Advanced functions 12

593 Pages·2008·242.293 MB·English

by  Wayne Erdman ... [et al.].

Authors Consultants Advisors
Wayne Erdman Assessment Consultant Kirsten Boucher
B.Math., B.Ed. Jacqueline Hill Durham District School Board
Toronto District School Board Durham District School Board Karen Frazer
Ottawa-Carleton District School Board
Antonietta Lenjosek
Technology Consultant
B.Sc., B.Ed. Janine LeBlanc
Dan Ciarmoli
Ottawa Catholic School Board Whitby, Ontario
Hamilton-Wentworth District
Roland W. Meisel School Board Anthony Silva
B.Sc., B.Ed., M.Sc. York Region District School Board
Port Colborne, Ontario Mathematical Processes Consultant Ken Stewart
Susan Siskind York Region District School Board
Jacob Speijer
Toronto, Ontario
B.Eng., M.Sc.Ed., P.Eng.
District School Board of Niagara Advisory Panel
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Wayne Erdman Derrick Driscoll
Contributing Authors Toronto District School Board Thames Valley District School Board
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Contents
Preface Chapter 3
Rational Functions 145
Chapter 1
Prerequisite Skills 146
Polynomial Functions 1
3.1 Reciprocal of a Linear Function 148
Prerequisite Skills 2
Extension: Asymptotes and the TI-83 Plus
1.1 Power Functions 4 or TI-84 Plus Graphing Calculator 156
1.2 Characteristics of Polynomial 3.2 Reciprocal of a Quadratic Function 157
Functions 15
3.3 Rational Functions of the Form
1.3 Equations and Graphs of Polynomial f(x) (cid:2) _ax (cid:3)_ b 168
Functions 30 cx (cid:3) d
1.4 Transformations 42 3.4 Solve Rational Equations and
1.5 Slopes of Secants and Average Inequalities 177
Rate of Change 53 3.5 Making Connections With Rational
1.6 Slopes of Tangents and Instantaneous Functions and Equations 186
Rate of Change 65 Review 192
Review 74 Practice Test 194
Practice Test 78 Chapters 1 to 3 Review 196
Task: Create Your Own Water Park 80 Task: ZENN and Now 198
Chapter 2 Chapter 4
Polynomial Equations and Inequalities 81 Trigonometry 199
Prerequisite Skills 82 Prerequisite Skills 200
2.1 The Remainder Theorem 84 4.1 Radian Measure 202
2.2 The Factor Theorem 94 4.2 Trigonometric Ratios and Special Angles 211
2.3 Polynomial Equations 104 4.3 Equivalent Trigonometric Expressions 220
2.4 Families of Polynomial Functions 113 4.4 Compound Angle Formulas 228
2.5 Solve Inequalities Using Technology 123 4.5 Prove Trigonometric Identities 236
2.6 Solve Factorable Polynomial Extension: Use The Geometer’s Sketchpad®
Inequalities Algebraically 132 to Sketch and Manipulate Three-Dimensional
Review 140 Structures in a Two-Dimensional
Representation 242
Practice Test 142
Review 244
Task: Can You Tell Just by Looking? 144
Practice Test 246
Task: Make Your Own Identity 248
iv MHR • Advanced Functions • Contents
Chapter 5 Chapter 7
Trigonometric Functions 249 Tools and Strategies for Solving
Prerequisite Skills 250 Exponential and Logarithmic Equations 361
5.1 Graphs of Sine, Cosine, and Tangent Prerequisite Skills 362
Functions 252 7.1 Equivalent Forms of Exponential
5.2 Graphs of Reciprocal Trigonometric Equations 364
Functions 261 7.2 Techniques for Solving Exponential
5.3 Sinusoidal Functions of the Form Equations 370
f(x) (cid:2) a sin [k(x (cid:4) d)] (cid:3) c and 7.3 Product and Quotient Laws of
f(x) (cid:2) a cos [k(x (cid:4) d)] (cid:3) c 270 Logarithms 378
Extension: Use a Graphing Calculator 7.4 Techniques for Solving Logarithmic
to Fit a Sinusoidal Regression to Equations 387
Given Data 280
7.5 Making Connections: Mathematical
5.4 Solve Trigonometric Equations 282 Modelling With Exponential and
5.5 Making Connections and Instantaneous Logarithmic Equations 393
Rate of Change 290 Review 408
Review 300 Practice Test 410
Practice Test 302 Task: Make Your Own Slide Rule 412
Chapters 4 and 5 Review 304
Task: Predators and Prey 306 Chapter 8
Combining Functions 413
Chapter 6 Prerequisite Skills 414
Exponential and Logarithmic Functions 307 8.1 Sums and Differences of Functions 416
Prerequisite Skills 308 8.2 Products and Quotients of Functions 429
6.1 The Exponential Function and 8.3 Composite Functions 439
Its Inverse 310
8.4 Inequalities of Combined Functions 450
6.2 Logarithms 323
8.5 Making Connections: Modelling
6.3 Transformations of Logarithmic With Combined Functions 461
Functions 331
Review 472
6.4 Power Law of Logarithms 341 Practice Test 474
6.5 Making Connections: Logarithmic Chapters 6 to 8 Review 476
Scales in the Physical Sciences 349 Task: Modelling a Damped Pendulum 478
Review 356 Course Review 479
Practice Test 358
Prerequisite Skills Appendix 484
Task: Not Fatal 360
Technology Appendix 505
Answers 524
Glossary 586
Index 595
Credits 600
Contents • MHR v
Preface
McGraw-Hill Ryerson Advanced Functions 12 is designed for students planning
to qualify for college or university. The book introduces new mathematical
principles while providing a wide variety of applications linking the
mathematical theory to real situations and careers.
Text Organization
Chapter 1 generalizes concepts of polynomial functions and introduces the
process of using secants and tangents to analyse rates of change. These concepts
are then integrated as appropriate throughout other chapters of the text.
In Chapter 2, you will combine your equation-solving skills with principles
of polynomial functions to solve polynomial equations and inequalities.
Chapter 3 focuses on properties of rational functions.
Chapter 4 extends your understanding of trigonometry by defi ning
trigonometric ratios of any angle using radians for angle measure. These
concepts are then used in Chapter 5 to analyse trigonometric functions.
Chapters 6 and 7 provide opportunities for you to explore and apply
concepts of exponents and logarithms. In Chapter 8, concepts from all
seven preceding chapters are integrated in the topic of combining functions.
Mathematical Processes
Reasoning and Proving This text integrates the seven mathematical processes: problem solving,
Representing Selecting Tools reasoning and proving, refl ecting, selecting tools and computational
Problem Solving strategies, connecting, representing, and communicating. These processes
Connecting Reflecting are interconnected and are used throughout the course. Some examples
Communicating and exercises are fl agged with a mathematical processes graphic to show
you which processes are involved in solving the problem.
Chapter Features
The Chapter Opener introduces what you will learn in the chapters. It
includes a list of the specifi c curriculum expectations that the chapter covers.
Prerequisite Skills reviews key skills from previous mathematical courses
and previous chapters in this book that are needed to be successful with the
current chapter. Examples and further practice are given in the Prerequisite
Skills Appendix on pages 484 to 504. The Chapter Problem is introduced
at the end of the Prerequisite Skills. Questions related to this problem are
identifi ed in the exercises, and the Chapter Problem Wrap-Up is found at
the end of the Chapter Review.
vi MHR • Advanced Functions • Preface
Prerequisite Skills 9. a) Use a unit circle, similar to the one shown, to Distance Between Two Points
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200 MHR • Advanced Functions • Chapter 4 Prerequisite Skills • MHR 201
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done using graphing calculators or dynamic geometry 1 2
2
software, but in most instances a choice of tools is given. 34
5
Worked Examples provide model solutions that show b) Descri6be any patterns you see in the values of y as x increases.
how the new concepts are used. They often include more
than one method, with and without technology. New
mathematical terms are highlighted and defi ned in context.
Refer to the Glossary on pages 586 to 594 for a full list of
defi nitions of mathematical terms used in the text.
The Key Concepts box summarizes the ideas in the lesson,
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and the Communicate Your Understanding questions allow KEY CONCEPTS
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more involved problems that require you to use several
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concepts from the preceding chapters. Some tasks may cats
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and 8.
A Practice Test is also included at the end of each chapter.
A Course Review follows the task at the end of Chapter 8. This comprehensive
selection of questions will help you to determine if you are ready for the
fi nal examination.
Preface • MHR vii
Assessment
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Solution The text shows examples of the use of the TI-83 Plus
a) t_Oann1 xa ginr aYp1h.ing calculator, enter the expression and TI-84 Plus graphing calculators, The Geometer’s
bx Tx)Ot UhU∈ hsr∈EneewseaD s eeen[ eno[i0p it n0pttgstYe,ohhphd ,eor2 imee le o2nioa8 πnrwtIhπiyf ]sntnp i]o tno tt oosorhoeh(cid:3) ecfifiYrecze tcn si cito u2(cid:2)nvtenigu nnrce.ttror1 setetrgaPsf a0rtrspa orisslai. cet hnpeaetl ascci a tensnluaettrp d iipreasatoop si xptYnnaenriirooctg .msoi tn xny,iutxa o i h atnmi(cid:3)xtnmoenht a i(cid:3)deZiad8ltn teei.y Ta en 1lttodryleht0yiuje rgeu.x mr x csvP i (cid:3)atniar(cid:3) nntetl eehs3 0 rse.v 2. f1a72l... STaalkrhgeeee tn bcTerhIawp- 8sa ay9dts ® Tttehi,mt eaa nng(Cidrua AmFdSae c)t 1 haa2olpc muplell aviDcteaoly,trn i doiasenm tusai.sic elF edSod trf aok ttreies ycctshioctnmrsoi™qpkuu eSetseos ra ft rthwea at re.
shown in worked examples.
264 MHR • Advanced Functions•Chat
Extension
These optional features extend the concepts of the preceding section using
technology or advanced mathematical techniques. They provide you with
interesting activities to challenge and engage you in new mathematical ideas.
Connections
This margin item includes
connections between topics in the course or to topics learned previously
interesting facts related to topics in the examples and exercises
suggestions for how to use the Internet to help you solve problems or to
research or collect information—direct links are provided on the Advanced
Functions 12 page on the McGraw-Hill Ryerson Web site.
Answers
Answers to the Prerequisite Skills, numbered sections, Chapter Review,
and Practice Test are provided on pages 524 to 585.
Responses for the Investigate, Communicate Your Understanding, and
Achievement Check questions and Chapter Problem Wrap-Up are provided
in the McGraw-Hill Ryerson Advanced Functions 12 Teacher’s Resource.
Full solutions to all questions, including proof questions, are on the
McGraw-Hill Ryerson Advanced Functions 12 Solutions CD-ROM.
viii MHR • Advanced Functions • Preface
1
Chapter
Polynomial Functions
Linear and quadratic functions are members of a larger
group of functions known as polynomial functions.
In business, the revenue, profi t, and demand can be
modelled by polynomial functions. An architect may
design bridges or other structures using polynomial
curves, while a demographer may predict population
trends using polynomial functions.
This chapter focuses on the properties and key
features of graphs of polynomial functions and their
transformations. You will also be introduced to the
concepts of average and instantaneous rate of change.
By the end of this chapter, you will
recognize a polynomial expression and the determine an equation of a polynomial function
equation of a polynomial function, and identify that satisfi es a given set of conditions (C1.7)
linear and quadratic functions as examples of investigate properties of even and odd polynomial
polynomial functions (C1.1) functions, and determine whether a given
compare, through investigation, the numeric, polynomial function is even, odd, or neither (C1.9)
graphical, and algebraic representations of investigate and recognize examples of a variety
polynomial functions (C1.2) of representations of average rate of change and
describe key features of the graphs of polynomial instantaneous rate of change (D1.1, D1.2, D1.3, D1.6)
functions (C1.3) calculate and interpret average rates of change of
distinguish polynomial functions from sinusoidal functions, given various representations of the
and exponential functions (C1.4) functions (D1.4)
investigate connections between a polynomial make connections between average rate of change
function given in factored form and the x-intercepts and the slope of a secant, and instantaneous rate
of its graph, and sketch the graph of a polynomial of change and the slope of a tangent (D1.7)
function given in factored form using its key recognize examples of instantaneous rates of
features (C1.5) change arising from real-world situations, and
investigate the roles of the parameters a, k, d, and c make connections between instantaneous rates
in functions of the form y (cid:2) af [k(x (cid:3) d)] (cid:4) c and of change and average rates of change (D1.5)
describe these roles in terms of transformations solve real-world problems involving average and
on the functions f(x) (cid:2) x 3 and f(x) (cid:2) x4 (C1.6) instantaneous rate of change (D1.9)
1
Prerequisite Skills
Function Notation b)
x y
1. Determine each value for the function (cid:3)1 (cid:3)8
f(x) (cid:2) (cid:3)4x (cid:4) 7. 0 (cid:3)2
a) f(0) b) f(3) c) f((cid:3)1) 1 (cid:3)1
d) f ( _1 ) e) f((cid:3)2x) f) f(3x) 2 5
2
3 7
2. Determine each value for the function
4 13
f(x) (cid:2) 2x2 (cid:3) 3x (cid:4) 1.
5 20
a) f(0) b) f(3) c) f((cid:3)1)
d) f ( _1 ) e) 3f(2x) f) f(3x) c) x y
2 (cid:3)4 (cid:3)12
Slope and y-intercept of a Line (cid:3)3 (cid:3)5
(cid:3)2 0
3. State the slope and the y-intercept of each line.
(cid:3)1 3
a) y(cid:2) 3x (cid:4) 2 b) 4y (cid:2) 6 (cid:3) 2x
0 4
c) 5x (cid:3) y (cid:4) 7 (cid:2) 0 d) y(cid:4) 6 (cid:2) (cid:3)5(x (cid:4) 1)
1 3
e) (cid:3)(x (cid:4) 4) (cid:2) 2(y (cid:3) 3)
2 0
Equation of a Line
Domain and Range
4. Determine an equation for the line that
satisfi es each set of conditions. 6. State the domain and range of each function.
a) The slope is 3 and the y-intercept is 5. Justify your answer.
b) The x-intercept is (cid:3)1 and the y-intercept is 4. a) y(cid:2) 2(x (cid:3) 3)2 (cid:4) 1
c) The slope is (cid:3)4 and the line passes through b) y(cid:2) _1_
x (cid:4) 5
the point (7, 3).
c) y(cid:2) √(cid:3)1 (cid:3)(cid:3) 2(cid:3)x
d) The line passes through the points (2, (cid:3)2)
and (1, 5).
Quadratic Functions
Finite Diff erences 7. Determine the equation of a quadratic function
that satisfi es each set of conditions.
5. Use fi nite differences to determine if each
a) x-intercepts 1 and (cid:3)1, y-intercept 3
function is linear, quadratic, or neither.
b) x-intercept 3, and passing through the
a)
x y
point (1, (cid:3)2)
(cid:3)2 (cid:3)7 c) x-intercepts (cid:3) _1 and 2, y-intercept (cid:3)4
(cid:3)1 (cid:3)5 2
0 (cid:3)3
1 (cid:3)1
2 1
3 3
4 5
2 MHR • Advanced Functions • Chapter 1