000200010271661095_CH00_FM_pi-xxii ii 6/28/12 5:14:05 PM6/28/12 5:14:05 PM000200010271661095_CH00_FM_pi-xxi.pdf 2

Cover Art: 9 Surf Studios; Mike Chew/Corbis; Ajosch/AFP/Getty Images

Taken from:

CME Project: Algebra 1By 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 1By 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

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, Nin 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

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 16 15 14 13 12

000200010271661095

MD

Cover Art: 9 Surf Studios; Mike Chew/Corbis; Ajosch/AFP/Getty Images

Taken from:

CME Project: Algebra 1By 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 1By 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

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, Nin 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 identifi cation 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 16 15 14 13 12

a

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.

ISBN 10: 1-256-74146-9ISBN 13: 978-1-256-74146-6

wto, Daniel Erman, Waterman

from the publisher.

iii

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

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

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

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

Chapter 1 Arithmetic to Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Chapter 2 Expressions and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Chapter 3 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Chapter 4 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

Chapter 5 Introduction to Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

Chapter 6 Exponents and Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

Chapter 7 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

Chapter 8 Quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694

TI-Nspire™ Technology Handbook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 790

Tables

Math Symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 800

Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 801

Formulas From Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 802

Properties and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806

Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853

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iv Algebra 1

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 1

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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, LowellJames Madden, Louisiana State University, Baton RougeJacqueline Miller, Education Development CenterJames Newton, University of MarylandRobert Segall, Greater Hartford Academy of Mathematics and ScienceGlenn 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 1 v

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

1.0 Habits of Mind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

The Tables of Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.02 Thinking About Negative Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.03 Extending the Addition Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.04 Extending the Multiplication Table . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.05 The Basic Rules of Arithmetic—Properties of Operations. . . . . . . . 28 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

The Number Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.07 Numbers Besides the Integers—Fractions . . . . . . . . . . . . . . . . . . . . 37 1.08 Decimals—Addresses on the Number Line . . . . . . . . . . . . . . . . . . . 41 1.09 Number Line Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.10 Number Line Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

The Algorithms of Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.11 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 1.12 Addition and Subtraction Algorithms . . . . . . . . . . . . . . . . . . . . . . . 61 1.13 Adding and Subtracting Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 67 1.14 Multiplication Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 1.15 Multiplying and Dividing Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . 77 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Project: Using Mathematical Habits Lo . . . ong Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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

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

1A

1B

1C

vi Algebra 1

Arithmetic to Algebra

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2A

2B

2D

2C

Contents vii

2 Expressions and EquationsChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.02 Modeling General Situations—Writing Expressions . . . . . . . . . . . . 93 2.03 Evaluating Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.04 Simplifying Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2.05 Rephrasing the Basic Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 2.06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 2.07 Reversing Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 2.08 Solving Equations by Backtracking . . . . . . . . . . . . . . . . . . . . . . . . . 126 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Solving Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 2.09 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 2.10 When Backtracking Does Not Work . . . . . . . . . . . . . . . . . . . . . . . . 138 2.11 The Basic Moves for Solving Equations . . . . . . . . . . . . . . . . . . . . . 143 2.12 Solutions of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 2.13 Focus on the Distributive Property . . . . . . . . . . . . . . . . . . . . . . . . . 153 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 2.14 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 2.15 Building Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 2.16 Solving Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 2.17 More Than One Variable— Solving in Terms of Each Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Project: Using Mathematical Habits Good Questions About Perfect Squares . . . . . . . . . . . . . . . . . . . . . . . . . 179

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Cumulative Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

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Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Introduction to Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 3.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 3.02 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 3.03 Distance and Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 3.04 Graphing Related Quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Statistical Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 3.05 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 3.06 Mean, Median, and Mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 3.07 Data Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 3.08 Paired Comparisons— Box-and-Whisker Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 3.09 Catogorical Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 3.10 Two-Variable Data— Scatter Plots . . . . . . . . . . . . . . . . . . . . . . . . . 253 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Equations and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 3.11 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 3.12 Equations as Point-Testers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 3.13 Graphing by Plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 3.14 Intersection of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Basic Graphs and Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 3.15 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 3.16 Two Basic Graphs: y = cx, y =

cx —

Direct and Inverse Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 3.17 Four More Basic Graphs: y = x2, y = x3, y = "x, y = u x u . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 3.18 Translating Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314

Project: Using Mathematical Habits Drawing With Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

3

viii Algebra 1

Graphs

3D

3C

3B

3A

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Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

All About Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 4.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 4.02 Pitch and Slope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 4.03 Rates of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 4.04 Collinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

Linear Equations and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 4.05 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 4.06 Equations of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 4.07 Jiffy Graphs: Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 4.08 Overtaking—Slope in Distance-Time Graphs. . . . . . . . . . . . . . . . . 363 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

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

Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 4.09 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 4.10 Solving Systems: Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 4.11 Slope and Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 4.12 Solving Systems: Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

Applications of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 4.13 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 4.14 Inequalities With One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398 4.15 Linear Trends in Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414

Project: Using Mathematical Habits Wireless Phone Plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418

Cumulative Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

4 Lines

4A

4B

4C

4D

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

Functions—The Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 5.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 5.02 Building Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428 5.03 Is It a Function? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 5.04 Naming Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 5.05 Function Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 5.06 Graphing Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

Functions, Graphs, and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 5.07 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 5.08 Constant Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 5.09 Recursive Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

Functions and Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 5.10 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 5.11 From Situations to Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 5.12 From Situations to Recursive Rules . . . . . . . . . . . . . . . . . . . . . . . . . 496 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

Project: Using Mathematical Habits Managing Money . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512

5

x Algebra 1

Introduction to Functions

5A

5B

5C

. . . and out comes 5 . . .

. . . or is it 6?

You put in 3 . . .

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49

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Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514

Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 6.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 6.02 Squares, Cubes, and Beyond— Some Basic Rules of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 6.03 More Basic Rules of Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 6.04 Zero and Negative Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 6.05 Scientifi c Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

Radicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 6.06 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 6.07 Defi ning Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 6.08 Arithmetic With Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 6.09 Conventions for Roots— Simplifi ed Forms for Radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 6.10 Rational and Irrational Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 561 6.11 Roots, Radicals, and the nth Root. . . . . . . . . . . . . . . . . . . . . . . . . . 568 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573

Exponential Expressions and Functions. . . . . . . . . . . . . . . . . . . 574 6.12 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 6.13 Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 6.14 Graphs of Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 584 6.15 Constant Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

Project: Using Mathematical Habits Calculating Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602

Cumulative Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604

Exponents and Radicals6

6A

6B

6C

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Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606

The Need for Identities—Equivalent Expressions . . . . . . . . . 608 7.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 7.02 Form and Function— Showing Expressions Are Equivalent . . . . . . . . . . . . . . . . . . . . . . . . 612 7.03 The Zero-Product Property — ab = 0 1 a = 0 or b = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619 7.04 Transforming Expressions— Introduction to Polynomial Factoring . . . . . . . . . . . . . . . . . . . . . . 625 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631

Polynomials and Their Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . 632 7.05 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 7.06 Anatomy of a Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636 7.07 Normal Form—Standard Representation of a Polynomial . . . . . . 643 7.08 Arithmetic With Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655

Factoring to Solve: Quadratics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 656 7.09 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 7.10 Factoring a Difference of Squares . . . . . . . . . . . . . . . . . . . . . . . . . 661 7.11 Factoring Monic Quadratics — When a = 1 in ax2 1 bx 1 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 7.12 Factoring by Completing the Square . . . . . . . . . . . . . . . . . . . . . . . 677 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687

Project: Using Mathematical Habits Differences of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692

7 Polynomials

7A

7B

7C

xii Algebra 1

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Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694

The Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696 8.01 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697 8.02 Making It Formal— Deriving the Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 699 8.03 Going the Other Way— Building a Quadratic Equation From Its Roots . . . . . . . . . . . . . . . . 706 8.04 Factoring Nonmonic Quadratics— When a 2 1 in ax2 1 bx 1 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716

Mid-Chapter Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717

Quadratic Graphs and Applications. . . . . . . . . . . . . . . . . . . . . . . 718 8.05 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719 8.06 Optimization—Finding Maximums and Minimums. . . . . . . . . . . . 721 8.07 Graphing Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727 8.08 Jiffy Graphs: Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745

Working With Quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746 8.09 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747 8.10 Solving by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 750 8.11 Inequalities With Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 755 8.12 Graphing Linear Inequalties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 8.13 Difference Tables of Quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . 773 Mathematical Refl ections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 781

Project: Using Mathematical Habits Iteration and Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782

Chapter Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784

Chapter Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786

Cumulative Review. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 788

8

8A

8B

8C

Contents xiii

Quadratics

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