Pre-Calculus 12 Textbook PDF

McGraw-Hill Ryerson Pre-Calculus 12 McGraw-Hill Ryerson Pre-Calculus Authors Bruce McAskill, B.Sc., B.Ed., M.Ed., Ph.D. Mathematics Consultant, Victoria, British Assessment Consultant...

McGraw-Hill Ryerson Pre-Calculus 12 McGraw-Hill Ryerson Pre-Calculus Authors Bruce McAskill, B.Sc., B.Ed., M.Ed., Ph.D. Mathematics Consultant, Victoria, British Assessment Consultant Chris Zarski, B.Ed., M.Ed. Wetaskiwin Regional Division No. 11, Advisors 12 John Agnew, School District 63 (Saanich), British Columbia Columbia Alberta Len Bonifacio, Edmonton Catholic Separate Wayne Watt, B.Sc., B.Ed., M.Ed. School District No. 7, Alberta Pedagogical Consultant Mathematics Consultant, Winnipeg, Manitoba Katharine Borgen, School District 39 Scott Carlson, B.Ed., B. Sc. Eric Balzarini, B.Sc., B.Ed., M.Ed. (Vancouver) and University of British Golden Hills School Division No. 75, School District 35 (Langley), British Columbia Columbia, British Columbia Alberta Blaise Johnson, B.Sc., B.Ed. Renée Jackson, University of Alberta, Alberta School District 45 (West Vancouver), British Aboriginal Consultant Gerald Krabbe, Calgary Board of Education, Columbia Chun Ong, B.A., B.Ed. Alberta Manitoba First Nations Education Resource Ron Kennedy, B.Ed. Gail Poshtar, Calgary Roman Catholic Centre, Manitoba Mathematics Consultant, Edmonton, Alberta Separate School District, Alberta Terry Melnyk, B.Ed. Differentiated Instruction Consultant Harold Wardrop, Brentwood College Edmonton Public Schools, Alberta Heather Granger School, Mill Bay (Independent), British Prairie South School Division No. 210, Columbia Chris Zarski, B.Ed., M.Ed. Saskatchewan Wetaskiwin Regional Division No. 11, Francophone Advisors Alberta Mario Chaput, Pembina Trails School Gifted and Career Consultant Division, Manitoba Contributing Author Rick Wunderlich School District 83 (North Okanagan/ Luc Lerminiaux, Regina School Division Gail Poshtar, B.Ed. Shuswap), British Columbia No. 4, Saskatchewan Calgary Roman Catholic Separate School District, Alberta Inuit Advisor Math Processes Consultant Senior Program Consultants Reg Fogarty Christine Purse, Mathematics Consultant, School District 83 (North Okanagan/ British Columbia Bruce McAskill, B.Sc., B.Ed., M.Ed., Ph.D. Mathematics Consultant, Victoria, British Shuswap), British Columbia Métis Advisor Columbia Technology Consultants Greg King, Northern Lights School Division Wayne Watt, B.Sc., B.Ed., M.Ed. Ron Kennedy No. 69, Alberta Mathematics Consultant, Winnipeg, Manitoba Mathematics Consultant, Edmonton, Alberta Technical Advisor Ron Coleborn Darren Kuropatwa, Winnipeg School School District 41 (Burnaby), British Division #1, Manitoba Columbia Toronto Montréal Boston Burr Ridge, IL Dubuque, IA Madison, WI New York San Francisco St. Louis Bangkok Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan New Delhi Santiago Seoul Singapore Sydney Taipei COPIES OF THIS BOOK McGraw-Hill Ryerson MAY BE OBTAINED BY Pre-Calculus 12 CONTACTING: Copyright © 2012, McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. McGraw-Hill Ryerson Ltd. All rights reserved. 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VICE-PRESIDENT, EDITORIAL: Beverley Buxton MATHEMATICS PUBLISHER: Jean Ford PROJECT MANAGER: Janice Dyer DEVELOPMENTAL EDITORS: Maggie Cheverie, Jackie Lacoursiere, Jodi Rauch MANAGER, EDITORIAL SERVICES: Crystal Shortt SUPERVISING EDITOR: Jaime Smith COPY EDITOR: Julie Cochrane PHOTO RESEARCH & PERMISSIONS: Linda Tanaka EDITORIAL ASSISTANT: Erin Hartley EDITORIAL COORDINATION: Jennifer Keay MANAGER, PRODUCTION SERVICES: Yolanda Pigden PRODUCTION COORDINATOR: Jennifer Hall INDEXER: Belle Wong INTERIOR DESIGN: Pronk & Associates COVER DESIGN: Michelle Losier ART DIRECTION: Tom Dart, First Folio Resource Group Inc. ELECTRONIC PAGE MAKE-UP: Tom Dart, Kim Hutchinson, First Folio Resource Group Inc. COVER IMAGE: Courtesy of Ocean/Corbis Acknowledgements There are many students, teachers, and administrators who the publisher, authors, and consultants of Pre-Calculus 12 wish to thank for their thoughtful comments and creative suggestions about what would work best in their classrooms. Their input and assistance have been invaluable in making sure that the Student Resource and its related Teacher’s Resource meet the needs of students and teachers who work within the Western and Northern Canadian Protocol Common Curriculum Framework. Reviewers Kristi Allen R. Paul Ledet Wetaskiwin Regional Public Schools School District 63 (Saanich) Alberta British Columbia Karen Bedard Amos Lee School District 22 (Vernon) School District 41 (Burnaby) British Columbia British Columbia Robert Burzminski Jay Lorenzen Medicine Hat Catholic Board of Education Horizon School District No. 205 Alberta Saskatchewan Tracy Connell Deanna Matthews School District 57 (Prince George) Edmonton Public Schools British Columbia Alberta Janis Crighton Dick McDougall Lethbridge School District No. 51 Calgary Catholic School District Alberta Alberta Cynthia L. Danyluk Yasuko Nitta Light of Christ Catholic School Division No. 16 School District 38 (Richmond) Saskatchewan British Columbia Kelvin Dueck Catherine Ramsay School District 42 (Maple Ridge/Pitt Meadows) River East Transcona School Division British Columbia Manitoba Pat Forsyth Dixie Sillito Elk Island Public Schools Prairie Rose School Division No. 8 Alberta Alberta Barbara Gajdos Jill Taylor Calgary Catholic School District Fort McMurray Public School District Alberta Alberta Murray D. Henry John J. Verhagen Prince Albert Catholic School Board No. 6 Livingstone Range School Division No. 68 Saskatchewan Alberta Christopher Hunter Jimmy Wu Curriculum and Instructional Services Centre School District 36 (Surrey) British Columbia British Columbia Jane Koleba School District 61 (Greater Victoria) British Columbia Contents A Tour of Your Textbook..................... vii Unit 2 Trigonometry........................ 162 Unit 1 Transformations and Chapter 4 Trigonometry and the Functions.............................................. 2 Unit Circle....................................................................164 4.1 Angles and Angle Measure........................... 166 Chapter 1 Function Transformations.................... 4 4.2 The Unit Circle.................................................... 180 1.1 Horizontal and Vertical Translations...............6 4.3 Trigonometric Ratios....................................... 191 1.2 Reflections and Stretches................................ 16 4.4 Introduction to Trigonometric 1.3 Combining Transformations............................. 32 Equations.............................................................. 206 1.4 Inverse of a Relation.......................................... 44 Chapter 4 Review......................................................... 215 Chapter 1 Review............................................................ 56 Chapter 4 Practice Test............................................. 218 Chapter 1 Practice Test................................................ 58 Chapter 5 Trigonometric Functions and Graphs..........................................................................220 Chapter 2 Radical Functions................................. 60 5.1 Graphing Sine and Cosine Functions........ 222 2.1 Radical Functions and Transformations..... 62 5.2 Transformations of Sinusoidal 2.2 Square Root of a Function..................................78 Functions.............................................................. 238 2.3 Solving Radical Equations Graphically........ 90 5.3 The Tangent Function..................................... 256 Chapter 2 Review............................................................ 99 5.4 Equations and Graphs of Trigonometric Chapter 2 Practice Test............................................. 102 Functions.............................................................. 266 Chapter 5 Review......................................................... 282 Chapter 3 Polynomial Functions........................104 Chapter 5 Practice Test............................................. 286 3.1 Characteristics of Polynomial Functions.............................................................. 106 Chapter 6 Trigonometric Identities...................288 6.1 Reciprocal, Quotient, and Pythagorean 3.2 The Remainder Theorem............................... 118 Identities............................................................... 290 3.3 The Factor Theorem........................................ 126 6.2 Sum, Difference, and Double-Angle 3.4 Equations and Graphs of Polynomial Identities............................................................... 299 Functions.............................................................. 136 6.3 Proving Identities.............................................. 309 Chapter 3 Review......................................................... 153 6.4 Solving Trigonometric Equations Chapter 3 Practice Test............................................. 155 Using Identities.................................................. 316 Chapter 6 Review......................................................... 322 Unit 1 Project Wrap-Up.................... 157 Chapter 6 Practice Test............................................. 324 Cumulative Review, Chapters 1–3.. 158 Unit 2 Project Wrap-Up.................... 325 Unit 1 Test........................................ 160 Cumulative Review, Chapters 4–6.. 326 Unit 2 Test........................................ 328 iv MHR Contents Unit 3 Exponential and Unit 4 Equations and Functions..... 426 Logarithmic Functions..................... 330 Chapter 9 Rational Functions.............................428 Chapter 7 Exponential Functions......................332 9.1 Exploring Rational Functions Using 7.1 Characteristics of Exponential Transformations................................................. 430 Functions.............................................................. 334 9.2 Analysing Rational Functions...................... 446 7.2 Transformations of Exponential 9.3 Connecting Graphs and Rational Functions.............................................................. 346 Equations.............................................................. 457 7.3 Solving Exponential Equations................... 358 Chapter 9 Review......................................................... 468 Chapter 7 Review......................................................... 366 Chapter 9 Practice Test............................................. 470 Chapter 7 Practice Test............................................. 368 Chapter 10 Function Operations.......................472 Chapter 8 Logarithmic Functions......................370 10.1 Sums and Differences of Functions.......... 474 8.1 Understanding Logarithms........................... 372 10.2 Products and Quotients of Functions...... 488 8.2 Transformations of 10.3 Composite Functions....................................... 499 Logarithmic Functions.................................... 383 Chapter 10 Review...................................................... 510 8.3 Laws of Logarithms......................................... 392 Chapter 10 Practice Test........................................... 512 8.4 Logarithmic and Exponential Equations.............................................................. 404 Chapter 11 Permutations, Combinations, Chapter 8 Review......................................................... 416 and the Binomial Theorem...................................514 11.1 Permutations...................................................... 516 Chapter 8 Practice Test............................................. 419 11.2 Combinations...................................................... 528 Unit 3 Project Wrap-Up.................... 421 11.3 Binomial Theorem............................................. 537 Cumulative Review, Chapters 7–8.. 422 Chapter 11 Review...................................................... 546 Chapter 11 Practice Test........................................... 548 Unit 3 Test........................................ 424 Unit 4 Project Wrap-Up.................... 549 Cumulative Review, Chapters 9–11.. 550 Unit 4 Test........................................ 552 Answers............................................. 554 Glossary............................................. 638 Index.................................................. 643 Credits............................................... 646 Contents MHR v  A Tour of Your Textbook Unit Opener Unit 2 Trigonometry Each unit begins with a two-page spread. The first page of the Unit Trigonometry is used extensively in our daily lives. For example, will you listen to music today? Most songs are recorded digitally Opener introduces what you will and are compressed into MP3 format. These processes all involve trigonometry. learn in the unit. The Unit Project is Your phone may have a built-in Global Positioning System (GPS) that uses trigonometry to introduced on the second page. Each tell where you are on Earth’s surface. GPS satellites send a signal to receivers such as the one in your phone. The signal Unit Project helps you connect the from each satellite can be represented using trigonometric functions. The receiver uses these signals to determine the math in the unit to real life using location of the satellite and then uses trigonometry to calculate your position. Unit 2 Project Applications of Trigonometry experiences that may interest you. In this project, you will explore trigonometric functions, and you applications. angle measurement, trigonometri will explore how they relate to c equations, and past and present In Chapter 4, you will research Looking Ahead the history of units of angle measure such as radians. In Chapter 5, you will gather information In this unit, you will solve problems about the application of periodic to the field of communications. functions involving... Finally, in Chapter 6, you will explore trigonometric identities in Mach the use of angle measures and the unit circle numbers. trigonometric functions and their At the end of the unit, you will graphs choose at least one of the following the proofs of trigonometric identities Research the history, usage, and options: relationship of types of units for the solutions of trigonometric equations Examine an application of periodic angle measure. functions in electronic communicat investigate why it is an appropriate ions and model. Apply the skills you have learned about trigonometric identities to Explore the science of forensics supersonic travel. through its applications of trigonometry. 162 MHR Unit 2 Trigonometry Unit 2 Trigonometry MHR 163 Project Corner boxes throughout the chapters help you gather information for your project. Some Project Corner boxes include questions to help you to begin thinking about and discussing your project. The Unit Projects in Units 1, 3, and 4 provide an opportunity for you to choose a single Project Wrap-Up at the end of the unit. The Unit Project in Unit 2 is designed for you to complete in pieces, chapter by chapter, throughout the unit. At the end of the unit, a Project Wrap-Up allows you to consolidate your work in a meaningful presentation. Chapter Opener CHAPTER Each chapter begins with a two-page 10 Function spread that introduces you to what Operations Throughout your mathematics courses, you have learned methods of interpreting you will learn in the chapter. a variety of functions. It is important to understand functional relationships to the between variables since they apply physical fields of engineering, business, sciences, and social sciences, to a few. name The opener includes information The relationships that exist between variables can be complex and can combining two or more functions. this chapter, you will learn how involve In to use about a career that uses the skills to various combinations of functions model real-world phenomena. covered in the chapter. A Web Link Did You Know? Career Link Wave interference occurs when two travel through the same medium The net amplitude at each point or more waves at the same time. of the resulting of the individual In 2004, researchers from universities Columbia, Alberta, Ontario, and in British Québec, as allows you to learn more about wave is the sum of the amplitudes in wave pools Council of waves. For example, waves interfere well as from the National Research and in noise-cancelling headphones. Canada, began using the Advanced Source (ALLS) to do fascinating Laser Light experiments. quadrillionth this career and how it involves the The ALLS is a femtosecond (one facility (10-15) of a second) multi-beam laser used in the dynamic investigation of matter in disciplines such as biology, medicine, such as mathematics you are learning. chemistry, and physics. Universities offer the University of British Columbia degrees students the chance to obtain advanced research. leading to careers involving laser We b Link Visuals on the chapter opener go about a career involving laser research, earn more a To learn spread show other ways the skills Key Terms gcentres and follow to www.mcgrawhill.ca/school/learnin composite function the links. Chapter 10 MHR 473 472 MHR Chapter 10 and concepts from the chapter are used in daily life. vi MHR A Tour of Your Textbook Three-Part Lesson 3.1 3. Compare the sets of graphs from step 1 to each other. Describe similarities their Each numbered section is organized and differences as in step 2. Characteristics of 4. Consider the cubic, quartic, and quintic graphs from step 1. Which Polynomial Functions graphs are y = x? similar to the graph of y = -x? in a three-part lesson: Investigate, Focus on... identifying polynomial functions analysing polynomial functions y = x2? y = -x2? Explain how they are similar. Link the Ideas, and Check Your A cross-section of a honeycomb has a pattern with one hexagon by six more hexagons. Surrounding surrounded Reflect and Respond 5. a) How are the graphs and functions equations of linear, cubic, and quintic these is similar? Understanding. a third ring of 12 hexagons, and b) How are the graphs and equations so on. The quadratic function f (r) models the of quadratic and quartic total functions similar? number of hexagons in a honeycomb, Did Yo Youu Know? where c) Describe the relationship r is the number of rings. Then, between the end behaviours of you can use Falher, Alberta is known graphs and the degree of the correspondin the the graph of the function to solve g function. questions as the “Honey Capital of 6. What is the relationship between about the honeycomb pattern. Canada.” The Fahler Honey the sign of the leading coefficient Festival is an annual event of a function equation and the end A quadratic function that models behaviour of the graph of this pattern will that celebrates beekeeping the function? be discussed later in this section. and francophone history in the region. 7. What is the relationship between the constant term in a function equation and the position of the graph of the function? 8. What is the relationship between the minimum and maximum Investigate Investigate Graphs of Polynomial Functions numbers of x-intercepts of the graph of the function? of a function with the degree Materials 1. Graph each set of functions on a different set of coordinate graphing calculator using graphing axes Link the Ideas technology. Sketch the results. or computer with The Investigate consists of short graphing software Type of Function linear Set A y=x Set B y = -3x Set C y=x+1 Set D The degree of a polynomial function exponent of the greatest power in one variable, x, is n, the of the variable x. The coefficient the greatest power of x is the leading of polynomial function quadratic y = x2 y = -2x coefficient, a , and the term 2 y = x2 - 3 y = x2 - x - 2 steps often accompanied by cubic whose value is not affected by the n a function of the form y = x3 y = -4x3 y = x3 - 4 y = x3 + 4x2 + x - 6 variable is the constant term, a 0. f (x) = anxn + a xn - 1 n-1 quartic y = x4 y = -2x 4 y = x4 + 2 In this chapter, the coefficients + an - 2 x n - 2 + … + a x 2 y = x + 2x - 7x2 - 8x + 12 4 3 an to a1 and the What power of x is 2 quintic constant a0 are restricted to integral associated with a ? + a1x + a0, where y = x5 y = -x5 y = x5 - 1 values. y = x5 + 3x 4 - 5x3 - 15x2 + 4x 0  n is a whole number + 12  x is a variable illustrations. It is designed to help 2. Compare the graphs within each set from step 1. Describe their  the coefficients an to similaritiesand differences in terms of a0 are real numbers end behaviour end behaviour examples are the behaviour of the degree of the function in one f (x) = 2x - 1, y-values of a function variable, x Recall that the degree f (x) = x2 + x - 6, and constant term you build your own understanding as |x| becomes very large leading coefficient number of x-intercepts of a polynomial is the greatest exponent of x. y = x3 + 2x2 - 5x - 6 106 MHR Chapter 3 of the new concept. 3.1 Characteristics of Polynomial Functions MHR 107 The Reflect and Respond questions help you to analyse and communicate what you are learning and draw conclusions. Link the Ideas The explanations in this section help you connect the concepts explored in the Investigate to the Examples. The Examples and worked Solutions show how to use the concepts. The Example 4 Determine Exact Trigonometri c Values for Angles Determine the exact value for each Method 2: Use a Quotient Identity sin 105° tan 105° = __ with Sine and Cosine expression. cos 105° π Examples include several tools to a) sin _ b) tan 105° 12 sin (60° + 45°) = ___ cos (60° + 45°) = sin ___ 60° cos 45° +___ cos 60° sin 45° Use sum identities with special angles. Could you use a difference of cos 60° cos 45° - sin 60° sin 45° angles identity here? Solution __ __ help you understand the work. ( )( ) ( )( ) __ √3 _ _ √2 1 _√2 a) Use the difference identity + _ for sine with two special angles. = ___2 2 __ __ 2 2 __ __ ( )( ) ( )( ) For example, because _ π 3π π =_ -_ 2π use _ π -_ 1 _ _ √2 √3 _ _ √2 2 2 - 2 12 12 ,. 12 4 6 2 π sin _ = sin _ π ( π ) π __ Words in green font help you -_ The special angles _ π __ 12 and _ could also √6 √2 4 6 be used. 3 4 _ +_ π π 4__ = __ 4__ = sin _ cos _ - cos π _ π _ Use sin (A - B) 4 __ 6 sin √2 _ √6 __ __ 4 6 = sin A cos B - cos A sin B. -_ = _ ( )( ) ( )( ) √2 _ √3 - _ √2 _ 1 4 __ 4 __ think through the steps. =_ √6 2 __ 2 √2 __ 2 2 ( √6 + √2 = __ 4 __ 4 __ __ √2 - √6 )( ) -_ __ __ 4__ 4 __ √6 + √2 √6 - √2 __ = __ __ __ How could you verify that = How could you verify this answer with √2 - √6 this is the same answer as in Different methods of solving the 4 a calculator? Method 1? b) Method 1: Use the Difference Your Turn Identity for Tangent Rewrite tan 105° as a difference Use a sum or difference identity of special angles. to find the exact values of tan 105° = tan (135° - 30°) Are there other ways of writing 105° as the a) cos 165° 11π b) tan _ same problem are sometimes Use the tangent difference identity, tan 135° - tan_ tan (135° - 30°) = ___ sum or difference of two special tan (A - B) = ___ 30° tan A - tan B 1 + tan A tan B. angles? 12 1 + tan 135° tan 30° Key Ideas shown. One method may make = ___ -1 - _ 1__ √3 You can use the sum and difference identities to simplify expressions and to ( ) 1__ determine exact trigonometric values 1 + (-1) _ for some angles. √3 Sum Identities more sense to you than the -1 - _ = __ 1-_ √3 1__ √3 1__ Simplify. sin (A + B) = sin A cos B + cos cos (A + B) = cos A cos B - sin A sin B A sin B Difference Identities sin (A - B) = sin A cos B - cos cos (A - B) = cos A cos B + sin A sin B A sin B tan A + tan B ( tan (A + B) = ___ )( tan A - tan B others. Or, you may develop = __ -1 - _ 1-_ __ √3 1__ √3 1__ _ -√3 __ -√__ 3 Multiply ) __ numerator and denominator by -√3. 1 - tan A tan B The double-angle identities are tan (A - B) = ___ 1 + tan A tan B special cases of the sum identities two angles are equal. The double-angle in three forms using the Pythagorean when the identity for cosine can be expressed √3 + 1 identity, cos2 A + sin2 A = 1. = __ another method that means more __ How could you rationalize the 1 - √3 denominator? Double-Angle Identities sin 2A = 2 sin A cos A cos 2A = cos2 A - sin2 A tan 2A = __ 2 tan A cos 2A = 2 cos2 A - 1 1 - tan2 A to you. cos 2A = 1 - 2 sin2 A 304 MHR Chapter 6 6.2 Sum, Difference, and Double-Angle Identities MHR 305 Each Example is followed by a l Your Turn. The Your Turn allows you to explore d d your understanding of the skills covered in the Example. After all the Examples are presented, the Key Ideas summarize the main new concepts. A Tour of Your Textbook MHR vii Check Your Understanding Key Ideas Practise: These questions allow you to check your understanding An exponential function of the form y y = cx, c > 0, c ≠ 1, y 8 8  is increasing for c > 1 y = 2x () x  is decreasing for 0 < c < 1 6 y= 1 _ 6 2 of the concepts. You can often do the first few questions by    has a domain of {x | x ∈ R} has a range of {y | y > 0, y ∈ R} has a y-intercept of 1 4 2 4 2 checking the Link the Ideas notes or by following one of the  has no x-intercept