Mathematics SL - Course Companion

2012·41.15 MB·english

by  Laurie Buchanan

O X F O R D I B D I P L O M A P R O G R A M M E
M at h e M at i c s
s ta N Da R D le v e l
COURSE COMPANION
Laurie Buchanan
Jim Fensom
Ed Kemp
Paul La Rondie
Jill Stevens

P155: Shutterstock/Upthebanner; P158: The Art Gallery Col-
lection/Alamy; P158: Mireille Vautier/Alamy; P159: Yayayoyo/
Shutterstock; P159: Travis Manley/Dreamstime.com; P159: Lou-
louphotos /Shutterstock; P161: James Harbal/Dreamstime.com;
Great Clarendon Street, Oxford OX2 6DP P162: Robyn Mackenzie/Dreamstime.com; P164: Science Photo
Oxford University Press is a department of the University of Library; P176: Nito/Shutterstock; P183: Gsplanet/Shutterstock;
P182: Gingergirl/Dreamstime.com; P: Christopher King/Dream-
Oxford. It furthers the University’s objective of excellence in
research, scholarship, and education by publishing worldwide in stime.com; P192: Art Directors & Trip/Alamy; P192: Alex Garaev/
Shutterstock; P193: Adam Eastland Italy/Alamy; P192: Lebrecht
Oxford New York Music And Arts Photo Library/Alamy; P195: Yellowj/Shutterstock;
P201: Science Source/Science Photo Library; P205: Glasscuter/
Auckland Cape Town Dar es Salaam Hong Kong Karachi
Dreamstime.com; P205: Jules2000 /Shutterstock; P214: Sheila
Kuala Lumpur Madrid Melbourne Mexico City Nairobi
Terry/Science Photo Library; P217: Sheila Terry/Science Photo
New Delhi Shanghai Taipei Toronto
Library; P221: Glowimages/Getty Images; P222: Ccat82/Shut-
 terstock; P253: Adrian Zenz/Dreamstime.com; P255: Mitchell
Gunn/Dreamstime.com; P256: Will & Deni Mcintyre/Corbis;
Argentina Austria Brazil Chile Czech Republic France
P256: Gravicapa/Fotolia.Com; P256: Brett Critchley/Dreamstime.
Greece Guatemala Hungary Italy Japan Poland Portugal

Singapore South Korea Switzerland Thailand Turkey
Dreamstime.com; P270: Pamela Tekiel/Dreamstime.com; P271:
Ukraine Vietnam
William Perry/Dreamstime.com; P274: Sergio Azenha/Alamy;
© Oxford University Press 2012 P275: Brenda Carson/Dreamstime.com; P280: Tina Norris/
Rex Features; P283: Maniec/Dreamstime.com; P282: Asdf_1/
The moral rights of the author have been asserted
Dreamstime.com; P289: Robodread/Dreamstime.com; P288:
Database right Oxford University Press (maker) Niday Picture Library/Alamy; P291: Sean Gladwell/Fotolia; P313:
Science Photo Library; P318: Andrew Brookes, National Physical
First published 2012
Laboratory/Science Photo Library; P318: Ted Foxx/Alamy; P330:
All rights reserved. No part of this publication may be Mediacolor's/Alamy; P330: Fromoldbooks.Org/Alamy; P333: Alex
reproduced, stored in a retrieval system, or transmitted, in any James Bramwell/Shutterstock; P334: Paulmerrett/Dreamstime.
form or by any means, without the prior permission in writing com; P349: Science Photo Library; P361: Maxx-Studio/Shut-
of Oxford University Press, or as expressly permitted by law, or terstock; P363: Lamb/Alamy; P363: Bcampbell65/Shutterstock;
under terms agreed with the approprate reprographics rights P372: Maridav/Shutterstock ; P373: Chris Harvey/Shutterstock;
organization. Enquiries concerning reproduction outside the P384: Michel Stevelmans/Shutterstock ; P384: Brandon Bourd-
scope of the above should be sent to the Rights Department, ages/Shutterstock; P385: Darren Baker/Dreamstime.com; P: Vik-
Oxford University Press, at the address above tor Pravdica/Dreamstime.com; P403: Tomadesign/Shutterstock;
P403: Tomadesign/Shutterstock; P403: Tomadesign/Shutterstock;
You must not circulate this book in any other binding or cover
P403: Konstantin Mironov/Shutterstock; P405: Darryl Brooks/
and you must impose this same condition on any acquirer
Shutterstock; P406: Rafa Irusta/Shutterstock; P436: Dmitry_K/
British Library Cataloguing in Publication Data Shutterstock; P444: Pagadesign/Istockphoto; P447: Mr.Xutakupu/
Shutterstock; P455: National Portrait Gallery London; P483:
Data available
Rorem/Shutterstock; P487: Cynthia Burkhardt/Shutterstock;
ISBN: 978-0-19-912935-5 P487: Lori Martin/Shutterstock; P488: Phb.Cz (Richard Semik)/
10 9 8 7 6 5 4 3 2 1 Shutterstock; P493: Mary Evans Picture Library/Alamy; P492:
Alan Haynes/Alamy; P497: Mythic Ink/Getty Images; P497: Noah
Printed by Vivar Printing Sdn Bhd, Malaysia
Berger/Associated Press; P506: Mythic Ink/Getty Images; P517:
Acknowledgments Science Photo Library; P521: Reuters Pictures; P526: Doodled-
ance/Shutterstock; P535: Stanth/Shutterstock; P536: Anke Van
The publishers would like to thank the following for permission
Wyk/Dreamstime.com; P528: Cla78/Shutterstock; P529: Vladimir
to reproduce photographs:
Yessikov/Shutterstock; P547: Monkey Business Images/Dream-
P3: Nasa; P4: Konstantin Chagin/Shutterstock; P4: Janine Wiedel stime.com; P552: Monkey Business Images/Dreamstime.com;
Photolibrary/Alamy; P13: Hulton Archive/Stringer/Getty Images; P555: Vladimir Voronin/Dreamstime.com; P557: Phase4photog-
P17: Trip/Art Director; P17: Lunar And Planetary Instotute; P31: raphy/Dreamstime.com; P558: Sculpies/Dreamstime.com; P555:
Nlshop/Shutterstock; P31: Itsmejust/Shutterstock; P33: Robert Mario Savoia/Shutterstock; P555: R. Gino Santa Maria/Shut-
Crow/Dreamstime.com; P33: Lane Erickson/Dreamstime.com; terstock; P554: James Weston/Shutterstock; P567: Beboy/Shut-
P41: Sean Nel/Shutterstock; P54: Blasbike/Dreamstime.com; terstock; P567: Buslik/Shutterstock; P567: Dadek/Shutterstock;
P56: Brad Remy/Dreamstime.com; P61: David Gee/Alamy; P61: P566: Scott Camazine/Science Photo Library; P566: Nasa/Science
Paulpaladin/Dreamstime.com; P61: Konstantin Androsov/Dream- Photo Library; P567: Mikkel Juul Jensen/Science Photo Library.
stime.com; P61: Lembit Ansperi/Dreamstime.com; P61: Julián
Cover Image: Joshua McCullough / Photo Library
Rovagnati/Dreamstime.com; P60: Ilya Postnikov/Dreamstime.
com; P61: Victor Habbick Visions/Science Photo Library; P63: Every effort has been made to contact copyright holders of
Martin Fischer/Dreamstime.com; P64: Mrshining/Dreamstime. 
com; P64: Science Photo Library; P73: Francesco Abrignani/Shut- will be pleased to rectify any errors or omissions at the earliest
terstock; P75: Ints Vikmanis/Shutterstock; P84: 3dimentii/Shut- opportunity.
terstock; P85: Nicemonkey/Shutterstock; P92: Supri Suharjoto/
Shutterstock; P98: Peter E Noyce/Alamy; P99: James Steidl/
Dreamstime.com; P99: Motorolka/Dreamstime.com; P98: Ste-
phen Gray/Shutterstock; P101: Viorel Dudau/Dreamstime.com;
P111: Pcheruvi/Dreamstime.com; P112: Classic Image/Alamy;
P132: Adisa/Shutterstock; P133: Robyn Mackenzie/Shutterstock;
P134: Nigel Spiers/Dreamstime.com; P138: Irochka/Dreamstime.
com; P139: Nasa Archive/Alamy; P141: Iofoto/Shutterstock;
P142: Science Source/Science Photo Library; P145: Shutterstock/
Patrik Dietrich; P149: Shutterstock/Plampy; P152: Shutterstock/
John Orsbun; P152: Shutterstock/Wavebreakmedia Ltd; P153:
Shutterstock/Filipe B. Varela; P155: Shutterstock/Luckyphoto;
Course Companion definition
The IB Diploma Programme Course Companions are use of a wide range of resources; international-
resource materials designed to provide students with mindedness; the IB learner profile and the IB Diploma
support through their two-year course of study. These Programme core requirements; theory of knowledge,
books will help students gain an understanding of what the extended essay, and creativity, action, service
is expected from the study of an IB Diploma (CAS).
Programme subject.
Each book can be used in conjunction with other
The Course Companions reflect the philosophy and materials and indeed, students of the IB are required
approach of the IB Diploma Programme and present and encouraged to draw conclusions from a variety of
content in a way that illustrates the purpose and aims resources. Suggestions for additional and further
of the IB. They encourage a deep understanding of reading are given in each book and suggestions for how
each subject by making connections to wider issues and to extend research are provided.
providing opportunities for critical thinking.
In addition, the Course Companions provide advice
The books mirror the IB philosophy of viewing the and guidance on the specific course assessment
curriculum in terms of a whole-course approach; the requirements and also on academic honesty protocol.
IB mission statement
The International Baccalaureate aims to develop programmes of international education and rigorous
inquiring, knowledgable and caring young people who assessment.
help to create a better and more peaceful world through
These programmes encourage students across the
intercultural understanding and respect.
world to become active, compassionate, and lifelong
To this end the IB works with schools, governments learners who understand that other people, with their
and international organizations to develop challenging differences, can also be right.
The IB Learner Profile
The aim of all IB programmes is to develop dignity of the individual, groups, and communities.
internationally minded people who, recognizing their They take responsibility for their own actions and the
common humanity and shared guardianship of the consequences that accompany them.
planet, help to create a better and more peaceful world.
Open-minded They understand and appreciate their
IB learners strive to be:
own cultures and personal histories, and are open to
Inquirers They develop their natural curiosity. They the perspectives, values, and traditions of other
acquire the skills necessary to conduct inquiry and individuals and communities. They are accustomed to
research and show independence in learning. They seeking and evaluating a range of points of view, and
actively enjoy learning and this love of learning will be are willing to grow from the experience.
sustained throughout their lives.
Caring They show empathy, compassion, and respect
Knowledgable They explore concepts, ideas, and towards the needs and feelings of others. They have a
issues that have local and global significance. In so personal commitment to service, and act to make a
doing, they acquire in-depth knowledge and develop positive difference to the lives of others and to the
understanding across a broad and balanced range of environment.
disciplines.
Risk-takers They approach unfamiliar situations and
Thinkers They exercise initiative in applying thinking uncertainty with courage and forethought, and have
skills critically and creatively to recognize and the independence of spirit to explore new roles, ideas,
approach complex problems, and make reasoned, and strategies. They are brave and articulate in
ethical decisions. defending their beliefs.
Communicators They understand and express ideas Balanced They understand the importance of
and information confidently and creatively in more intellectual, physical, and emotional balance to achieve
than one language and in a variety of modes of personal well-being for themselves and others.
communication. They work effectively and willingly in
Reflective They give thoughtful consideration to their
collaboration with others.
own learning and experience. They are able to assess
Principled They act with integrity and honesty, with a and understand their strengths and limitations in order
strong sense of fairness, justice, and respect for the to support their learning and personal development
iii
A note on academic honesty
It is of vital importance to acknowledge and What constitutes malpractice?
appropriately credit the owners of information
Malpractice is behavior that results in, or may
when that information is used in your work. After
result in, you or any student gaining an unfair
all, owners of ideas (intellectual property) have
advantage in one or more assessment component.
property rights. To have an authentic piece of
Malpractice includes plagiarism and collusion.
work, it must be based on your individual and
Plagiarism is defined as the representation of the
original ideas with the work of others fully
ideas or work of another person as your own.
acknowledged. Therefore, all assignments, written
The following are some of the ways to avoid
or oral, completed for assessment must use your
plagiarism:
own language and expression. Where sources are
used or referred to, whether in the form of direct ● W ords and ideas of another person used to
support one’s arguments must be
quotation or paraphrase, such sources must be
acknowledged.
appropriately acknowledged.
● Passages that are quoted verbatim must be
How do I acknowledge the work
enclosed within quotation marks and
of others? acknowledged.
The way that you acknowledge that you have ● CD-ROMs, email messages, web sites on the
used the ideas of other people is through the use Internet, and any other electronic media must
of footnotes and bibliographies. be treated in the same way as books and
journals.
Footnotes (placed at the bottom of a page) or
endnotes (placed at the end of a document) are to ● The sources of all photographs, maps,
be provided when you quote or paraphrase from illustrations, computer programs, data, graphs,
another document, or closely summarize the audio-visual, and similar material must be
information provided in another document. You acknowledged if they are not your own work.
do not need to provide a footnote for information ● Works of art, whether music, film, dance,
that is part of a “body of knowledge”. That is, theatre arts, or visual arts, and where the
definitions do not need to be footnoted as they creative use of a part of a work takes place,
are part of the assumed knowledge. must be acknowledged.
Bibliographies should include a formal list of Collusion is defined as supporting malpractice by
the resources that you used in your work. another student. This includes:
“Formal” means that you should use one of the
● allowing your work to be copied or submitted
several accepted forms of presentation. This
for assessment by another student
usually involves separating the resources that you
● duplicating work for different assessment
use into different categories (e.g. books,
components and/or diploma requirements.
magazines, newspaper articles, Internet-based
resources, CDs and works of art) and providing Other forms of malpractice include any action
that gives you an unfair advantage or affects the
full information as to how a reader or viewer of
results of another student. Examples include,
your work can find the same information. A
taking unauthorized material into an examination
bibliography is compulsory in the extended essay.
room, misconduct during an examination, and
falsifying a CAS record.
iv
About the book
The new syllabus for Mathematics Standard where a GDC may be used. Questions are
Level is thoroughly covered in this book. Each designed to increase in difficulty, strengthen
chapter is divided into lesson size sections with analytical skills and build confidence through
the following features: understanding. Internationalism, ethics and
applications are clearly integrated into every
Investigations Exploration suggestions
section and there is a TOK application page that
concludes each chapter.
Examiner's tip Theory of Knowledge
It is possible for the teacher and student to work
through in sequence but there is also the
Did you know? Historical exploration
flexibility to follow a different order. Where
Mathematics is a most powerful, valuable appropriate the solutions to examples using the
instrument that has both beauty in its own study TI-Nspire calculator are shown. Similar solutions
and usefulness in other disciplines. The Sumerians using the TI-84 Plus and Casio FX-9860GII are
developed mathematics as a recognized area of included on the accompanying interactive CD
teaching and learning about 5,000 years ago and it which includes a complete ebook of the text,
has not stopped developing since then. prior learning, GDC support, an interactive
glossary, sample examination papers, internal
The Course Companion will guide you through
assessment support, and ideas for the exploration.
the latest curriculum with full coverage of all
topics and the new internal assessment. The Mathematics education is a growing, ever
emphasis is placed on the development and changing entity. The contextual, technology
improved understanding of mathematical concepts integrated approach enables students to become
and their real life application as well as proficiency adaptable, life-long learners.
in problem solving and critical thinking. The
Note: US spelling has been used, with IB style for
Course Companion denotes questions that would
mathematical terms.
be suitable for examination practice and those
About the authors
Laurie Buchanan has been teaching mathematics in the IB curriculum review board and is an online
Denver, Colorado for over 20 years. She is a team workshop developer for IB.
leader and a principal examiner for mathematics
Paul La Rondie has been teaching IB Diploma
SL Paper One and an assistant examiner for Paper
Programme mathematics at Sevenoaks School for
Two. She is also a workshop leader and has worked
10 years. He has been an assistant examiner and
as part of the curriculum review team.
team leader for both papers in Mathematics SL
Jim Fensom has been teaching IB mathematics and an IA moderator. He has served on the IB
courses for nearly 35 years. He is currently curriculum review board and is an online
Mathematics Coordinator at Nexus International workshop developer for IB.
School in Singapore. He is an assistant examiner
Jill Stevens has been teaching IB Diploma
for Mathematics HL.
Programme mathematics at Trinity High School,
Edward Kemp has been teaching IB Diploma Euless, Texas for nine years. She is an assistant
Programme mathematics for 20 years. He is examiner for Mathematics SL, a workshop leader
currently the head of mathematics at Ruamrudee and has served the IB in curriculum review. Jill
International School in Thailand. He is an has been a reader and table leader for the
assistant examiner for IB mathematics, served on College Board AP Calculus exam.
v
Contents
Chapter  Functions  Chapter  Patterns, sequences and
1.1 Introducing functions 4 series 
1.2 The domain and range of a relation on a 6.1 Patterns and sequences 162
Cartesian plane 8 6.2 Arithmetic sequences 164
1.3 Function notation 13 6.3 Geometric sequences 167
1.4 Composite functions 14 6.4 Sigma (Σ) notation and series 170
1.5 Inverse functions 16 6.5 Arithmetic series 172
1.6 Transforming functions 21 6.6 Geometric series 175
6.7 Convergent series and sums to infinity 178
Chapter  Quadratic functions and 6.8 Applications of geometric and
equations  arithmetic patterns 181
2.1 Solving quadratic equations 34 6.9 Pascal’s triangle and the binomial
2.2 The quadratic formula 38 expansion 184
2.3 Roots of quadratic equations 41
2.4 Graphs of quadratic functions 43 Chapter  Limits and derivatives 
2.5 Applications of quadratics 53 7.1 Limits and convergence 196
7.2 The tangent line and derivative of xn 200
Chapter  Probability  7.3 More rules for derivatives 208
3.1 Definitions 64 7.4 The chain rule and higher order
3.2 Venn diagrams 68 derivatives 215
3.3 Sample space diagrams and the 7.5 Rates of change and motion in a line 221
product rule 77 7.6 The derivative and graphing 230
3.4 Conditional probability 85 7.7 More on extrema and optimization
3.5 Probability tree diagrams 89 problems 240
Chapter  Exponential and logarithmic Chapter  Descriptive statistics 
functions  8.1 Univariate analysis 256
4.1 Exponents 103 8.2 Presenting data 257
4.2 Solving exponential equations 107 8.3 Measures of central tendency 260
4.3 Exponential functions 109 8.4 Measures of dispersion 267
4.4 Properties of logarithms 115 8.5 Cumulative frequency 271
4.5 Logarithmic functions 118 8.6 Variance and standard deviation 276
4.6 Laws of logarithms 122
4.7 Exponential and logarithmic equations 127 Chapter  Integration 
4.8 Applications of exponential and 9.1 Antiderivatives and the indefinite
logarithmic functions 131 integral 291
9.2 More on indefinite integrals 297
Chapter  Rational functions  9.3 Area and definite integrals 302
5.1 Reciprocals 142 9.4 Fundamental Theorem of Calculus 309
5.2 The reciprocal function 143 9.5 Area between two curves 313
5.3 Rational functions 147 9.6 Volume of revolution 318
9.7 Definite integrals with linear motion and
other problems 321
vi
Chapter  Bivariate analysis  Chapter  Probability distributions 
10.1 Scatter diagrams 334 15.1 Random variables 520
10.2 The line of best fit 339 15.2 The binomial distribution 527
10.3 Least squares regression 345 15.3 The normal distribution 538
10.4 Measuring correlation 349
Chapter  The Exploration 
Chapter  Trigonometry  16.1 About the exploration 556
11.1 Right-angled triangle trigonometry 363 16.2 Internal assessment criteria 557
11.2 Applications of right-angled triangle 16.3 How the exploration is marked 562
trigonometry 369 16.4 Academic Honesty 562
11.3 Using the coordinate axes in 16.5 Record keeping 563
trigonometry 373 16.6 Choosing a topic 564
11.4 The sine rule 380 16.7 Getting started 568
11.5 The cosine rule 386
11.6 Area of a triangle 389 Chapter  Using a graphic display
11.7 Radians, arcs and sectors 391 calculator 
1 Functions 572
Chapter  Vectors  2 Differential calculus 598
3 Integral calculus 606
12.1 Vectors: basic concepts 407
4 Vectors 608
12.2 Addition and subtraction of vectors 420
5 Statistics and probability 612
12.3 Scalar product 426
12.4 Vector equation of a line 430
12.5 Application of vectors 437 Chapter  Prior learning 
1 Number 633
Chapter  Circular functions  2 Algebra 657
13.1 Using the unit circle 448 3 Geometry 673
13.2 Solving equations using the unit circle 454 4 Statistics 699
13.3 Trigonometric identities 456
13.4 Graphing circular functions 462 Chapter  Practice papers 
13.5 Translations and stretches of Practice paper 1 708
trigonometric functions 469 Practice paper 2 712
13.6 Combined transformations with sine
and cosine functions 478 Answers 
13.7 Modeling with sine and
cosine functions 483 Index 
Chapter  Calculus with trigonometric
functions 
14.1 Derivatives of trigonometric functions 496
14.2 More practice withderivatives 500
14.3 Integral of sine and cosine 505
14.4 Revisiting linear motion 510
vii
What's on the CD?
The material on your CD-ROM includes the entire student book as an eBook,
as well as a wealth of other resources specifically written to support your
learning. On these two pages you can see what you will find and how it will
help you to succeed in your Mathematics Standard Level course.
The whole print text is presented as a user-friendly eBook for use in class and at home.
Extra content can be found in the Contents
menu or attached to specic pages.
This icon appears in the book
wherever there is extra content.
Navigation is straightforward either through
the Contents Menu, or through the Search
and Go to page tools.
A range of tools enables you to zoom in and
out and to annotate pages with your own
notes.
The glossary provides comprehensive Extension material is included for each
coverage of the language of the subject and chapter containing a variety of extra
explains tricky terminology. It is fully editable exercises and activities. Full worked
making it a powerful revision tool. solutions to this material are also provided.
viii
CASIO 9860-GII
Practice paper 2
Simultaneous and quadratic equations
1.5 Solving simultaneous linear equations
Tolerance Asian European
Low 30 80
Medium 50 40
High 40 20
Practice paper 2 Using a graphic display calculator
Practice exam papers will help you to fully Alternative GDC instructions for all material in
prepare for your examinations. Worked the book is given for the TI-84 Plus and Casio-
solutions can be found on the website 9860-GII calculators, so you can be sure you will
www.oxfordsecondary.co.uk/ibmathsl be supported no matter what calculator you use.
Powerpoint presentations cover detailed
worked solutions for the practice papers
in the book, showing common errors and
providing hints and tips.
(, 0) 0 (3, 0)
What's on the website?
Visit www.oxfordsecondary.co.uk/
ibmathsl for free access to the full
worked solutions to each and every
question in the Course Companion.
www.oxfordsecondary.co.uk/ibmathsl
also offers you a range of GDC
activities for the TI-Nspire to help
support your understanding.


Functions
CHAPTER OBJECTIVES:
2.1 Functions: domain, range, composite, identity and inverse functions
2.2 Graphs of functions, by hand and using GDC, their maxima and minima,
asymptotes, the graph of f −1(x)
2.3 Transformations of graphs, translations, reections, stretches and composite
transformations
Before you start
You should know how to: Skills check
1 Plot coordinates. y 1 a Plot these points on a coordinate plane.
e.g. Plot the 2 A(1, 3), B(5, −3), C(4, 4), D(−3, 2),
C D
points A(4, 0), B(0, −3), 1 E(2, −3), F(0, 3).
A y
C(−1, 1) and D(2, 1) –2 –1 0 1 2 3 4 x b Write down the 2 A
–1
on a coordinate plane. coordinates of 1.5
–2
–3 B points A to H H 1 E
–4
0.5
2 Substitute values into an expression.
C D B
e.g. Given x = 2, y = 3 and z = −5, 0 x
–2 –1 1 2 3
find the value of a 4x + 2y b y2 − 3z 0.5
–1
a 4x + 2y = 4(2) + 2(3) = 8 + 6 = 14 G
–0.5
b y2 − 3z = (3)2 −3(−5) = 9 + 15 = 24
F
–2
3 Solve linear equations.
2 Given that x = 4, y = 6 and z = −10, find
e.g. Solve 6 − 4x = 0

6 − 4x = 0 ⇒ 6 = 4x a 4x + 3y b z2 − 3y c y − z d

1.5 = x ⇒ x = 1.5 3 Solve
y 
4 Use your GDC to graph 6 a 3x − 6 = 6 b 5x + 7 = −3 c 
4
a function. 4 Graph these functions on your GDC
2
e.g. Graph within the given domain. Then
0 x
f (x) = 2x − 1, –3 ≤ x ≤ 3 –6 –4 –2 2 4 6 sketch the functions on paper.
–4 a y = 2x − 3, −4 ≤ x ≤ 7
–6
b y = 10 − 2x, −2 ≤ x ≤ 5
–8
5 Expand linear binomials. c y = x2 – 3, –3 ≤ x ≤ 3.
e.g. Expand (x + 3) (x − 2) 5 Expand
= x 2 + x − 6 a (x + 4) (x + 5) b (x − 1) (x − 3)
c (x + 5) (x − 4)
 Functions