D
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
K Y O X
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
B F J N Q B
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E − Y
C = IBT B QPTJUJWF QPXFS B− D − − = × × C = C C B ÷ = × C J
JJ
JJJ Y
JW C B
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
D
E J
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G −. × H . × I × J . ×
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C . ×
H . × −
J − × − L . × Essential Mathematics for the Australian Curriculum Year 9 2ed
C . ×
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K −. × M −. × Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
B
C
D
E
F
G L
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J
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C . × E . ×
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F
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B CPUI = D CPUI =
C CPUI = E CPUI =
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
4IPSUBOTXFS RVFTUJPOT B C Y Z D B C " # " # E × F × G × × B Y C B C D D N O C F Y Z G B B N C B
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
B JTPTDFMFT USJBOHMF D FRVJMBUFSBM USJBOHMF F BDVUF BOHMFE USJBOHMF H SJHIUBOHMFE USJBOHMF B J ∠#"$
C PCUVTF BOHMF USJBOHMF
B ◦ C ◦ D ∠8:; + B◦ + C◦ = ◦ BOHMF TVN PG B USJBOHMF
JW ◦ JJ BDVUF
JJJ "OTXFST XJMM WBSZ D J ∠9:;
JW ◦ JJ SFþFY
JJJ "OTXFST XJMM WBSZ E J ∠435
JW ◦ JJ TUSBJHIU BOHMF
JJJ "OTXFST XJMM WBSZ F J ∠30#
JW ◦ JJ PCUVTF
JJJ "OTXFST XJMM WBSZ G J ∠"0#
JW ◦ JJ SJHIU
F ◦ B J ◦
G ◦ JJ ◦
C J ◦ D J ◦
JJ ◦ JJ ◦
E J ◦ F J ◦
JJ ◦ JJ ◦
G J ◦ H J ◦
JJ ◦ JJ ◦
I J ◦
JJ ◦
J J ◦ K J ◦
JJ ◦ JJ ◦
B $ G /
C 4 H 4
B B = C B = F B = G B =
JW ◦ D ◦
∠9:; + ∠8:; = ◦ TUSBJHIU MJOF
= ◦ − (B + C + D)◦ = ◦ − ◦
E ◦
= ◦ B Y = Y = .
C Y = Y =
D Y = Y = E (Y + ) + Y = Y = F Y + = Y = G Y + = Y =
&YFSDJTF #
D / I /
E $
D B =
E B =
I FRVJMBUFSBM ◦
B T = C U = E B = Y =
D S = F B = C =
G D = E = B ◦ C ◦
D ◦
E ◦
F ◦ B Y =
G ◦ H ◦ I ◦ C Y = D Y =
B ◦ F ◦
C ◦ G ◦
D ◦ H ◦
E ◦ I ◦
J ◦ ∠"0# + ∠"#0 = ◦ (FYUFSJPS BOHMF PG USJBOHMF) (SFþFY) Y =
5IF FYUFSJPS BOHMFT CFDPNF ◦ − B◦ ◦ − C◦ ◦ − D◦ TUSBJHIU MJOF
= ◦ − B◦ − C◦ − D◦
E FRVJMBUFSBM ◦ F PCUVTF JTPTDFMFT ◦ G SJHIUBOHMFE JTPTDFMFT ◦
∠"0# = ◦
∴ ∠9:; = B◦ + C◦ -FU UIF JOUFSJPS BOHMFT PG BOZ USJBOHMF CF B◦ C◦ BOE D◦ /PX B + C + D =
&YUFSJPS TVN = ( − B)◦ + ( − C)◦ + ( − D)◦
B PCUVTF JTPTDFMFT ◦ C BDVUF TDBMFOF ◦ D SJHIUBOHMFE TDBMFOF ◦
H PCUVTF TDBMFOF ◦ J BDVUF TDBMFOF ◦
∆"0# JT JTPDFMFT TJEFT FRVBM ∠"0# = ◦ ∴ ∠0"# = ◦ CBTF BOHMFT PG JTPTDFMFT USJBOHMF
JJ PCUVTF
JJJ "OTXFST XJMM WBSZ C J ∠132
JJJ "OTXFST XJMM WBSZ B ◦ C ◦
"0 = #0 SBEJJ
E JTPTDFMFT USJBOHMF G TDBMFOF USJBOHMF
F 4
B FRVBM C FRVBM D TVQQMFNFOUBSZ ** B ◦ BMUFSOBUF BOHMFT JO ** MJOFT ** C ◦ DPJOUFSJPS BOHMFT JO ** MJOFT ** D ◦ DPSSFTQPOEJOH BOHMFT JO ** MJOFT * * E ◦ BMUFSOBUF BOHMFT JO ** MJOFT F ◦ WFSUJDBMMZ PQQPTJUF ** G ◦ DPSSFTQPOEJOH BOHMFT PO ** MJOFT H ◦ TVQQMFNFOUBSZ BOHMFT
** I ◦ DPSSFTQPOEJOH BOHMFT PO ** MJOFT * * J ◦ DPJOUFSJPS BOHMFT PO ** MJOFT
B OP BMUFSOBUF BOHMFT BSF OPU FRVBM C ZFT DPSSFTQPOEJOH BOHMFT BSF FRVBM
D ZFT BMUFSOBUF BOHMFT BSF FRVBM E OP DPJOUFSJPS BOHMFT EPOU BEE UP ◦ F ZFT DPJOUFSJPS BOHMFT BEE UP ◦ G ZFT DPSSFTQPOEJOH BOHMFT BSF FRVBM H OP DPSSFTQPOEJOH BOHMFT BSF OPU FRVBM I OP BMUFSOBUF BOHMFT BSF OPU FRVBM J OP DPJOUFSJPS BOHMFT EP OPU BEE UP ◦ B B = C = C D = E = D F = G = H = E B = C = F B = C = D = G F = G = E = B Y = Z = C U = [ = D V = W = X = E Y = F Y = G Y = B ◦ C ◦ F
◦
G
D ◦
E ◦
◦
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
C
D
B ◦ − B◦ D ◦ − (B◦ + C◦ )
B QBSBMMFMPHSBN SFDUBOHMF LJUF
C ◦ − B◦ E ◦ − (B◦ + C◦ )
C SFDUBOHMF TRVBSF E TRVBSF SIPNCVT LJUF
F B◦ + D◦ G ◦ − D◦ ∠"#$ = ◦
B B
∠#$% = ∠"#$ + ∠#$% = ◦ ** ∴ "# ** %$ BT DPJOUFSJPS BOHMFT BSF TVQQMFNFOUBSZ
BOHMFT PG UIF USJBOHMF ** B ∠#"& = ◦ − B◦ BMUFSOBUF BOHMFT BOE "# ** %&
◦
= ◦ − D◦ − ◦ + B◦ = −D◦ + B◦
= B◦ − D◦ C ∠"#% = ◦ − (B◦ + C◦ ) (BOHMF TVN PG USJBOHMF ∆"#%)
D & = E ◦ O B J POF C O −
JJ UXP
( − F) = TVN PG FYUFSJPS BOHMFT JT ◦
++++−(B + C + D + E + F) = − (B + C + D + E + F) =
D DPOTUSVDU 9: UISPVHI # QBSBMMFM UP "& ** ∴ ∠"#: = B◦ (BMUFSOBUF BOHMFT, "& ** 9:) ** ∴ ∠$#: = C◦ (BMUFSOBUF BOHMFT, %$ ** 9:)
B + C + D + E + F = B B + C + D + E + F + G = C B + C + D + E + F + G + H =
∴ ∠"#$ = B◦ + C◦
&YFSDJTF %
E $POTUSVDU 9: UISPVHI " QBSBMMFM UP &%
** ∠9"% = ◦ − C◦ (DPJOUFSJPS BOHMFT, &% ** 9:) ∠%"# = ◦ − B◦ (SFWPMVUJPO)
∴ ∠9"# = ◦ − B◦ − (◦ − C◦ ) = ◦ + C◦ − B◦
** ∠"#$ = ∠9"# (BMUFSOBUF BOHMFT BOE 9: ** #$)
∴ ∠"#$ = ◦ + C◦ − B◦
B TJ[F B J 9:
D
E
C ◦
D ◦
B QBSBMMFM C SJHIU B DPOWFY RVBESJMBUFSBM
D USBQF[JVN E FRVBM C OPODPOWFY IFYBHPO
D OPODPOWFY IFQUBHPO B C
D
E
D
E
B F
C G
B C F C = B = H C = B = J
D E G C = B = I C = B =
F
D 4"4 3)4 ""4
JJJ ∠$ C ∆%&' ≡ ∆456
F 444 G 3)4 B Y = , Z =
H ""4 I 444 C Y = , Z =
D ∆".1 ≡ ∆$#% B 4"4 C ""4
D B = , C = F Y = ., C =
C
G
C ∆"#$ ≡ ∆456 JJ 9; JJJ :;
C J ∠" JJ ∠# B ∆"#$ ≡ ∆'()
&YFSDJTF $
F
JJJ ýWF
( − B) + ( − C) + ( − D) + ( − E) +
∴ ∠"#$ = B◦ + C◦
B
C * = (O − ) O
B 4 = (O − )
∠"#$ + ∠"#% = ◦ (TUSBJHIU MJOF)
G B ◦
E
QBSBMMFMPHSBNT
◦
∠"#$ = − D −( − B) (BOHMF TVN PG B USJBOHMF)
D D
FRVBM 3FDUBOHMFT TRVBSFT BOE SIPNCJ IBWF UIFTF QSPQFSUJFT BOE NPSF BOE BSF UIFSFGPSF BMM
◦
C BMUFSOBUF BOHMFT BSF FRVBM PO QBSBMMFM MJOFT D ∆"#$ => B + C + D = BOE UIFTF BSF UIF UISFF
◦
C C
F G " QBSBMMFMPHSBN IBT PQQPTJUF TJEFT QBSBMMFM BOE
◦
B DPJOUFSJPS BOHMFT PO QBSBMMFM MJOFT BEE UP
D TRVBSF SFDUBOHMF
"OTXFST
B
E ∆#.8 ≡ ∆4-, D 3)4 E 4"4
E B = , C = G B = , Y = ., C =
H Z = ., B = I B = ., C = B ∆"#$ ≡ ∆456 3)4
C ∆%&' ≡ ∆()* 444
D ∆"#$ ≡ ∆%&' 4"4
E ∆"#$ ≡ ∆()* ""4
F ∆"$# ≡ ∆%&' 4"4
∆%&' ≡ ∆#31
∆-./ ≡ ∆,*+ ∆#$% ≡ ∆'()
∆./0 ≡ ∆321 B #$ = C #$ = OP − UIFZ DBO BMM CF EJGGFSFOU TJ[FT POF NJHIU IBWF BMM TJEFT DN BOE BOPUIFS BMM TJEFT DN B POF HJWFO UIF PUIFS QBJS BSF WFSUJDBMMZ PQQPTJUF C ""4 B 444
C FRVBM
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
B POF HJWFO #" = #$ BOE TJEF #% JT DPNNPO
E ∠"%# = ∠$%# DPSSFTQPOEJOH BOHMFT JO DPOHSVFOU USJBOHMFT
CVU ∠"%# + ∠$%# = ◦ TUSBJHIU BOHMF
∴ ∠"%# = ∠$%# = ◦ BOE "$ JT QFSQFSEJDVMBS UP %# BoF "OTXFST XJMM WBSZ
B #% C "$ D "$ 0" = 0# SBEJJ PG DJSDMF DFOUSF 0
B
** ∠"#% = ∠$%# (BMUFSOBUF BOHMFT, "# ** $%) ** ∠"%# = ∠$#% (BMUFSOBUF BOHMFT, "% ** $#)
∴ ∆"#% ≡ ∆$%# ""4
L 0" = 0$ SBEJJ
0# JT DPNNPO "# = $# HJWFO
∴ ∆"0# ≡ ∆$0# 444
M 0" = 0% BOE 0# = 0$ SBEJJ
&YFSDJTF &
B ∠&$% B 444
#$ = %" HJWFO
∴ ∆"#$ ≡ ∆$%" 444
K #% JT DPNNPO
C 4"4 D ∆"#% ≡ ∆$#%
C ∠$#" C ∠#.$
D ∠%&$
"% = $% (HJWFO) ◦
∠%"# = ∠%$# = (HJWFO) %# JT DPNNPO ∴ ∆"#% ≡ ∆$#% 3)4
C "$ JT DPNNPO
∠"0# = ∠$0% (WFSUJDBMMZ PQQPTJUF) ∆"0# ≡ ∆$0% (4"4) B %$ = #$ HJWFO
&$ = "$ HJWFO
∠%$& = ∠#$" WFSUJDBMMZ PQQPTJUF
∴ ∆"#$ ≡ ∆&%$ 4"4
C ∠&%$ = ∠"#$ DPSSFTQPOEJOH BOHMFT JO DPOHSVFOU USJBOHMFT
** ∴ "# ** %& BMUFSOBUF BOHMFT BSF FRVBM
B "& = $% HJWFO
#& = #% HJWFO
"% = "# HJWFO
∠%"$ = ∠#"$ HJWFO
∠"#& = ∠$#% WFSUJDBMMZ PQQPTJUF XJUI ∠"#& HJWFO ◦
∴ ∆"%$ ≡ ∆"#$ 4"4
D "$ JT DPNNPO ∠"%$ = ∠"#$ (HJWFO) ∠%"$ = ∠#"$ (HJWFO) ∴ ∆"%$ ≡ ∆"#$ (""4) E "$ JT DPNNPO "% = "# HJWFO
%$ = #$ HJWFO
∴ ∆"#& ≡ ∆$#% 3)4
C ∠&"# = ∠%$# DPSSFTQPOEJOH BOHMFT JO DPOHSVFOU USJBOHMFT
** ∴ "& ** $% BMUFSOBUF BOHMFT FRVBM
B %# JT DPNNPO "# = $% HJWFO
"% = $# HJWFO
∴ ∆"#% ≡ ∆$%# 444
∴ ∆"%$ ≡ ∆"#$ 444
F "$ = %$ HJWFO
#$ = &$ HJWFO
∠"$# ≡ ∠%$& WFSUJDBMMZ PQQPTJUF
∴ ∆"#$ ≡ ∆%&$ 4"4
G "$ = &$ HJWFO
C ∠"%# = ∠$#% DPSSFTQPOEJOH BOHMFT JO DPOHSVFOU USJBOHMFT
** ∴ "%** #$ BMUFSOBUF BOHMFT FRVBM
B 0# = 0$ SBEJJ
** ∠$"# = ∠$&% BMUFSOBUF BOHMFT "# ** %&
∠"$# = ∠&$% WFSUJDBMMZ PQQPTJUF
** (PS ∠$#" = ∠$%& (BMUFSOBUF BOHMFT "# ** %&)) ∴ ∆"#$ ≡ ∆&%$ ""4
H %$ = #$ HJWFO
** ∠&%$ = ∠"#$ (BMUFSOBUF BOHMFT %& ** "#) ∠%$& = ∠#$" (WFSUJDBMMZ PQQPTJUF)
** (PS ∠%&$ = ∠#"$ (BMUFSOBUF BOHMFT "# ** %&))
∴ ∆$%& ≡ ∆$#" (""4) I #% JT DPNNPO "% = $% HJWFO
∠"%# = ∠$%# (HJWFO) ∴ ∆"#% ≡ ∆$#% (4"4) J "$ JT DPNNPO
0" = 0% SBEJJ
∠"0# = ∠%0$ WFSUJDBMMZ PQQPTJUF
∴ ∆"0# ≡ ∆%0$ 4"4
C ∠"#0 = ∠%$0 DPSSFTQPOEJOH BOHMFT JO DPOHSVFOU USJBOHMFT
** ∴ "# ** $% BMUFSOBUF BOHMFT FRVBM
B #% JT DPNNPO "% = $% HJWFO
∠"%# = ∠$%# HJWFO
∴ ∆"#% ≡ ∆$#% 4"4
C ∠"#% = ∠$#% DPSSFTQPOEJOH BOHMFT JO DPOHSVFOU USJBOHMFT
BOE ∠"#% + ∠$#% = ◦ TUSBJHIU MJOF
∴ ∠"#% = ∠$#% = ◦ BOE "$ JT QFSQFOEJDVMBS UP #%
"# = $% HJWFO
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
∠"#% = ∠$#% HJWFO ◦
∠"%# = ∠$%# HJWFO
USJBOHMFT
∠%&$ + ∠#&$ = ◦ TUSBJHIU MJOF
∴ ∆"#% ≡ ∆$#% ""4
C "% = $% DPSSFTQPOEJOH TJEF JO DPOHSVFOU USJBOHMFT
∴ ∠%&$ = ∠#&$ = ◦ BOE "& = $& DPSSFTQPOEJOH TJEFT JO DPOHSVFOU
∴ ∆"$% JT JTPTDFMFT FRVBM TJEFT
$POTJEFS ∆0"% BOE ∆0#%
USJBOHMFT
∴ "$ CJTFDUT #% BU ◦
0% JT DPNNPO 0" = 0# SBEJJ
A
"% = #% HJWFO
∴ ∆0"% ≡ ∆0#% 444
M
∠0%" = ∠0%# = ◦ DPSSFTQPOEJOH BOHMFT JO DPOHSVFOU USJBOHMFT BSF FRVBM BOE TVQQMFNFOUBSZ UP B TUSBJHIU
B C N MFU ∆"#$ CF BOZ FRVJMBUFSBM USJBOHMF "# = $# = "$ 4UFQ POF
MJOF
∴ 0$⊥"# $POTJEFS ∆"%$ BOE ∆$#" "$ JT DPNNPO
+PJO $ UP . UIF NJEQPJOU PG "# 1SPWF ∆$". ≡ ∆$#. 444
** ∠%"$ = ∠#$" BMUFSOBUF BOHMFT "% ** #$
** ∠%$" = ∠#"$ BMUFSOBUF BOHMFT %$ ** "#
∴ ∠$". = ∠$#. DPSSFTQPOEJOH BOHMFT JO DPOHSVFOU USJBOHMFT
∴ ∆"%$ ≡ ∆$#" ""4
4P "% = #$ "# = %$ BSF FRVBM DPSSFTQPOEJOH TJEFT JO
4UFQ UXP +PJO " UP / UIF NJEQPJOU PG $#
DPOHSVFOU USJBOHMFT "# = %$ PQQPTJUF TJEFT PG QBSBMMFMPHSBN
1SPWF ∆"/$ ≡ ∆"/# ∴ ∠"$/ = ∠"#/ DPSSFTQPOEJOH BOHMFT JO DPOHSVFOU
∠"&# = ∠$&% WFSUJDBMMZ PQQPTJUF
** ∠#"& = ∠%$& BMUFSOBUF BOHMFT %$ ** "#
USJBOHMFT
/PX ∠$"# = ∠"#$ = ∠"$#
∴ ∆"#& ≡ ∆$%& ""4
BOE BT ∠$"# + ∠"#$ + ∠"$# = ◦ BOHMF TVN PG ∆"#$
4P "& = $& BOE #& = %& DPSSFTQPOEJOH TJEFT JO DPOHSVFOU USJBOHMFT A
"OTXFST
∠%&$ = ∠#&$ DPSSFTQPOEJOH BOHMFT JO DPOHSVFOU
B %# JT DPNNPO
∠$"# = ∠"#$ = ∠"$# = ◦
B
1SPHSFTT RVJ[
D C $POTJEFS ∆"%$ BOE ∆#$% "% = $# BOE %$ JT DPNNPO PQQPTJUF TJEFT PG B SFDUBOHMF BSF FRVBM
∠"%$ = ∠#$% = ◦ BOHMFT PG B SFDUBOHMF
∴ ∆"%$ ≡ ∆#$% 4"4
4P "$ = #% DPSSFTQPOEJOH TJEFT JO DPOHSVFOU USJBOHMFT
∴ 5IF EJBHPOBMT PG B SFDUBOHMF BSF FRVBM B $POTJEFS ∆"#& BOE ∆$%& "# = $% TJEFT PG B SIPNCVT
** ∠"#& = ∠$%& BMUFSOBUF BOHMFT "# **
%$** ∠#"& = ∠%$& BMUFSOBUF BOHMFT "# **
%$∴ ∆"#& ≡ ∆$%& ""4
C $POTJEFS ∆%$& BOE ∆#$& $& JT DPNNPO %$ = #$ TJEFT PG B SIPNCVT
%& = #& DPSSFTQPOEJOH TJEFT JO DPOHSVFOU USJBOHMFT
∴ ∆%$& ≡ ∆#$& 444
B ◦
B F
C ◦ C G
D
E
B :FT BT UIFSF JT B QBJS PG FRVBM BMUFSOBUF BOHMFT C /P BT UIF QBJS PG DPJOUFSJPS BOHMFT BSF OPU TVQQMFNFOUBSZ B B = BMUFSOBUF BOHMFT FRVBM JO QBSBMMFM MJOFT
C = DPSSFTQPOEJOH BOHMFT FRVBM JO QBSBMMFM MJOFT
C X = BMUFSOBUF BOHMFT FRVBM JO QBSBMMFM MJOFT
D Y = BOHMF TVN PG B RVBESJMBUFSBM
E Y = FYUFSJPS BOHMF PG B SFHVMBS IFYBHPO
B ""4 C 4"4 "$ JT DPNNPO %$ = #$ HJWFO ∠"$% = ∠"$# HJWFO ◦
∴ ∆"#$ ≡ ∆"%$ 4"4
"# = "% DPSSFTQPOEJOH TJEFT JO DPOHSVFOU USJBOHMFT
∴ ∆"#% JT JTPTDFMFT
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
B "& = %$ HJWFO
B DN = LN
∠"#& = ∠%#$ WFSUJDBMMZ PQQPTJUF BOHMFT
∠&"# = ∠$%# BMUFSOBUF BOHMFT "& ∥ $%
B /" C J M
∴ ∆"#& ≡ ∆%#$ ""4
C ◦
D J
D DN E DN
B ∠'
B EPVCMF F ZFT
JJJ M
JJ
JJJ
&YFSDJTF ( C ∠%
D ()
E "&
C EPVCMF
D EPVCMF
E
F
B 0"′ JT B RVBSUFS PG 0" C 0%′ JT B RVBSUFS PG 0% D E ZFT B ZFT
JJ M
E [FSP
&YFSDJTF '
C DN
C DN
D N B "′ #′ $′ TIPVME IBWF TJEFT UIBU PG "#$ C "′ #′ $′ TIPVME IBWF TJEFT EPVCMF UIBU PG "#$ C J "′ #′ $′ %′ TIPVME IBWF TJEFT MFOHUIT UIBU PG "#$% JJ "′ #′ $′ %′ TIPVME IBWF TJEF MFOHUIT UJNFT UIBU PG "#$% B J JJ C J JJ
B &
C $
F ∠"
JJJ
E #$
B 4"4 """ BOE 3)4 B """ C 3)4
C TIBQF TJ[F D 444 E 4"4
F 3)4 G """ !!! B ∆"#$ !!! ∆()* !!! D ∆"#$ !!! ∆"%& !!! F ∆"%$ !!! ∆"&#
H 4"4 I 444 !!! C ∆"#$ !!! ∆./0 !!! E ∆)'( !!! ∆)+* !!! G ∆"#% !!! ∆&
%$ B """ B 3)4
B J C J
JJJ
D %'
G ∠&
C C
JJ JJ
B """ B J """
C JJ JJ JJ
D J
JJ
JJJ
C J """ D J """
E J F J G J
JJ
JJJ
B B ∆%&'
JJ JJ
JJJ JJJ
∠"$# = ◦ """ ∠89: = ◦ OPU TJNJMBS BT BOHMFT OPU FRVBM
B J C J
JJ
D J E J
JJ
B J N C J N
JJ N JJ N
D J N B DN
D D ∆"#$
E ∆%&'
QBJST PG FRVBM BMUFSOBUF BOHMFT BSF BMXBZT GPSNFE """ JO DPOHSVFODZ B TJEF MFOHUI JT OFFEFE GPS UIF USJBOHMFT
JJ
UP CF UIF TBNF TJ[F JO TJNJMBSJUZ JU JT OPU OFFEFE B 5SJBOHMF 0SJHJOBM *NBHF -FOHUI TDBMF GBDUPS
JJ
JJ N C DN
C C ∆%&'
JJJ N
D N
B B > C B Y = − PS Y = CPUI WBMJE BT CPUI BSF JOUFHFST B C J D J B C J
JJ JJ
NJOJNVN NJOJNVN NBYJNVN NBYJNVN NJOJNVN NBYJNVN NJOJNVN NJOJNVN NBYJNVN
K
Y=
NJOJNVN
L M
Y= Y = −
NBYJNVN NBYJNVN
B
Y Z
JJ
B " = ( + Y) = Y + Y + C DN
−
B
Y
− Y − =
C
− Y − =
D Y − Y − = E Y − Y − = B Y = JT OPU B TPMVUJPO C Y = − JT B TPMVUJPO D Y = JT B TPMVUJPO
E Y = − JT B TPMVUJPO
Z = Y −
C D
x-intercept Y
−
DN
1SPHSFTT RVJ[
D Y = − E Y = −
Y= Y= Y= Y= Y= Y = − Y= Y= Y=
D XJEUI >
C Y=
C Y=
B C D E F G H I J
D XJEUI >
− - = DN 8 = DN
C QBSBCPMB D JOUFSDFQUT E WFSUFY G [FSP JJ 5ZQF PG UVSOJOH QPJOU
B Y (Y + ) = E - = N 8 = DN − −
B IJHIFTU F MPXFTU
J "YJT PG TZNNFUSZ
B Y (Y + ) = C , − E - = DN 8 = DN
E Y + Y + =
− =
B Y=
&YFSDJTF %
C Y − Y − =
&YFSDJTF &
J Y = −
Y
"OTXFST
E Y − Y − = G Y + Y + =
W ZJOUFSDFQU
C Y − Y − =
JW YJOUFSDFQUT
D Y + Y − = F Y − Y + =
JJJ 5VSOJOH QPJOU
B Y − Y + =
, −
, −
−,
, −
" # , −
− − − OPOF − −
− − −
− OPOF
−
−, −
−
y 4 3 2 1
−3 −2 −1−1O −2
Axis of symmetry x=0 x-intercept x 1 2 3 y-intercept Minimum turning point (0, –1)
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
B
Y Z
−
−
Z = − Y C D
−
y y-intercept Maximum turning point (0, 9) Axis of symmetry x=0
10 8 6 4 2
Y Z
−
−
− −
Z = Y + Y −
−
Zero x-intercepts
4 2
DPSSFDU TPMVUJPO JT Z = − + × = − + = C −(−) = − × = − *U JT OPU − = + TP
x-intercept
DPSSFDU TPMVUJPO JT Z = − − (−) = − − = − B Y = −, Y =
x
1 2 3 y-intercept = –3
C Y=
D J JOýOJUF JJ POF B J Y = Y =
Minimum turning point (−1, −4)
C D
Y Z
− −
Z = −Y + Y+
y-intercept
−
3 2 1
x
1 2 3 x-intercept
UIF XBZ EPXO C J T JJ N
B Z = Y Z = Y Z = Y C Z = −Y Z = −Y Z = − Y D
F Z=
−Y
JJ Z = Y JJJ Z = −Y
JW Z = − Y JJJ Z = Y
G J Z = −Y JJ Z = −Y
JJJ T
C C Z = Y −
E Z = − Y + G Z = Y − Y −
F J Z = Y
B N C N D T E KPVSOFZ UBLFT T MPOHFS UP HP EPXO UP UIF HSPVOE
Y
&YFSDJTF '
E Y=
B J T T JJ POF UJNF JT PO UIF XBZ VQ BOE UIF PUIFS JT PO
B Y = B Z = Y
E OPOF − JT UIF NJOJNVN ZWBMVF TP UIFSF BSF OP WBMVFT PG Z MFTT UIBO −
"OTXFST XJMM WBSZ
Maximum turning point (0.5, 2.25)
Axis of symmetry x = 0.5
B Y = C (, −)
JJJ OPOF JJ Y = − Y =
C UXP B QBSBCPMB JT TZNNFUSJDBM D POF UIJT JT UIF NJOJNVN UVSOJOH QPJOU
−
y
−3 −2 −1−1O −2 x-intercept −3
E /P
B − = − × = − *U JT OPU (−) = TP
−3 −2 −1−2O x-intercept −4
B
One x-intercept
y
Axis of symmetry x = −1
Two x-intercepts
D :FT
− −
C D
C :FT
x
O −3 −2 −1−1 1 2 3 −2 x-intercept x-intercept B
B :FT
D Z = − Y
− Y
B QPTJUJWF C OFHBUJWF B Z = −Y C Z = −Y E Z = Y B Y − − Z
Y
−
−
Z Y Z
−
−
D Z = Y
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
D J
y 12 y = 3x
Y
−
Z
2
−
"OTXFST
C
JJ 4FF QBSU B JJJ BYJT PG TZNNFUSZ Y = UVSOJOH QPJOU
8
JW XJEFS E J Y
−
Z
4
y = x2
1
y = 13 x 2
−
JJ 4FF QBSU B JJJ BYJT PG TZNNFUSZ Y = UVSOJOH QPJOU
−2 −1−1O
JW XJEFS B J Y Z JJ
x
1 2
− −
y
D 'PS BMM HSBQIT UIF UVSOJOH QPJOU JT B NJOJNVN BU
Y
−
−
Z JJ
−
−
x
BOE UIF BYJT PG TZNNFUSZ JT Y = E J OBSSPXFS JJ XJEFS B J
− −
1
y = − 3x 2
y = − 12 x 2 −4
y 20
y = 5x 2 −8
y = −2x 2
16 y = 4x 2
−12
y = −3x 2
JJJ BYJT PG TZNNFUSZ Y = UVSOJOH QPJOU Y BOE
12
JW OBSSPXFS
ZJOUFSDFQU C J
Y Z
−
−
JJ 4FF QBSU B
8
− −
−
−
JJJ BYJT PG TZNNFUSZ Y = UVSOJOH QPJOU Y BOE ZJOUFSDFQU JW OBSSPXFS D J
4
Z
y = 14 x 2
−2 −1 O
x
1 2
−
−
JW OBSSPXFS
Y Z
−
−
−
−
− −
JW XJEFS − −
−
−
JJ 4FF QBSU B JJJ BYJT PG TZNNFUSZ Y = UVSOJOH QPJOU Y BOE
JJ 4FF QBSU B JJJ BYJT PG TZNNFUSZ Y = UVSOJOH QPJOU Y BOE ZJOUFSDFQU
−
ZJOUFSDFQU E J
JJJ BYJT PG TZNNFUSZ Y = UVSOJOH QPJOU
JW OBSSPXFS Y Z
−
JJ 4FF QBSU B JJJ BYJT PG TZNNFUSZ Y = UVSOJOH QPJOU Y BOE
y = 15 x 2
1
C J
Y
B "
ZJOUFSDFQU C )
J E W G
JJ D WJ C
JW XJEFS JJJ B
JW F
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
B SFþFDUJPO JO UIF YBYJT EJMBUJOH CZ B GBDUPS PG GSPN UIF
"OTXFST
YBYJT C SFþFDUJPO JO UIF YBYJT EJMBUJOH CZ B GBDUPS PG GSPN UIF
b a
YBYJT D SFþFDUJPO JO UIF YBYJT EJMBUJOH CZ B GBDUPS PG GSPN UIF YBYJT E SFþFDUJPO JO UIF YBYJT EJMBUJOH CZ B GBDUPS PG GSPN UIF YBYJT F SFþFDUJPO JO UIF YBYJT EJMBUJOH CZ B GBDUPS PG GSPN UIF YBYJT G SFþFDUJPO JO UIF YBYJT EJMBUJOH CZ B GBDUPS PG GSPN UIF YBYJT B Z = −Y C Z = Y D Z = −Y E Z = − Y /P CFDBVTF CPUI USBOTGPSNBUJPOT BSF NVMUJQMZJOH UP AB BOE NVMUJQMJDBUJPO JT DPNNVUBUJWF CD = DC Z = BY IBT ZBYJT BT BYJT PG TZNNFUSZ TP JU JT TZNNFUSJDBM BCPVU UIF ZBYJT B Z = Y C Z = Y D Z = Y E Z = Y F Z = Y G Z = Y H Z = Y I Z = −Y Z = Y
y y-intercept Minimum turning point (0, 3)
6 4
c
2
−2 −1 O
1 2
−2 E Z = −Y + G Z = −Y −
F Z = −Y +
y 5 −3 −2 −1 O d −5 e −10 f
y-intercept Maximum turning point (0, 4) y-intercept Maximum turning point x (0, 1) 1 2 3 y-intercept Maximum turning point (0, –5)
B Z = (Y − ) D Z = (Y +
C Z = (Y − )
)
y a
c
&YFSDJTF (
B J (−, ) C J D MFGU E SJHIU
B J (, −) C J −
b
20
y-intercept = 16
15 JJ
JJ −
y-intercept = 4
Minimum turning point (−3, 0)
JJ
JJ
−10
5
−5
O
Minimum −5 turning point (2, 0)
JJ (−, −) JJ
D J POF MFGU POF EPXO JJ UXP SJHIU UISFF VQ B C − D − E B Z = Y + D Z = Y −
y-intercept = 9
10
D EPXO E VQ
B J
C J −
y-intercept Minimum turning point (0, 1) x y-intercept Minimum turning point (0, −2)
E Z = −(Y − ) G Z = −(Y + )
5
10
x
Minimum turning point (4, 0) F Z = −(Y + )
C Z = Y +
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
−10
O
−5
5
−10
x 10 y-intercept = −4 y-intercept = −9 y-intercept = −36
−30
y y-intercept = 11
20 15 10 5 −4
2
6
4
x
G Z = −(Y − ) +
2
y
20 15 10 y-intercept = 0 5 −6 −4 −2−5O −10
2 4 6
x
Minimum turning point (1, −1)
2
y
2 4 6 8 10
x
y-intercept = −22
y 5
y
x
x
Maximum turning point (5, 3)
H Z = −(Y + ) −
−6
2
4
y-intercept = −3
−10 −8 −6 −4 −2−5O −10 −15 −20 −25
y-intercept = 1 10 5 −2 −5O −4 Minimum −10 turning point (−2, −3)
x
4
Maximum turning point (2, 1)
10 5
C Z = (Y − ) −
D Z = (Y + ) −
−2 e −5O −10 −15 −20
−4
Minimum turning point (3, 2)
−2 −5O −10
y
10 5
B Z = (Y − ) +
y-intercept = 8
−2 −5O −10
F Z = −(Y − ) +
d
−40
e
20 15 10 5
Minimum turning point (−1, 7) −4
−20
f
y
Maximum turning point (3, 0)
"OTXFST
E Z = (Y + ) +
Maximum turning point y (−2, 0) Maximum 10 turning point (−6, 0)
2 4 −2 −5O y-intercept = −13 −10 Maximum turning point −15 −4
x
(−3, −4)
I Z = −(Y + ) −
y Maximum turning point 5 (−1, −5)
−6
−4
−2 −5 −10 −15
x 2 4 y-intercept = −6
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
&YFSDJTF )
"OTXFST
J Z = −(Y − ) −
y
5 −2 −5O y-intercept = −15 −10 −4
2
J K W L
JJ E WJ H
JY M Y J B Z = (Y − ) + )
D Z = (Y + − F Z = −(Y − ) −
8
JJJ C WJJ B
JW I WJJJ F
B − F −
YJ G YJJ D C Z = (Y + ) +
B J − C J −
)
E Z = −(Y − + G Z = −(Y + ) +
−2 −1 O
(2, 9)
1 2 3 4 5 6 7 8 9 10
B J Z = −(Y + ) + JJJ Z = (Y − ) −
W Z = −Y − C J (−, ) JJ (−, )
W (, −) WJ (, −) B (I, L) C Z = BI + L
JJ Z = (Y + ) + JW Z = −(Y − ) −
WJ Z = Y − JJJ (, −) JW (, −)
B ( − Y) = (−(Y − )) = (−) (Y − ) = (Y − ) TP HSBQIT BSF UIF TBNF C HSBQIT BSF UIF TBNF B Z = Y + D Z = Y − B Z = −Y +
C Z = Y −
E Z = Y − C Z = −Y +
D Z = −Y + E Z = −Y + B Z = (Y + ) PS Z = (Y − ) C Z = (Y − ) PS Z = (Y − ) D Z = (Y + ) PS Z = (Y − )
E Z = Y PS Z = (Y − ) B Z = −(Y − ) + C Z = (Y + ) D Z = −(Y − ) F Z = (Y + ) +
E Z = −(Y + ) + G Z = (Y − ) −
E I
D −
J
E −
JJ JJ
D J − E J −
JJ JJ −
H J − I J −
JJ − JJ
JJ − JJ −
y
Axis of symmetry x-intercept x = 2.5 (5, 0)
5
x
(7, −4)
D − H −.
C − G −
J J JJ B Z = Y (Y − )
(10, 5)
−7 (−2, −7)
H (Y + )(Y + ) J (Y − )(Y + )
C G −
F J − G J −
Z = (Y − ) − , UVSOJOH QPJOU (, −) C y
5
G (Y + )(Y − ) I (Y + )(Y + )
B F −
B Z = −(Y − ) + , UVSOJOH QPJOU (, )
9
x
K (Y − )(Y + ) L (Y − )(Y − ) = (Y − ) M (Y + )(Y + ) = (Y + )
Maximum turning point (3, −6)
−15 −20
6
4
B Y (Y + ) C Y (Y − ) D Y (Y − ) E (Y + )(Y − ) F (Y + )(Y − )
O −4 −2 y-intercept x-intercept−5 (0, 0) −10
2
4
6
8
x
Minimum turning point (2.5, −6.25)
C Z = Y (Y + )
y Axis of symmetry x = −0.5 O x −4 −3 −2 −1 −2 1 2 3 Minimum −4 y-intercept x-intercept turning point −6 (0, 0) (−0.5, −0.25) 4 2
x-intercept (−1, 0)
D Z = Y (Y − )
y 4 2 −2 −2 y-intercept x-intercept −4 (0, 0) −6
Axis of symmetry x = 1.5 x-intercept (3, 0)
O 2
4
6
x
Minimum turning point (1.5, −2.25)
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
E Z = Y ( + Y)
I Z = −Y ( + Y)
−4 −3 −2
1 2 3 y-intercept x-intercept (0, 0)
Minimum −5 turning point (−1, −1)
x
−5 Axis of symmetry x = −1 −10 J Z = −Y (Y − )
F Z = Y ( + Y)
y Axis of symmetry x = −2.5 −8
O −5
Minimum turning point (−2.5, −6.25) G Z = Y ( − Y)
−10
2 4 y-intercept x-intercept (0, 0)
5
2
x
4
y 20 15
5 O
−10
−5
y-intercept x-intercept (0, 0) 5
−5
Axis of symmetry x = 0.5
−4
10
y-intercept 4 (0, 2) 3 x-intercept 2 1 (1, 0) −2 −1−1O Axis of −2 symmetry −3 x = 1.5
x-intercept (2, 0) x 1 2 3 4 Minimum turning point (1.5, −0.25)
C Z = (Y − )(Y − )
Axis of symmetry x = −4
y
6
10 x-intercept (−8, 0)
−2
y
6 x-intercept Axis of (3, 0) symmetry x = 1.5
H Z = −Y (Y + )
−3 −2 −1
B Z = (Y − )(Y − )
Maximum turning point (1.5, 2.25)
O −2 y-intercept x-intercept −5 (0, 0) −10
x
Maximum turning point (0.5, 0.25) x-intercept (1, 0) x O1 2 3 4
y
−6
y
Maximum turning point (−4, 16)
y-intercept x-intercept 2 (0, 0)
5
−2
−6 −4 x-intercept (−5, 0)
"OTXFST
Maximum y turning point y-intercept (−1, 1) 5 x-intercept x-intercept (−2, 0) (0, 0) O x 1 2 3 −4 −3 −2 −1
y Axis of symmetry 10 x = −1 5 x-intercept (−2, 0) O
x
Axis of symmetry x=2
y-intercept 4 (0, 3) 2 x-intercept (1, 0) −2
O −2 −4
x-intercept (3, 0) 2
4
6
x
Minimum turning point (2, −1)
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
D Z = (Y + )(Y − )
B Z = Y −
y 5
x-intercept (−3, 0)
6
x-intercept (1, 0) x 4 2
O −4 −2 Minimum −5 turning point y-intercept (−1, −4) (0, −3) Axis of −10 symmetry x = −1
y
4 x-intercept (−1, 0) −4
−2
2
x-intercept (1, 0)
O
2
−2
E Z = (Y + )
−4
y 8 6 4 2
Axis of symmetry x = −2
C Z = − Y
y-intercept (0, 4)
−5 −4 −3 −2 −1−2O Minimum −4 turning point x-intercept (−2, 0)
1 2
−4
y 4
−2
G Z = (Y + )(Y − )
O
2
4
x
x-intercept (3, 0)
y
y-intercept (0, 1) x 1 2 Minimum turning point x-intercept ( −1, 0) y
Axis of symmetry 5 x = –1 −5 −4 −3 −2 −1 O x-intercept −5 (−4, 0) Turning point −10 minimum (−1, −9)
5
D Z = Y + Y
2
−3 −2 −1
y-intercept Minimum turning point (0, −1)
y-intercept Maximum turning point (0, 9)
−5 x-intercept (−3, 0)−10
F Z = (Y + )
Axis of symmetry x = −1
−2
x
y 10
x
4
1 2
x-intercept (2, 0) x
x-intercept (−5, 0) −6
6 y-intercept 4 x-intercept (0, 0) 2 −2 −2O −4 −6 Minimum −8 turning point (−2.5, −6.25) −4
2
x
y-intercept (0, –8)
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
E Z = Y − Y
y
y
10
5 4 y-intercept 3 x-intercept 2 (0, 0) 1
5
−2 −1−1 −2 −3 −4 −5
1 2 3 x-intercept (3, 0)
B − F −
D
E −
C N C N
5IFSF JT POMZ POF YJOUFSDFQU TP UIJT NVTU CF UIF UVSOJOH QPJOU B Z = D C Z = BC C ( ) (− ) UVSOJOH QPJOU YDPPSEJOBUF + (−) = 51 −
D Z = (Y − ) − , I = , L = −
x-intercept (2, 0) x 1 2 3
C Z = (Y + )(Y − ) D Z = (Y + )(Y − ) E Z = −(Y + )(Y − ) F Z = −(Y + )(Y + ) G Z = (Y + )(Y − ) H Z = − (Y + )(Y − )
Minimum turning point (1, −4)
G Z = Y − Y −
1SPCMFNT BOE DIBMMFOHFT y
5
0 −2 y-intercept −5 (0, −4) −10
B UIFZ BSF TRVBSF OVNCFST C = B − C − D +
x-intercept (4, 0) 4
2
6
x
TFDPOET O(O + ) O =
Minimum turning point (1.5, −6.25)
Y= B VOJUT VOJUT VOJUT B L
H Z = Y + Y −
x-intercept (−4, 0) 5
y x-intercept (3, 0)
−2 −5O −10 −15 Minimum turning point −20 (−0.5, −12.25)
−6
x
6
B ( − )
y-intercept x-intercept 4 2 (0, 0)
x-intercept (−1, 0)
C G
B N B N
y
−2 −1−2O −4 −6
4
2
−5
Minimum turning point (1.5, −4.5)
F Z = Y − Y
O
−2
x
y-intercept (0, 9) Minimum turning point x-intercept (3, 0)
"OTXFST
J Z = Y − Y +
−4
2
B − − − − C B = , −, , −, , −, , −
4
Y = −, , , Y = −, −, ,
x
.VMUJQMFDIPJDF RVFTUJPOT
y-intercept (0, −12)
I Z = Y + Y +
y Minimum turning point x-intercept (−1, 0)
8 6 4 2
−4 −3 −2 −1−2O
C VOJUT
%
#
$
"
%
#
&
$
"
#
4IPSUBOTXFS RVFTUJPOT
y-intercept (0, 1) 1 2 3 4
B Z = Y − Y − Y − − Z
−
−
−
−
x
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
"OTXFST
C
D Z = (Y + )
y
y Minimum turning point 8 (−2, 0) 6 4 y-intercept 2 x −6 −5 −4 −3 −2 −1−20 1 2
15 12 9 6 3 −3 −2 −1−3O
1 2 3 4
B − F −
C G
B − F −
C D − √ √ G −
x
D −
E −
E Z = −(Y − )
y
E − H − −
2
I − J B Y − Y = Y = , C Y − Y = Y = D Y + Y + = Y = −, − E Y − Y − = Y = −, F Y − Y + = Y = , G Y + Y − = Y = −,
B Y + Y − = Y = VOJUT C Y + Y − = Y = VOJUT −Y
B Z= C Z= E Z = (Y − ) B Z = Y +
y
+ D Z = (Y + F Z = Y −
y y-intercept Minimum turning point (0, 2) x 1 2 3
6 4 2 −3 −2 −1−2O C Z = −Y −
)
10 8 y-intercept 6 4 2 −3 −2 −1−2
Minimum turning point (1, 3) 1 2 3 4 5 6
G Z = (Y + ) −
y
x
y-intercept = 4
5 −5 −4 −3 −2 −1 O
y 2
−4 −3 −2 −1−2O −4 −6 −8 −10
x
−2 −1−2O 1 2 3 4 5 6 7 8 −4 −6 −8 y-intercept = −9 −10 −12 F Z = (Y − ) +
B J Y= JJ NJOJNVN −
C J Y = − JJ NBYJNVN −
Y
Maximum turning point (3, 0)
x
1 2 3 4 y-intercept Maximum turning point (0, −5)
1 2
x
Minimum −5 turning point (−2, −4) B Z = Y USBOTMBUFE VQ VOJUT C Z = Y SFþFDUFE JO UIF YBYJT BOE USBOTMBUFE EPXO VOJUT D Z = Y USBOTMBUFE VOJUT MFGU E Z = Y SFþFDUFE JO UIF YBYJT UIFO USBOTMBUFE VOJUT SJHIU F Z = Y USBOTMBUFE VOJU SJHIU UIFO VOJUT VQ G Z = Y EJMBUFE CZ B GBDUPS PG GSPN UIF YBYJT USBOTMBUFE VOJUT MFGU BOE UIFO EPXO VOJUT
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
JJJ
h
y
y-intercept 10 =12
20 15 10 5
x-intercept
5
−3 −2 −1 O −5
1 2 3 4 5 6 7 −2 −1 C J
JJ
JJJ
JW h
x 1 2 3 4 5 6 7 8
Minimum turning point (4, −4)
C Z = Y + Y +
−2 −1
y
x-intercept
20 15 y-intercept 10 = 16 5
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1−5O −10 Minimum −15 turning point (−5, −9) D Z = Y + Y −
x-intercept
*OEJDFT BOE TVSET
x
1 2
.VMUJQMFDIPJDF RVFTUJPOT $
x-intercept
x-intercept
C " = Y + Y
D Y = Y = − OPU WBMJE BT Y >
E #BTF = N )FJHIU = N B J
%
&
$
x
C Y Z
B C B . × − LH C
D . × TFDPOET B C √ E
D Y − D Y Z C . × LH
E . × − TFDPOET √ √ D +
&YUFOEFESFTQPOTF RVFTUJPO B J C . ×
JJ . ×
D . × −
x
(FPNFUSZ .VMUJQMFDIPJDF RVFTUJPOT
$
# "
&YUFOEFESFTQPOTF RVFTUJPOT B Y+
B B C
B N
1 2 3 −8 −7 −6 −5 −4 −3 −2 −1−5 y-intercept −10 Minimum −15 turning point (−2, −9)
#
4IPSUBOTXFS RVFTUJPO
y
10 5
t
4FNFTUFS SFWJFX
x-intercept
−8 −7 −6 −5 −4 −3 −2 −1−5O 1 2 3 4 5 − 10 −15 y-intercept Minimum −20 turning point (−1, −16) E Z = Y + Y −
x-intercept
1 2 3 4 5 6 7 8 9 10
D J $POOPST SPDLFU T JJ 4BNT SPDLFU N IJHIFS JJJ N UIF TBNF IFJHIU
y
10 5
t
20 15 10 5
x-intercept
−10
"OTXFST
B Z = Y − Y +
JJ .BYJNVN BU
& #
4IPSUBOTXFS RVFTUJPOT
B B = C = D Y = Z =
C B = C = E Y =
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
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Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.
Essential Mathematics for the Australian Curriculum Year 9 2ed
Cambridge University Press ISBN 978-1-107-57007-8 © Greenwood et al. 2015 Photocopying is restricted under law and this material must not be transferred to another party.