cme project: algebra 2

ISBN 10: 1-256-74147-7ISBN 13: 978-1-256-74147-3

Cover Art: 9 Surf Studios; Alamy/RubberBall

Taken from:

CME Project: Algebra 2By the CME Project Development TeamCopyright ©2009 by Educational Development Center, Inc.Published by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

CME Common Core Additional Lessons: Algebra 2By the CME Project Development TeamCopyright ©2012 by Educational Development Center, Inc.Published by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

CME Project Development TeamLead Developer: Al Cuoco

Core Development Team: Anna Baccaglini-Frank, Jean Benson, Nancy Antonellis D’Amato, Daniel Erman, Brian Harvey, Wayne Harvey, Bowen Kerins, Doreen Kilday, Ryota Matsuura, Stephen Maurer, Sarah Sword, Audrey Ting, and Kevin Waterman

Others who contributed include Steve Benson, Paul D’Amato, Robert Devaney, Andrew Golay, Paul Goldenberg, Jane Gorman, C. Jud Hill, Eric Karnowski, Helen Lebowitz, Joseph Leverich, Melanie Palma, Mark Saul, Nina Shteingold, and Brett Thomas.

All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher.

This special edition published in cooperation with Pearson Learning Solutions.

All trademarks, service marks, registered trademarks, and registered service marks are the property of their respective owners and are used herein for identification purposes only.

Pearson Learning Solutions, 501 Boylston Street, Suite 900, Boston, MA 02116A Pearson Education Companywww.pearsoned.com

Printed in the United States of America

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000200010271661100

MD

The Center for Mathematics Education Project was developed at Education Development Center, Inc. (EDC) within the Center for Mathematics Education (CME), with partial support from the National Science Foundation.

Education Development Center, Inc.Center for Mathematics EducationNewton, Massachusetts

This material is based upon work supported by the National Science Foundation under Grant No. ESI-0242476, Grant No. MDR-9252952, and Grant No. ESI-9617369. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Cover Art: Courtesy of 9 Surf Studios; Jim Cummins/Corbis; Stockbyte/Getty Images, Inc.

Taken from:CME Project: Geometry, Algebra 2, Algebra 1, PrecalculusBy the CME Project Development TeamCopyright © 2009 by Education Development Center, Inc.Published by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

CME Common Core Additional Lessons: Geometry, Precalculus, Algebra 2, Algebra 1By the CME Project Development TeamCopyright © 2012 by Education Development Center, Inc.Published by Pearson Education, Inc.Upper Saddle River, New Jersey 07458

CME Project Development Team

Lead Developers: Al Cuoco and Bowen Kerins

Core Development Team: Anna Baccaglini-Frank, Jean Benson, Nancy Antonellis D’Amato, Daniel Erman, Paul Goldenberg, Brian Harvey, Wayne Harvey, Doreen Kilday, Ryota Matsuura, Stephen Maurer, Nina Shteingold, Sarah Sword, Audrey Ting, Kevin Waterman.

All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher.

This special edition published in cooperation with Pearson Learning Solutions.

All trademarks, service marks, registered trademarks, and registered service marks are the property of their respective owners and are used herein for identification purposes only.

Pearson Learning Solutions, 501 Boylston Street, Suite 900, Boston, MA 02116A Pearson Education Companywww.pearsoned.com

Printed in the United States of America

1 2 3 4 5 6 7 8 9 10 XXXX 17 15 14 13 12

000200010271285469

CP

ISBN 10: 1-256-51345-8ISBN 13: 978-1-256-51345-2

000200010271285469_CH00_FM_i-1.indd ii 6/18/12 7:08 AM000200010271661100_CH00_FM_pi-xxi.indd 2 7/10/12 3:43 AM

iii

Contents in BriefIntroduction to the CME Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

CME Project Student Handbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

Go Online . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi

Chapter 1 Fitting Functions to Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Chapter 2 Functions and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Chapter 3 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Chapter 4 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

Chapter 5 Exponential and Logarithmic Functions . . . . . . . . . . . . . . . . 422

Chapter 6 Graphs and Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

Chapter 7 Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612

Chapter 8 Introduction to Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . 706

TI-Nspire™ Technology Handbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804

Tables

Math Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829

Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830

Formulas From Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 831

Properties and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835

Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904

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Introduction to the CME Project

The CME Project, developed by EDC’s Center for Mathematics Education, is a new NSF-funded high school program, organized around the familiar courses of algebra 1, geometry, algebra 2, and precalculus . The CME Project provides teachers and schools with a third alter native to the choice between traditional texts driven by basic skill development and more pro gressive texts that have unfamiliar organizations . This program gives teachers the option of a problem-based, student-centered program, organized around the mathematical themes with which teachers and parents are familiar . Furthermore, the tremendous success of NSF-funded middle school programs has left a need for a high school program with similar rigor and pedagogy . The CME Project fills this need .

The goal of the CME Project is to help students acquire a deep understanding of mathematics . Therefore, the mathematics here is rigorous . We took great care to create lesson plans that, while challenging, will capture and engage students of all abilities and improve their mathematical achievement .

The Program’s Approach The organization of the CME Project provides students the time and focus they need to develop fundamental mathematical ways of thinking . Its primary goal is to develop in students robust mathematical proficiency .

• The program employs innovative instructional methods, developed over decades of classroom experience and informed by research, that help students master mathematical topics .

• One of the core tenets of the CME Project is to focus on developing students’ Habits of Mind, or ways in which students approach and solve mathematical challenges .

• The program builds on lessons learned from high-performing countries: develop an idea thoroughly and then revisit it only to deepen it; organize ideas in a way that is faithful to how they are organized in mathematics; and reduce clutter and extraneous topics .

• It also employs the best American models that call for grappling with ideas and problems as preparation for instruction, moving from concrete problems to abstractions and general theories, and situating mathematics in engaging contexts .

• The CME Project is a comprehensive curriculum that meets the dual goals of mathematical rigor and accessibility for a broad range of students .

About CMEEDC’s Center for Mathematics Education, led by mathematician and teacher Al Cuoco, brings together an eclectic staff of mathematicians, teachers, cognitive scientists, education researchers, curriculum developers, specialists in educational technology, and teacher educators, internationally known for leadership across the entire range of K–16 mathematics education . We aim to help students and teachers in this country experience the thrill of solving problems and building theories, understand the history of ideas behind the evolution of mathematical disciplines, and appreciate the standards of rigor that are central to mathematical culture .

CME PROJECT

iv CME Project • Algebra 2


National Advisory Board The National Advisory Board met early in the project, providing critical feedback on the instructional design and the overall organization . Members include

Richard Askey, University of Wisconsin Edward Barbeau, University of Toronto Hyman Bass, University of MichiganCarol Findell, Boston University Arthur Heinricher, Worcester Polytechnic InstituteRoger Howe, Yale UniversityBarbara Janson, Janson AssociatesKenneth Levasseur, University of Massachusetts, Lowell James Madden, Louisiana State University, Baton Rouge Jacqueline Miller, Education Development CenterJames Newton, University of MarylandRobert Segall, Greater Hartford Academy of Mathematics and Science Glenn Stevens, Boston UniversityHerbert Wilf, University of PennsylvaniaHung-Hsi Wu, University of California, Berkeley

Core Mathematical Consultants Dick Askey, Ed Barbeau, and Roger Howe have been involved in an even more substantial way, reviewing chapters and providing detailed and critical advice on every aspect of the program . Dick and Roger spent many hours reading and criticizing drafts, brainstorming with the writing team, and offering advice on everything from the logical organization to the actual numbers used in problems . We can’t thank them enough .

Teacher Advisory Board The Teacher Advisory Board for the CME Project was essential in help ing us create an effective format for our lessons that embodies the philosophy and goals of the program . Their debates about pedagogi cal issues and how to develop mathematical top ics helped to shape the distinguishing features of the curriculum so that our lessons work effective ly in the classroom . The advisory board includes

Jayne Abbas, Richard Coffey, Charles Garabedian, Dennis Geller, Eileen Herlihy, Doreen Kilday, Gayle Masse, Hugh McLaughlin, Nancy McLaughlin, Allen Olsen, Kimberly Osborne, Brian Shoemaker, and Benjamin Sinwell

Field-Test Teachers Our field-test teachers gave us the benefit of their classroom experi ence by teaching from our draft lessons and giv ing us extensive, critical feedback that shaped the drafts into realistic, teachable lessons . They shared their concerns, questions, challenges, and successes and kept us focused on the real world . Some of them even welcomed us into their classrooms as co-teachers to give us the direct experience with students that we needed to hone our lessons . Working with these expert professionals has been one of the most gratifying parts of the development—they are “highly qualified” in the most profound sense .

California Barney Martinez, Jefferson High School, Daly City; Calvin Baylon and Jaime Lao, Bell Junior High School, San Diego; Colorado Rocky Cundiff, Ignacio High School, Ignacio; Illinois Jeremy Kahan, Tammy Nguyen, and Stephanie Pederson, Ida Crown Jewish Academy, Chicago; Massachusetts Carol Martignette, Chris Martino, and Kent Werst, Arlington High School, Arlington; Larry Davidson, Boston University Academy, Boston; Joe Bishop and Carol Rosen, Lawrence High School, Lawrence; Maureen Mulryan, Lowell High School, Lowell; Felisa Honeyman, Newton South High School, Newton Centre; Jim Barnes and Carol Haney, Revere High School, Revere; New Hampshire Jayne Abbas and Terin Voisine, Cawley Middle School, Hooksett; New Mexico Mary Andrews, Las Cruces High School, Las Cruces; Ohio James Stallworth, Hughes Center, Cincinnati; Texas Arnell Crayton, Bellaire High School, Bellaire; Utah Troy Jones, Waterford School, Sandy; Washington Dale Erz, Kathy Greer, Karena Hanscom, and John Henry, Port Angeles High School, Port Angeles; Wisconsin Annette Roskam, Rice Lake High School, Rice Lake .

Special thanks go to our colleagues at Pearson, most notably Elizabeth Lehnertz, Joe Will, and Stewart Wood . The program benefits from their expertise in every way, from the actual mathematics to the design of the printed page .

Contributors to the CME Project

CME Project • Algebra 2 v


1Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1 .01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1 .02 Two Types of Definitions—

Closed-Form and Recursive Function Definitions . . . . . . . . . . . . . . 8 1 .03 Constant Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1 .04 Tables and Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1 .05 Deeper Differences—

Difference Tables for Polynomial Functions . . . . . . . . . . . . . . . . . . 29 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Fitting and Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 1 .06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1 .07 Fitting Lines to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1 .08 The Line of Best Fit, Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1 .09 The Line of Best Fit, Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Mid-Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

More About Recursive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 1 .10 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 1 .11 Monthly Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 1 .12 The Factorial Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Project: Using Mathematical Habits More on Monthly Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

1A

1B

1C

vi

Fitting Functions to Tables


2A

2B

2D

2C

Contents vii

2 Functions and Polynomials

Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

2 .0 Polynomial Basics—Optional Review . . . . . . . . . . . . . . . . . . . . . . . 90

About Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2 .01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2 .02 Getting Precise About Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2 .03 Algebra With Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2 .04 Inverses: Doing and Undoing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2 .05 Graphing Inverse and Piecewise Functions . . . . . . . . . . . . . . . . . . 125 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Making It Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 2 .06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 2 .07 Lagrange Interpolation—

Fitting Polynomial Functions to Tables . . . . . . . . . . . . . . . . . . . . . 143 2 .08 Agreeing to Disagree—

Finding Functions with Specific Values . . . . . . . . . . . . . . . . . . . . . 151 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Mid-Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Factors, Roots, and Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 2 .09 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 2 .10 Polynomial Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 2 .11 The Factor Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Advanced Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 2 .12 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 2 .13 Quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 2 .14 Factoring Cubics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 2 .15 Pippins and Cheese—More Factoring Techniques . . . . . . . . . . . . 188 2 .16 Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Project: Using Mathematical Habits Heron’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

Cumulative Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

cme09a2se_a08760HS1/18/08


3

viii Algebra 2

Complex Numbers

3C

3B

3A

Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Introduction to Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . 210 3 .01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 3 .02 Extending Number Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 3 .03 Making the Extension: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 3 .04 Extension to Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 3 .05 Reciprocals and Division—Using Complex Conjugates . . . . . . . . 230 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

The Complex Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 3 .06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 3 .07 Graphing Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 3 .08 Arithmetic in the Complex Plane— The Geometry Behind the Operations . . . . . . . . . . . . . . . . . . . . . . 246 3 .09 Magnitude and Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

Mid-Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Complex Numbers, Geometry, and Algebra . . . . . . . . . . . . . . 268 3 .10 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 3 .11 Multiplying Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 3 .12 Conjugates and Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 3 .13 Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

Project: Using Mathematical Habits Factoring a Sequence of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

!21 .


Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 4 .01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 4 .02 Solving Systems Systematically . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 4 .03 Solving Again, in Matrix Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 4 .04 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 4 .05 Basic Matrix Operations—Addition, Subtraction,

and Scalar Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 4 .06 Dot Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 4 .07 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 4 .08 Matrix Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 4 .09 Matrix Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

Mid-Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

Applications of Matrix Multiplication . . . . . . . . . . . . . . . . . . . . 370 4 .10 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 4 .11 Geometric Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 4 .12 Transition Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 4 .13 Probability Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 4 .14 Experiments and Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Project: Using Mathematical Habits More Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419


4 Linear Algebra

4A

4B

4C

Contents ix


Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

Working with Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 5 .01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 5 .02 Laws of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 5 .03 Zero and Negative Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 5 .04 Sequences and Operations—Inserting Arithmetic and Geometric Means . . . . . . . . . . . . . . . . . . . . . . . . . 438 5 .05 Defining Rational Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 5 .06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 5 .07 Graphs of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 454 5 .08 Tables of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 462 5 .09 Properties of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . 470 5 .10 Exponential Functions, One-to-One . . . . . . . . . . . . . . . . . . . . . . . 477 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

Mid-Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 5 .11 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 5 .12 Defining Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 5 .13 Laws of Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 5 .14 Graphing Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 504 5 .15 The Logarithmic Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

Project: Using Mathematical Habits Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

5

x Algebra 2

Exponential and Logarithmic Functions

5A

5B

5C


Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

Transforming Basic Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 6 .01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527 6 .02 More Basic Graphs— . . . . . . . . . . . . . . 530 6 .03 Translating Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 6 .04 Scaling and Reflecting Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

Affine Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558 6 .05 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 6 .06 Introducing Affine Transformations . . . . . . . . . . . . . . . . . . . . . . . . 562 6 .07 Transforming Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

Mid-Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581

Graphing Using Affine Transformations . . . . . . . . . . . . . . . . . . 582 6 .08 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 6 .09 Replacing the Axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586 6 .10 Advanced Affine Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 597 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

Project: Using Mathematical Habits A Group of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608


Graphs and Transformations

6

6A

6B

6C

Contents xi

x2 1 y2= 1, y = x3

4 x


7 Sequences and Series

7A

7B

7C

xii Algebra 2

7D

Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612

The Need to Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 7 .01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 7 .02 Gauss’s Method and Euclid’s Method . . . . . . . . . . . . . . . . . . . . . . 619 7 .03 Ways to Visualize Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 7 .04 The Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631

Sum Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632 7 .05 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 7 .06 Definite and Indefinite Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635 7 .07 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640 7 .08 Tables and Figurate Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652

Mid-Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

Arithmetic and Geometric Sequences and Series . . . . . . . . . 654 7 .09 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 7 .10 Arithmetic Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . 658 7 .11 Geometric Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . . . . 665 7 .12 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675 7 .13 Repeating Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685

Pascal’s Triangle and the Binomial Theorem . . . . . . . . . . . . . . 686 7 .14 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 7 .15 Pascal’s Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689 7 .16 The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698

Project: Using Mathematical Habits The Line of Best Fit Contains the Centroid . . . . . . . . . . . . . . . . . . . . . . 699

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705

∑

∑


Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706

8 .0 Right Triangle Trigonometry—Optional Review . . . . . . . . . . . . . . . 708

Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714 8 .01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 8 .02 Extending the Domain, Part 1—0° to 360° . . . . . . . . . . . . . . . . . . 718 8 .03 Extending the Domain, Part 2—All Real Numbers . . . . . . . . . . . . 724 8 .04 The Pythagorean . . . . . . . . . . . . . 730 8 .05 Solving Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 735 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 739

Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . 740 8 .06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741 8 .07 Graphing Cosine and Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743 8 .08 Graphing the Tangent Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 748 8 .09 The Angle-Sum Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 760

Mid-Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761

Applications to Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762 8 .10 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 8 .11 The Area of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766 8 .12 The Law of Sines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 8 .13 The Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 780 8 .14 Heron’s Formula—Using Side Lengths to

Find the Area of a Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790 Mathematical Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796

Project: Using Mathematical Habits Brahmagupta’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801

Cumulative Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802

8

8A

8B

8C

Contents xiii

Introduction to Trigonometry

Identity—cos2 u 1 sin2 u = 1


cme project: algebra 2

Documents