i

Jerald Murdock

Ellen Kamischke

Eric Kamischke

SECOND EDITION

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Project EditorElizabeth DeCarli

Project AdministratorTamar Wolins

Consulting EditorsHeather Dever, Joan Lewis, Nathaniel Lewis, Sharon Taylor

Editorial ConsultantsDavid Rasmussen

Editorial AssistantsChrista Edwards, Juliana Tringali

Mathematical Content ReviewersLarry Copes, Inver Grove Heights, Minnesota

James Wilson, Ph.D., University of Georgia, Athens, Georgia

Multicultural and Equity ReviewerSwapna Mukhopadhyay, Ph.D., Portland State University, Portland, Oregon

Teacher ReviewersRhonda Ball, Troy High School, Troy, Ohio

Joan Conway, Waseca High School, Waseca, Minnesota

Judy Hicks, Ralston Valley High School, Arvada, Colorado

Paul Mlakar, St. Mark’s School of Texas, Dallas, Texas

Jennifer North Morris, Santa Rosa High School, Santa Rosa, California

William Putnam, John Marshall High School, Rochester, Minnesota

Scientific Content ReviewerArt Fortgang, Mills High School, Millbrae, California

Accuracy CheckersDudley Brooks, Cavan Fang

Editorial Production SupervisorKristin Ferraioli

Production EditorHolly Rudelitsch

CopyeditorMargaret Moore

Production DirectorChristine Osborne

Production CoordinatorAnn Rothenbuhler

Cover DesignersJill Kongabel, Jeff Williams

Text DesignerMarilyn Perry

Art EditorJason Luz, Maya Mélenchuk

Photo ResearcherMarcy Lunetta

IllustratorsJuan Alvarez, Robert Arnow, Pamela Hobbs, William Pasini, Sue Todd

Technical Art ICC Macmillan Inc., Matt Perry, Jason Luz

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Executive EditorJosephine Noah

PublisherSteven Rasmussen

© 2010 by Key Curriculum Press. All rights reserved.

No part of this publication may be reproduced, stored in aretrieval system, or transmitted, in any form or by any means, electronic, photocopying, recording, or otherwise, without the prior written permission of the publisher.

®The Geometer’s Sketchpad, Dynamic Geometry, and Key Curriculum Press are registered trademarks of Key Curriculum Press. ™The Discovering Mathematics logo and Sketchpad are trademarks of Key Curriculum Press.

™Fathom Dynamic Data and the Fathom logo are trademarks of KCP Technologies.

All other trademarks are held by their respective owners.

This material is based upon work supported by the National Science Foundation under award number MDR9154410. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Key Curriculum Press1150 65th StreetEmeryville, CA 94608editorial@keypress.comwww.keypress.com

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 09

ISBN 978-1-55953-984-5

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Acknowledgments

C reating a textbook and its supplementary materials is a team effort involving many individuals and groups. We are especially grateful to thousands of

Discovering Advanced Algebra and Discovering Algebra teachers and students, to teachers who participated in the summer institutes and workshops, and to manuscript readers, all of whom provided suggestions, reviewed material, located errors, and most of all, encouraged us to continue with the project.

Our students, their parents, and our administrators at Interlochen Arts Academy have played an important part in the development of this book. Most importantly, we wish to thank Carol Murdock, our parents, and our children for their love, encouragement, and support.

As authors we are grateful to the National Science Foundation for supporting our initial technology-and-writing project that led to the 1998 publication of Advanced Algebra Through Data Exploration. This second edition of Discovering Advanced Algebra has been developed and shaped by what we learned during the writing and publication of earlier versions of this text and Discovering Algebra and by our work with so many students, parents, and teachers who were searching for a more meaningful algebra curriculum.

Over the course of our careers, many individuals and groups have been instrumental in our development as teachers and authors. The Woodrow Wilson National Fellowship Foundation provided the initial impetus for involvement in leading workshops. Publications and conferences produced by the National Council of Teachers of Mathematics and Teachers Teaching with Technology have guided the development of this curriculum. Individuals such as Ron Carlson, Helen Compton, Frank Demana, Arne Engebretsen, Paul Foerster, Christian Hirsch, Glenda Lappan, Richard Odell, Heinz-Otto Peitgen, James Sandefur, James Schultz, Dan Teague, Charles VonderEmbse, Bert Waits, and Mary Jean Winter have inspired us.

The development and production of Discovering Advanced Algebra has been a collaborative effort between the authors and the staff at Key Curriculum Press. We truly appreciate the cooperation and valuable contributions offered by the Editorial and Production Departments at Key Curriculum Press. Finally, a special thanks to Key’s publisher, Steven Rasmussen, for encouraging and publishing a technology-enhanced Discovering Mathematics series that offers groundbreaking content and learning opportunities.

Jerald Murdock

Ellen Kamischke

Eric Kamischke

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A Note from the Publisher

F or more than 30 years, Key Curriculum Press has developed mathematics materials for students and teachers. The mathematics that we learn and teach in school

has changed to reflect advances in technology, new discoveries in mathematics and science, and changing societal needs. However, in spite of the changes in mathematics and mathematics education, we have found one truth that has not changed: Students learn mathematics best when they understand the concepts behind it. Discovering Advanced Algebra asks you to discuss mathematics, engage in hands-on activities, analyze and develop mathematical models for real-world situations, and make connections between different mathematical representations.

This diverse approach to teaching provides access for students of all learning styles. Kinesthetic learners make connections between their movement and graphs and equations representing motion. Visual learners appreciate the engaging graphics and art as well as the emphasis on multiple representations of relations and functions. Auditory learners enhance their understanding as they explore and discuss concepts with other students. Whatever your learning style, you’ll benefit from deeply exploring mathematical concepts, discussing your ideas with other students, and applying your discoveries to solving real-world problems.

Investigations are at the heart of this book. Through the investigations, you will concretely explore interesting problems and generalize concepts. And if you forget a concept, formula, or procedure, you can always re-create it—because you developed it yourself the first time! You’ll find that this approach allows you to form a deep and conceptual understanding of advanced algebra topics. As you progress through the text, you’ll see that graphing calculators and other technologies are used to explore patterns and to make, test, and generalize conjectures. The use of technology allows you to explore more interesting problems and to move easily among different representations.

As Glenda Lappan, mathematics professor at Michigan State University and former president of the National Council of Teachers of Mathematics, said about the first edition of this book, “Students coming out of a year with this text . . . will know the mathematics they know in deeper, more flexible ways. They will have developed a set of mathematical habits of mind that will serve them very well as students or users of mathematics. They will emerge with a sense of mathematics as a search for regularity that allows prediction.”

If you are a student, we believe that what you learn this year will serve you well inlife. If you are a parent, we believe you will enjoy watching your student develop mathematical confidence. And if you are a teacher, we believe Discovering AdvancedAlgebra will greatly enrich your classroom. The professional team at Key Curriculum Press wishes you success and joy in the lifetime of mathematics ahead of you. We look forward to hearing about your experiences.

Steven Rasmussen, PublisherKey Curriculum Press

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A Note to Students from the Authors xiii

0.1 Pictures, Graphs, and Diagrams 2 Investigation: Polar Bear Crossing the Arctic 30.2 Symbolic Representation 7 Investigation: Problems, Problems, Problems 100.3 Organizing Information 14 Investigation: Who Owns the Zebra? 17Chapter 0 Review 22 Take Another Look 25 Assessing What You’ve Learned 26

Refreshing Your Skills for Chapter 1: Differences and Ratios 281.1 Recursively Defined Sequences 30 Investigation: Monitoring Inventory 331.2 Modeling Growth and Decay 40 Investigation: Looking for the Rebound 411.3 A First Look at Limits 48 Investigation: Doses of Medicine 481.4 Graphing Sequences 54 Investigation: Match Them Up 54Exploration: Graphs of Sequences 62 Activity: Repeat After Me 621.5 Loans and Investments 64 Investigation: Life’s Big Expenditures 64 Project: The Pyramid Investment Plan 68Exploration: Refining the Growth Model 69 Activity: Cornering the Market 70Chapter 1 Review 73 Take Another Look 76 Assessing What You’ve Learned 77

CHAPTER

1Sequences 27

CHAPTER

0Problem Solving with Algebra 1

Contents

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Refreshing Your Skills for Chapter 2: Measures of Central Tendency and Dot Plots 79

2.1 Box Plots 81 Investigation: Pulse Rates 84Exploration: Precision, Accuracy, and Significant Figures 89 Activity: Measuring Up 892.2 Measures of Spread 93 Investigation: A Good Design 942.3 Histograms and Percentile Ranks 103 Investigation: Eating on the Run 106 Project: Stem-and-Leaf Plots 112Exploration: Census Microdata 113 Activity: Different Ways to Analyze Data 114Chapter 2 Review 116 Take Another Look 119 Assessing What You’ve Learned 120

Refreshing Your Skills for Chapter 3: Linear Relationships 1223.1 Linear Equations and Arithmetic Sequences 124 Investigation: Match Point 1263.2 Revisiting Slope 131 Investigation: Balloon Blastoff 1323.3 Fitting a Line to Data 138 Investigation: The Wave 141 Project: Talkin’ Trash 1443.4 The Median-Median Line 145 Investigation: Airline Schedules 148 Project: Counting Forever 1513.5 Prediction and Accuracy 152 Investigation: Spring Experiment 153Exploration: Residual Plots and Least Squares 159 Activity: A Good Fit? 1593.6 Linear Systems 161 Investigation: Population Trends 1623.7 Substitution and Elimination 167 Investigation: What’s Your System? 169Chapter 3 Review 174 Mixed Review 176 Take Another Look 179 Assessing What You’ve Learned 180

CHAPTER

3Linear Models and Systems 121

CHAPTER

2Describing Data 78

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Refreshing Your Skills for Chapter 4: Solving Equations 1824.1 Interpreting Graphs 184 Investigation: Graph a Story 1864.2 Function Notation 190 Investigation: To Be or Not to Be (a Function) 192 Project: Step Functions 1974.3 Lines in Motion 198 Investigation: Movin’ Around 1984.4 Translations and the Quadratic Family 205 Investigation: Make My Graph 2074.5 Reflections and the Square Root Family 213 Investigation: Take a Moment to Reflect 213Exploration: Rotation as a Composition of Transformations 220 Activity: Revolution 2204.6 Dilations and the Absolute-Value Family 221 Investigation: The Pendulum 2244.7 Transformations and the Circle Family 229 Investigation: When Is a Circle Not a Circle? 2314.8 Compositions of Functions 237 Investigation: Looking Up 238 Project: Piecewise Pictures 244Chapter 4 Review 245 Take Another Look 247 Assessing What You’ve Learned 248

Refreshing Your Skills for Chapter 5: Working with Positive Square Roots 250

5.1 Exponential Functions 252 Investigation: Radioactive Decay 252 Project: The Cost of Living 2585.2 Properties of Exponents and Power Functions 259 Investigation: Properties of Exponents 2595.3 Rational Exponents and Roots 266 Investigation: Getting to the Root 266 Project: Powers of 10 2745.4 Applications of Exponential and Power Equations 2755.5 Building Inverses of Functions 280 Investigation: The Inverse 2805.6 Logarithmic Functions 287 Investigation: Exponents and Logarithms 2875.7 Properties of Logarithms 293 Investigation: Properties of Logarithms 294

CHAPTER

4Functions, Relations, and Transformations 181

CHAPTER

5Exponential, Power, and Logarithmic Functions 249

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5.8 Applications of Logarithms 300 Investigation: Cooling 303 Project: Income by Gender 306Exploration: The Number e 307 Activity: Continuous Growth 307 Project: All About e 308Chapter 5 Review 309 Take Another Look 312 Assessing What You’ve Learned 312

Refreshing Your Skills for Chapter 6: Properties of Real Numbers 3146.1 Matrix Representations 318 Investigation: Chilly Choices 3196.2 Matrix Operations 325 Investigation: Find Your Place 3286.3 Solving Systems with Inverse Matrices 335 Investigation: The Inverse Matrix 3376.4 Row Reduction Method 3446.5 Systems of Inequalities 352 Investigation: Paying for College 3526.6 Linear Programming 360 Investigation: Maximizing Profit 360 Project: Nutritional Elements 366Chapter 6 Review 367 Mixed Review 369 Take Another Look 373 Assessing What You’ve Learned 374

Refreshing Your Skills for Chapter 7: Polynomial Expressions 3767.1 Polynomial Degree and Finite Differences 378 Investigation: Free Fall 3817.2 Equivalent Quadratic Forms 386 Investigation: Rolling Along 3897.3 Completing the Square 395 Investigation: Complete the Square 3967.4 The Quadratic Formula 403 Investigation: How High Can You Go? 405 Project: Calculator Program for the Quadratic Formula 408

CHAPTER

6Matrices and Linear Systems 313

CHAPTER

7Quadratic and Other Polynomial Functions 375

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7.5 Complex Numbers 409 Investigation: Complex Arithmetic 411 Project: The Mandelbrot Set 4157.6 Factoring Polynomials 416 Investigation: The Box Factory 4187.7 Higher-Degree Polynomials 423 Investigation: The Largest Triangle 4247.8 More About Finding Solutions 430Chapter 7 Review 437 Take Another Look 439 Assessing What You’ve Learned 440

Refreshing Your Skills for Chapter 8: The Distance Formula 4428.1 Using the Distance Formula 444 Investigation: Bucket Race 4448.2 Circles and Ellipses 452 Investigation: A Slice of Light 4588.3 Parabolas 463 Investigation: Fold a Parabola 4678.4 Hyperbolas 470 Investigation: Passing By 474Exploration: Constructing the Conic Sections 479 Activity: From Circles to the Ellipse 4798.5 The General Quadratic 481 Investigation: Systems of Conic Equations 4858.6 Introduction to Rational Functions 490 Investigation: The Breaking Point 4908.7 Graphs of Rational Functions 498 Investigation: Predicting Asymptotes and Holes 4998.8 Operations with Rational Expressions 505 Project: Cyclic Hyperbolas 511Chapter 8 Review 512 Take Another Look 514 Assessing What You’ve Learned 514

CHAPTER

8Conic Sections and Rational Functions 441

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CHAPTER

9Series 515

Refreshing Your Skills for Chapter 9: Arithmetic and Geometric Sequences 516

9.1 Arithmetic Series 518 Investigation: Arithmetic Series Formula 5209.2 Infinite Geometric Series 525 Investigation: Infinite Geometric Series Formula 5269.3 Partial Sums of Geometric Series 532 Investigation: Geometric Series Formula 533Exploration: Seeing the Sum of a Series 538 Activity: A Geometric Series 538Chapter 9 Review 540 Mixed Review 541 Take Another Look 543 Assessing What You’ve Learned 544

Refreshing Your Skills for Chapter 10: Basic Probability 54610.1 Randomness and Probability 548 Investigation: Flip a Coin 549Exploration: Geometric Probability 558 Activity: The Coin Toss Problem 55810.2 Counting Outcomes and Tree Diagrams 560 Investigation: The Multiplication Rule 561Exploration: The Law of Large Numbers 569 Activity: A Repeat Performance 56910.3 Mutually Exclusive Events and Venn Diagrams 571 Investigation: Addition Rule 57310.4 Random Variables and Expected Value 579 Investigation: “Dieing” for a Four 57910.5 Permutations and Probability 586 Investigation: Name That Tune 58610.6 Combinations and Probability 595 Investigation: Winning the Lottery 59810.7 The Binomial Theorem and Pascal’s Triangle 602 Investigation: Pascal’s Triangle and Combination Numbers 602Chapter 10 Review 611 Take Another Look 614 Assessing What You’ve Learned 614

CHAPTER

10Probability 545

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CHAPTER

11Applications of Statistics 615

Refreshing Your Skills for Chapter 11: Statistics 61611.1 Experimental Design 618 Investigation: Designing a Study 61911.2 Probability Distributions 624 Investigation: Pencil Lengths 625 Project: Simpson’s Paradox 63311.3 Normal Distributions 634 Investigation: The Normal Curve 637Exploration: Normally Distributed Data 643 Activity: Is This Normal? 64311.4 z-Values and Confidence Intervals 645 Investigation: Keeping Score 645Exploration: Prediction Intervals 652 Activity: Quality Control 65311.5 Bivariate Data and Correlation 655 Investigation: Looking for Connections 656 Project: Correlation vs. Causation 66311.6 The Least Squares Line 664 Investigation: Spin Time 665Exploration: Nonlinear Regression 672 Activity: How Does It Fit? 672Chapter 11 Review 674 Take Another Look 677 Project: Making It Fit 677 Assessing What You’ve Learned 678

Refreshing Your Skills for Chapter 12: Special Right Triangles 68012.1 Right Triangle Trigonometry 682 Investigation: Steep Steps 68612.2 The Law of Sines 690 Investigation: Oblique Triangles 69012.3 The Law of Cosines 698 Investigation: Around the Corner 70012.4 Extending Trigonometry 705 Investigation: Extending Trigonometric Functions 70512.5 Introduction to Vectors 712 Investigation: Vector Addition and Subtraction 71312.6 Parametric Equations 720 Investigation: Parametric Walk 722 Project: Catapult 728Chapter 12 Review 729Take Another Look 732Assessing What You’ve Learned 732

CHAPTER

12Trigonometry 679

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CHAPTER

13Trigonometric Functions 733

Refreshing Your Skills for Chapter 13: Geometry of the Circle 73413.1 Defining the Circular Functions 736 Investigation: Paddle Wheel 73713.2 Radian Measure 744 Investigation: A Radian Protractor 744Exploration: Circular Functions 754 Activity: Tracing Parent Graphs 75413.3 Graphing Trigonometric Functions 757 Investigation: The Pendulum II 760 Project: Design a Picnic Table 76713.4 Inverses of Trigonometric Functions 768 Investigation: Exploring the Inverses 76913.5 Modeling with Trigonometric Equations 775 Investigation: A Bouncing Spring 777 Project: Sunrise, Sunset 78213.6 Fundamental Trigonometric Identities 783 Investigation: Pythagorean Identities 78413.7 Combining Trigonometric Functions 789 Investigation: Sound Wave 790Exploration: Polar Coordinates 796 Activity: Rose Curves 797Chapter 13 Review 799 Mixed Review 801 Take Another Look 804 Assessing What You’ve Learned 805

Selected Hints and Answers 806

Glossary 852

Index 866

Photo Credits 879

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Jerald Murdock

Eric Kamischke

Ellen Kamischke

The goal of this stage of your mathematical journey is to develop advanced algebraic tools and the mathematical power that will help you participate

fully in life beyond school. On this journey you will make connections between algebra and the world around you.

Important decision-making situations will confront you in life, and your ability to use mathematics and algebra can help you make informed decisions. You’ll need skills that can evolve and be adapted to new situations. You’ll need to interpret numerical information and use it as a basis for making decisions. And you’ll need to find ways to solve problems that arise in real life, not just in textbooks. Success in algebra is a gateway to many varied career opportunities.

Hopefully, you’ve already found out that learning algebra is more than memorizing facts, theories, and procedures. With your teacher as a guide, you’ll learn algebra by doing mathematics. You’ll make sense of important algebraic concepts, learn essential skills, and discover how to use algebra.

During the Investigations, you will talk about algebra, share ideas, and learn from and with your fellow group members. You will also work and communicate with your teammates to strengthen your understanding of the mathematical concepts. To enjoy and gain respect in your role as a learner, honor differences among classmates, listen carefully when others are sharing, stay focused during the process, be responsible and respectful, and share your own ideas and suggestions.

The right technology can help you explore new ideas and answer questions that come up along the way. Using a graphing calculator, you will be able to manipulate large amounts of data quickly so that you can see the overall picture. Throughout the text you can refer to Calculator Notes for information that will help you use this tool. Technology will play an important role in your life and future career. Learning to use your graphing calculator efficiently and being able to interpret its output will prepare you to use other technologies successfully in situations to come.

The book itself will be a guide, leading you to explore ideas and ponder questions. Read it carefully—with paper, pencil, and calculator close at hand—and take good notes. Concepts and problems you have encountered before can help you solve new problems. Work through the Examples and answer the questions that are asked along the way. Some Exercises require a great deal of thought. Don’t give up. Make a solid attempt at each problem that is assigned. Sometimes you’ll make corrections and fill in details later, after you discuss a problem in class. Features called Project, Improving Your . . . Skills, and Take Another Look will challenge you to extend your learning and to apply it in creative ways.

Just as this book is your guide, your notebook can be a log of your travels through Algebra 2. In it you will record your notes and your work. You may also want to keep a journal of your personal impressions along the way. And just as every trip results in a photo album, you can place some of your especially notable accomplishments in a portfolio that highlights your trip. Collect pieces of work in your portfolio as you go, and refine the contents as you make progress on your journey. Each chapter ends with Assessing What You’ve Learned. This feature

A Note to Students from the Authors

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suggests ways to review your progress and prepare for what comes next: organizing your notebook, writing in your journal, updating your portfolio, and other ways to reflect on what you have learned.

You should expect hard work, interesting activities, and occasional frustration. Yet, as you gain more algebra skills, you’ll overcome obstacles and be rewarded with a deeper understanding of mathematics, an increased confidence in your own problem-solving abilities, and the opportunity to be creative. From time to time, look back to reflect on where you have been. We hope that your journey through Discovering Advanced Algebra will be a meaningful and rewarding experience.

And now it is time to begin. You are about to discover some pretty fascinating things.

Jerald Murdock

Ellen Kamischke

Eric Kamischke

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