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Mathematics in Action: Algebraic, Graphical, and Trigonometric Problem Solving, 4th Edition PDF

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Mathematics in Action Algebraic, Graphical, and Trigonometric Problem Solving Fourth Edition The Consortium for Foundation Mathematics Ralph Bertelle Columbia-Greene Community College Judith Bloch University of Rochester Roy Cameron SUNY Cobleskill Carolyn Curley Erie Community College—South Campus Ernie Danforth Corning Community College Brian Gray Howard Community College Arlene Kleinstein SUNY Farmingdale Kathleen Milligan Monroe Community College Patricia Pacitti SUNY Oswego Rick Patrick Adirondack Community College Renan Sezer LaGuardia Community College Patricia Shuart Polk State College—Winter Haven,Florida Sylvia Svitak Queensborough Community College Assad J.Thompson LaGuardia Community College Addison-Wesley Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Editorial Director, Mathematics: Christine Hoag Editor in Chief: Maureen O’Connor NOTICE: Content Editor: Courtney Slade This work is protected by U.S. Assistant Editor: Mary St. Thomas copyright laws and Senior Managing Editor: Karen Wernholm is provided solely for Production Project Manager: Beth Houston the use of college Senior Designer/Cover Designer: Barbara Atkinson instructors in review- Interior Designer: Studio Montage ing course materials Digital Assets Manager: Marianne Groth for classroom use. Production Coordinator: Katherine Roz Dissemination or sale Associate Producer: Christina Maestri of this work, or any Associate Marketing Manager: Tracy Rabinowitz part (including on the Marketing Coordinator: Alicia Frankel World Wide Web), Senior Author Support/Technology Specialist: Joe Vetere will destroy the integrity of the work Rights and Permissions Advisor: Michael Joyce and is not permitted. Senior Manufacturing Buyer: Carol Melville The work and materi- Production Management/Composition: PreMediaGlobal als from it should Cover photo: Eric Michaud/iStockphoto never be made avail- able to students Many of the designations used by manufacturers and sellers to distinguish their except by instructors products are claimed as trademarks. Where those designations appear in this book, using the accompany- and Addison-Wesley was aware of a trademark claim, the designations have been ing text in their printed in initial caps or all caps. classes. All recipi- ents of this work are Library of Congress Cataloging-in-Publication Data expected to abide by Mathematics in action: Algebraic, graphical, and trigonometric problem these restrictions solving / the Consortium for Foundation Mathematics. — 4th ed. and to honor the p. cm. intended pedagogical Includes index. purposes and the ISBN-13: 978-0-321-69861-2 (student ed.) needs of other ISBN-10: 0-321-69861-4 (student ed.) instructors who rely ISBN-13: 978-0-321-69290-0 (instructor ed.) on these materials. ISBN-10: 0-321-69290-X (instructor ed.) 1. Algebra—Textbooks. I. Consortium for Foundation Mathematics. II. Title: Algebraic, graphical, and trigonometric problem solving. QA152.3.M38 2012 512—dc22 2009052062 Copyright ©2012, 2008, 2004, 2001 Pearson Education, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. For information on obtaining per- mission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 75 Arlington Street, Suite 300, Boston, MA 02116, fax your request to 617-848-7047, or e-mail at http://www.pearsoned.com/legal/permissions.htm. 1 2 3 4 5 6 7 8 9 10—EB—14 13 12 11 10 Contents Preface xiv To the Student xx CHAPTER 1 Function Sense 1 Cluster 1 Modeling with Functions 1 Activity 1.1 Parking Problems 1 Objectives: 1.Identify inputand outputin situations involving two variable quantities. 2.Identify a functional relationship between two variables. 3.Identify the independentand dependentvariables. 4.Use a table to numerically representa functional relationship between two variables. 5.Write a function using function notation. Activity 1.2 Fill ’er Up 11 Objectives: 1.Determine the equation (symbolic representation) thatdefines a function. 2.Determine the domain and range of a function. 3.Identify the independentand the dependentvariables of a function. Activity 1.3 Graphically Speaking 18 Objectives: 1.Representa function verbally,symbolically,numerically,and graphically. 2.Distinguish between a discrete function and a continuous function. 3.Graph a function using technology. Activity 1.4 Stopping Short 28 Objectives: 1.Use a function as a mathematical model. 2.Determine when a function is increasing,decreasing,or constant. 3.Use the vertical line testto determine if a graph represents a function. Activity 1.5 Graphs Tell Stories 37 Objectives: 1.Describe in words whata graph tells you abouta given situation. 2.Sketch a graph thatbestrepresents the situation described in words. iii iv Contents 3.Identify increasing,decreasing,and constantparts of a graph. 4.Identify minimum and maximum points on a graph. WhatHave I Learned? 44 How Can I Practice? 46 Cluster 2 Linear Functions 51 Activity 1.6 Walking for Fitness 51 Objective: 1.Determine the average rate of change. Activity 1.7 Depreciation 58 Objectives: 1.Interpretslope as an average rate of change. 2.Use the formula to determine slope. 3.Discover the practical meaning of vertical and horizontal intercepts. 4.Develop the slope-interceptform of an equation of a line. 5.Use the slope-interceptformula to determine vertical and horizontal intercepts. 6.Determine the zeros of a function. Activity 1.8 A New Computer 69 Objectives: 1.Write a linear equation in the slope-interceptform,given the initial value and the average rate of change. 2.Write a linear equation given two points,one of which is the vertical intercept. 3.Use the point-slope form to write a linear equation given two points, neither of which is the vertical intercept. 4.Compare slopes of parallel lines. Activity 1.9 Skateboard Heaven 80 Objectives: 1.Write an equation of a line in standard form Ax + By = C. 2.Write the slope-interceptform of a linear equation given the standard form. 3.Determine the equation of a horizontal line. 4.Determine the equation of a vertical line. Activity 1.10 College Tuition 88 Objectives: 1.Constructscatterplots from sets of data pairs. 2.Recognize when patterns of points in a scatterplothave a linear form. 3.Recognize when the pattern in the scatterplotshows thatthe two variables are positively related or negatively related. 4.Estimate and draw a line of bestfitthrough a setof points in a scatterplot. 5.Use a graphing calculator to determine a line of bestfitby the least-squares method. Contents v 6.Measure the strength of the correlation (association) by a correlation coefficient. 7.Recognize thata strong correlation does notnecessarily imply a linear or a cause-and-effectrelationship. WhatHave I Learned? 99 How Can I Practice? 100 Cluster 3 Systems of Linear Equations,Inequalities,and Absolute Value Functions 105 Activity 1.11 Moving Out 105 Objectives: 1.Solve a system of 2 * 2linear equations numerically and graphically. 2.Solve a system of 2 * 2linear equations using the substitution method. 3.Solve an equation of the form ax + b = cx + dfor x. Activity 1.12 Healthy Lifestyle 117 Objectives: 1.Solve a 2 * 2linear system algebraically using the substitution method and the addition method. 2.Solve equations containing parentheses. Activity 1.13 Manufacturing Cell Phones 124 Objective: 1.Solve a 3 * 3linear system of equations. Activity 1.14 Earth Week 129 Objective: 1.Solve a linear system of equations using matrices. Activity 1.15 How Long Can You Live? 136 Objectives: 1.Solve linear inequalities in one variable numerically and graphically. 2.Use properties of inequalities to solve linear inequalities in one variable algebraically. 3.Solve compound inequalities algebraically. 4.Use interval notation to representa setof real numbers described by an inequality. Activity 1.16 Sales Commission 147 Objectives: 1.Graph a piecewise linear function. 2.Write a piecewise linear function to representa given situation. 3.Graph a function defined by y = ƒx - cƒ. WhatHave I Learned? 157 How Can I Practice? 158 Chapter 1 Summary 162 Chapter 1 Gateway Review 167 vi Contents CHAPTER 2 The Algebra of Functions 177 Cluster 1 Addition,Subtraction,and Multiplication of Polynomial Functions 177 Activity 2.1 Spending and Earning Money 177 Objectives: 1.Identify a polynomial expression. 2.Identify a polynomial function. 3.Add and subtractpolynomial expressions. 4.Add and subtractpolynomial functions. Activity 2.2 The Dormitory Parking Lot 188 Objectives: 1.Multiply two binomials using the FOIL method. 2.Multiply two polynomial functions. 3.Apply the property of exponents to multiply powers having the same base. Activity 2.3 Stargazing 199 Objectives: 1.Convertscientific notation to decimal notation. 2.Convertdecimal notation to scientific notation. 3.Apply the property of exponents to divide powers having the same base. 4.Apply the property of exponents a0 = 1,where a Z 0. 5.Apply the property of exponents a-n = 1 ,where a Z 0.and nis any n a real number. Activity 2.4 The Cube of a Square 207 Objectives: 1.Apply the property of exponents to simplify an expression involving a power to a power. 2.Apply the property of exponents to expand the power of a product. 3.Determine the nth rootof a real number. 4.Write a radical as a power having a rational exponentand write a base to a rational exponentas a radical. WhatHave I Learned? 216 How Can I Practice? 217 Cluster 2 Composition and Inverses of Functions 223 Activity 2.5 Inflated Balloons 223 Objectives: 1.Determine the composition of two functions. 2.Explore the relationship between f1g1x22and g1 f1x22. Activity 2.6 Finding a Bargain 228 Objective: 1.Solve problems using the composition of functions. Activity 2.7 Study Time 232 Objectives: 1.Determine the inverse of a function represented by a table of values. 2.Use the notation f -1to representan inverse function. Contents vii 3.Use the property f1 f -11x22 = f -1f1x22 = xto recognize inverse functions. 4.Determine the domain and range of a function and its inverse. Activity 2.8 Temperature Conversions 238 Objectives: 1.Determine the equation of the inverse of a function represented by an equation. 2.Describe the relationship between graphs of inverse functions. 3.Determine the graph of the inverse of a function represented by a graph. 4.Use the graphing calculator to produce graphs of an inverse function. WhatHave I Learned? 247 How Can I Practice? 248 Chapter 2 Summary 252 Chapter 2 Gateway Review 255 CHAPTER 3 Exponential and Logarithmic Functions 261 Cluster 1 Exponential Functions 261 Activity 3.1 The Summer Job 261 Objectives: 1.Determine the growth factor of an exponential function. 2.Identify the properties of the graph of an exponential function defined by y = bx,where b 7 1. 3.Graph an increasing exponential function. Activity 3.2 Half-Life of Medicine 269 Objectives: 1.Determine the decay factor of an exponential function. 2.Graph a decreasing exponential function. 3.Identify the properties of an exponential function defined by y = bx, where b 7 0and b Z 1. Activity 3.3 Cell Phones 277 Objectives: 1.Determine the growth and decay factor for an exponential function represented by a table of values or an equation. 2.Graph an exponential function defined by y = abx,where b 7 0 and b Z 1,a Z 0. 3.Determine the doubling and halving time. Activity 3.4 Population Growth 289 Objectives: 1.Determine the annual growth or decay rate of an exponential function represented by a table of values or an equation. 2.Graph an exponential function having equation y = a11 + r2x Z 0. Activity 3.5 Time Is Money 297 Objective: 1.Apply the compound interestand continuous compounding formulas to a given situation. viii Contents Activity 3.6 Continuous Growth and Decay 305 Objectives: 1.Discover the relationship between the equations of exponential functions defined by y = abtand the equations of continuous growth and decay exponential functions defined by y = aekt. 2.Solve problems involving continuous growth and decay models. 3.Graph base eexponential functions. Activity 3.7 Bird Flu 314 Objectives: 1.Determine the regression equation of an exponential function thatbest fits the given data. 2.Make predictions using an exponential regression equation. 3.Determine whether a linear or exponential model bestfits the data. WhatHave I Learned? 322 How Can I Practice? 323 Cluster 2 Logarithmic Functions 329 Activity 3.8 The Diameter of Spheres 329 Objectives: 1.Define logarithm. 2.Write an exponential statementin logarithmic form. 3.Write a logarithmic statementin exponential form. 4.Determine log and ln values using a calculator. Activity 3.9 Walking Speed of Pedestrians 337 Objectives: 1.Determine the inverse of the exponential function. 2.Identify the properties of the graph of a logarithmic function. 3.Graph the natural logarithmic function. Activity 3.10 Walking Speed of Pedestrians,continued 344 Objectives: 1.Compare the average rate of change of increasing logarithmic,linear, and exponential functions. 2.Determine the regression equation of a natural logarithmic function having the equation y = a + bln xthatbestfits a setof data. Activity 3.11 The Elastic Ball 354 Objectives: 1.Apply the log of a productproperty. 2.Apply the log of a quotientproperty. 3.Apply the log of a power property. 4.Discover change-of-base formula. Activity 3.12 Prison Growth 363 Objective: 1.Solve exponential equations both graphically and algebraically. WhatHave I Learned? 370 How Can I Practice? 372 Chapter 3 Summary 375 Chapter 3 Gateway Review 378 Contents ix CHAPTER 4 Quadratic and Higher-Order Polynomial Functions 385 Cluster 1 Introduction to Quadratic Functions 385 Activity 4.1 Baseball and the Willis Tower 385 Objectives: 1.Identify functions of the form f1x2 = ax2 + bx + cas quadratic functions. 2.Explore the role of cas itrelates to the graph of f1x2 = ax2 + bx + c. 3.Explore the role of aas itrelates to the graph of f1x2 = ax2 + bx + c. 4.Explore the role of bas itrelates to the graph of f1x2 = ax2 + bx + c. Note:a Z 0in Objectives 1–4. Activity 4.2 The ShotPut 395 Objectives: 1.Determine the vertex or turning pointof a parabola. 2.Identify the vertex as the maximum or minimum. 3.Determine the axis of symmetry of a parabola. 4.Identify the domain and range. 5.Determine the y-interceptof a parabola. 6.Determine the x-intercept(s) of a parabola using technology. 7.Interpretthe practical meaning of the vertex and intercepts in a given problem. Activity 4.3 Per Capita Personal Income 406 Objectives: 1.Solve quadratic equations graphically. 2.Solve quadratic equations numerically. 3.Solve quadratic inequalities graphically. Activity 4.4 Sir Isaac Newton 412 Objectives: 1.Factor expressions by removing the greatestcommon factor. 2.Factor trinomials using trial and error. 3.Use the Zero-Productprinciple to solve equations. 4.Solve quadratic equations by factoring. Activity 4.5 Price of Gold 419 Objective: 1.Solve quadratic equations by the quadratic formula. Activity 4.6 HeatIndex 428 Objectives: 1.Determine quadratic regression models using a graphing calculator. 2.Solve problems using quadratic regression models. Activity 4.7 Complex Numbers 433 Objectives: 1.Identify the imaginary uniti = 1-1. 2.Identify a complex number. 3.Determine the value of the discriminantb2 - 4ac.

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The third book of the Mathematics in Action series, Algebraic, Graphical, and Trigonometric Problem Solving, Fourth Edition, illustrates how mathematics arises naturally from everyday situations through updated and revised real-life activities and the accompanying practice exercises. Along with the
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