College algebra

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■ Linear Functions A linear function is a function of the form f 1x2 = b + mx ■ The graph of f is a line with slope m and y-intercept b y y b b x x Ï=b ■ Ï=b+mx Exponential Functions An exponential function is a function of the form f 1x2 = Ca x ■ The graph of f has one of the shapes shown ■ If a 1, then a is called the growth factor and r = a - is called the growth rate ■ If a 1, then a is called the decay factor and r = a - is called the decay rate y y 1 Ï=a˛, a>1 ■ x x Ï=a˛, 00 ■ x y=a(x-h)™+k a>0, h>0, k>0 Power Functions A power function is a function of the form f 1x = Cx p ■ Graphs of some power functions are shown Positive powers y y y x x x x Ï=x£ Ï=≈ y Ï=x¢ Ï=x∞ Fractional powers y y y x x £x Ï=œ ∑ Ï=œ∑ x y x x ∞x Ï= œ∑ ¢x Ï= œ ∑ Negative powers y y y x Ï= x1 y x Ï=≈ x Ï=x£ x Ï= x¢ College Algebra CONCEPTS AND CONTEXTS ABOUT THE AUTHORS JAMES STEWART received his MS from Stanford University and his PhD from the University of Toronto He did research at the University of London and was influenced by the famous mathematician George Polya at Stanford University Stewart is Professor Emeritus at McMaster University and is currently Professor of Mathematics at the University of Toronto His research field is harmonic analysis and the connections between mathematics and music James Stewart is the author of a bestselling calculus textbook series published by Brooks/Cole, Cengage Learning, including Calculus, Calculus: Early Transcendentals, and Calculus: Concepts and Contexts; a series of precalculus texts; and a series of high-school mathematics textbooks LOTHAR REDLIN grew up on Vancouver Island, received a Bachelor of Science degree from the University of Victoria, and received a PhD from McMaster University in 1978 He subsequently did research and taught at the University of Washington, the University of Waterloo, and California State University, Long Beach He is currently Professor of Mathematics at The Pennsylvania State University, Abington Campus His research field is topology SALEEM WATSON received his Bachelor of Science degree from Andrews University in Michigan He did graduate studies at Dalhousie University and McMaster University, where he received his PhD in 1978 He subsequently did research at the Mathematics Institute of the University of Warsaw in Poland He also taught at The Pennsylvania State University He is currently Professor of Mathematics at California State University, Long Beach His research field is functional analysis PHYLLIS PANMAN received a Bachelor of Music degree in violin performance in 1987 and a PhD in mathematics in 1996 from the University of Missouri at Columbia Her research area is harmonic analysis As a graduate student she taught college algebra and calculus courses at the University of Missouri She continues to teach and tutor students in mathematics at all levels, including conducting mathematics enrichment courses for middle school students Stewart, Redlin, and Watson have also published Precalculus: Mathematics for Calculus, Algebra and Trigonometry, and Trigonometry About the Cover Each of the images on the cover appears somewhere within the pages of the book itself—in real-world examples, exercises, or explorations The many and varied applications of algebra that we study in this book highlight the importance of algebra in understanding the world around us, and many of these applications take us to places where we never thought mathematics would go The global montage on the cover is intended to echo this universal reach of the applications of algebra College Algebra CONCEPTS AND CONTEXTS James Stewart McMaster University and University of Toronto Lothar Redlin The Pennsylvania State University Saleem Watson California State University, Long Beach Phyllis Panman Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States College Algebra: Concepts and Contexts James Stewart, Lothar Redlin, Saleem Watson, Phyllis Panman Acquisitions Editor: Gary Whalen Developmental Editor: Stacy Green Assistant Editor: Cynthia Ashton Editorial Assistant: Guanglei Zhang Media Editor: Lynh Pham Marketing Manager: Myriah Fitzgibbon Marketing Assistant: Angela Kim Marketing Communications Manager: Katy Malatesta Content Project Manager: Jennifer Risden Creative Director: Rob Hugel Art Director: Vernon Boes Print Buyer: Judy Inouye Rights Acquisitions Account Manager, Text: Roberta Broyer Rights Acquisitions Account Manager, Image: Don Schlotman Production Service: Martha Emry Text Designer: Lisa Henry Art Editor: Martha Emry Photo Researcher: Bill Smith Group Copy Editor: Barbara Willette Illustrator: Jade Myers, Matrix Art Services; Network Graphics Cover Designer: Larry Didona Cover Images: giant trees (Cate Frost/Shutterstock.com 2009); black-browed albatross (Armin Rose/ Shutterstock com 2009); church (Vladislav Gurfinkel/Shutterstock com 2009); student in chemistry lab (Laurence Gough/Shutterstock 2009); five skydivers performing formations (Joggie Botma/Shutterstock.com 2009); Shanghai at sunset (David Roos/Shutterstock.com 2009); combine harvester working on wheat crop (Stephen Mcsweeny/Shutterstock.com 2009); howler monkeys (Christopher Marin/ Shutterstock com 2009); family making sand castle (Magdalena Bujak/ Shutterstock.com 2009); Easter Island (Vladimir Korostyshevskiy/ Shutterstock.com 2009); giardia (Sebastian Kaulitzki/Shutterstock.com 2009); female in handcuffs (Jack Dagley Photography/Shutterstock.com 2009); red eye tree frog (Luis Louro/Shutterstock.com 2009); humpback whale (Josef78/Shutterstock.com 2009); businessman in car (Vladimir Mucibabic/Shutterstock.com 2009); origami birds (Slash331/Shutterstock.com 2009); pine forest (James Thew/Shutterstock.com 2009); house finch (Steve Byland/Shutterstock.com 2009); mother with baby (Lev Dolgachov/Shutterstock.com 2009); bacteria (Tischenko Irina/Shutterstock.com 2009); Dos Amigos pumping plant (Aaron Kohr/Shutterstock.com 2009); polar bears (Keith Levit/Shutterstock.com 2009); combine harvester (Orientaly/Shutterstock.com 2009); streptococcus (Sebastian Kaulitzki/Shutterstock.com 2009); jumping girl (Studio1One/Shutterstock.com 2009); traffic (Manfred Steinbach/Shutterstock.com 2009); hybrid car (Jonathan Larsen/Shutterstock.com 2009); woman driving car (Kristian Sekulic/Shutterstock com 2009); woman hiding money under mattress (cbarnesphotography/Shutterstock.com 2009); Mount Kilimanjaro (Peter Zaharov/Shutterstock.com 2009) Compositor: S4Carlisle Publishing Services Printed in Canada 13 12 11 10 09 © 2011 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Further permissions questions can be e-mailed to permissionrequest@cengage.com Library of Congress Control Number: 2009934974 ISBN-13: 978-0-495-38789-3 ISBN-10: 0-495-38789-4 Brooks/Cole 20 Davis Drive Belmont, CA 94002-3098 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan Locate your local office at www.cengage.com/global Cengage Learning products are represented in Canada by Nelson Education, Ltd To learn more about Brooks/Cole, visit www.cengage.com/brookscole Purchase any of our products at your local college store or at our preferred online store www.CengageBrain.com CONTENTS PROLOGUE: Algebra and Alcohol P1 Data, Functions, and Models chapter 1.1 Making Sense of Data Analyzing One-Variable Data • Analyzing Two-Variable Data 1.2 Visualizing Relationships in Data 12 Relations: Input and Output • Graphing Two-Variable Data in a Coordinate Plane • Reading a Graph 1.3 Equations: Describing Relationships in Data 25 Making a Linear Model from Data • Getting Information from a Linear Model 1.4 Functions: Describing Change 35 Definition of Function • Which Two-Variable Data Represent Functions? • Which Equations Represent Functions? • Which Graphs Represent Functions? • Four Ways to Represent a Function 1.5 Function Notation: The Concept of Function as a Rule 52 Function Notation • Evaluating Functions—Net Change • The Domain of a Function • Piecewise Defined Functions 1.6 Working with Functions: Graphs and Graphing Calculators 64 Graphing a Function from a Verbal Description • Graphs of Basic Functions • Graphing with a Graphing Calculator • Graphing Piecewise Defined Functions 1.7 Working with Functions: Getting Information from the Graph 74 Reading the Graph of a Function • Domain and Range from a Graph • Increasing and Decreasing Functions • Local Maximum and Minimum Values 1.8 Working with Functions: Modeling Real-World Relationships 88 Modeling with Functions • Getting Information from the Graph of a Model 1.9 Making and Using Formulas 101 What Is a Formula? • Finding Formulas • Variables with Subscripts • Reading and Using Formulas ■ chapter CHAPTER Review 113 CHAPTER Test 126 EXPLORATIONS Bias in Presenting Data 128 Collecting and Analyzing Data 134 Every Graph Tells a Story 138 Linear Functions and Models 2.1 Working with Functions: Average Rate of Change 141 142 Average Rate of Change of a Function • Average Speed of a Moving Object • Functions Defined by Algebraic Expressions v vi CONTENTS 2.2 Linear Functions: Constant Rate of Change 153 Linear Functions • Linear Functions and Rate of Change • Linear Functions and Slope • Using Slope and Rate of Change 2.3 Equations of Lines: Making Linear Models 165 Slope-Intercept Form • Point-Slope Form • Horizontal and Vertical Lines • When Is the Graph of an Equation a Line? 2.4 Varying the Coefficients: Direct Proportionality 177 Varying the Constant Coefficient: Parallel Lines • Varying the Coefficient of x: Perpendicular Lines • Modeling Direct Proportionality 2.5 Linear Regression: Fitting Lines to Data 189 The Line That Best Fits the Data • Using the Line of Best Fit for Prediction • How Good Is the Fit? The Correlation Coefficient 2.6 Linear Equations: Getting Information from a Model 201 Getting Information from a Linear Model • Models That Lead to Linear Equations 2.7 Linear Equations: Where Lines Meet 210 Where Lines Meet • Modeling Supply and Demand ■ chapter CHAPTER Review 219 CHAPTER Test 228 EXPLORATIONS When Rates of Change Change 229 Linear Patterns 233 Bridge Science 237 Correlation and Causation 239 Fair Division of Assets 242 Exponential Functions and Models 3.1 247 Exponential Growth and Decay 248 An Example of Exponential Growth • Modeling Exponential Growth: The Growth Factor • Modeling Exponential Growth: The Growth Rate • Modeling Exponential Decay 3.2 Exponential Models: Comparing Rates 261 Changing the Time Period • Growth of an Investment: Compound Interest 3.3 Comparing Linear and Exponential Growth 272 Average Rate of Change and Percentage Rate of Change • Comparing Linear and Exponential Growth • Logistic Growth: Growth with Limited Resources 3.4 Graphs of Exponential Functions 286 Graphs of Exponential Functions • The Effect of Varying a or C • Finding an Exponential Function from a Graph 3.5 Fitting Exponential Curves to Data 295 Finding Exponential Models for Data • Is an Exponential Model Appropriate? • Modeling Logistic Growth CHAPTER Review 303 CHAPTER Test 311 vii CONTENTS ■ chapter EXPLORATIONS Extreme Numbers: Scientific Notation 312 So You Want to Be a Millionaire? 315 Exponential Patterns 316 Modeling Radioactivity with Coins and Dice 320 Logarithmic Functions and Exponential Models 4.1 Logarithmic Functions 323 324 Logarithms Base 10 • Logarithms Base a • Basic Properties of Logarithms • Logarithmic Functions and Their Graphs 4.2 Laws of Logarithms 334 Laws of Logarithms • Expanding and Combining Logarithmic Expressions • Change of Base Formula 4.3 Logarithmic Scales 342 Logarithmic Scales • The pH Scale • The Decibel Scale • The Richter Scale 4.4 The Natural Exponential and Logarithmic Functions 350 What Is the Number e? • The Natural Exponential and Logarithmic Functions • Continuously Compounded Interest • Instantaneous Rates of Growth or Decay • Expressing Exponential Models in Terms of e 4.5 Exponential Equations: Getting Information from a Model 364 Solving Exponential and Logarithmic Equations • Getting Information from Exponential Models: Population and Investment • Getting Information from Exponential Models: Newton’s Law of Cooling • Finding the Age of Ancient Objects: Radiocarbon Dating 4.6 Working with Functions: Composition and Inverse 377 Functions of Functions • Reversing the Rule of a Function • Which Functions Have Inverses? • Exponential and Logarithmic Functions as Inverse Functions ■ chapter CHAPTER Review 393 CHAPTER Test 400 EXPLORATIONS Super Origami 401 Orders of Magnitude 402 Semi-Log Graphs 406 The Even-Tempered Clavier 409 Quadratic Functions and Models 5.1 Working with Functions: Shifting and Stretching 413 414 Shifting Graphs Up and Down • Shifting Graphs Left and Right • Stretching and Shrinking Graphs Vertically • Reflecting Graphs 5.2 Quadratic Functions and Their Graphs 428 The Squaring Function • Quadratic Functions in General Form • Quadratic Functions in Standard Form • Graphing Using the Standard Form A50 Answers to Selected Exercises and Chapter Tests D.4 Exercises ■ page T89 x above (a) - 1, 0, 1, (b) 3- 1, 04 ´ 31, 34 (a) 1, (b) (1, 4) - - 132 16 No real solution 10 No real solution 11 ; 52 12 2.61 13 3.00, 4.00 14 0.25, 0.50 15 1.00, 2.00, 3.00 16 - 1.00, - 0.25, 0.25 17 1.62 18 0, 2.31 19 0.75 20 - 0.31, 0.81 21 No solution 22 No solution 23 - 2.00, 5.004 24 - 2.00, 0.254 25 26 27 28 29 30 31 32 1- q, 1.00 ´ 2.00, 3.004 1- 1.00, - 0.25 ´ 1- 0.25, q 1- 1.00, ´ 11.00, q 1- q, - 0.5354 ´ 30.535, q 3- 3.00, 6.004 1- q, - 3.002 ´ 1- 2.00, q 1- 1.00, 2.002 ´ 15.00, q 1- q, - 2.00 ´ 1- 2.00, 4.00 INDEX Absolute value, T11 distance between points on real line and, T12 Absolute value function, 70 Addition of algebraic expressions, T26–T27 graphical, of functions, 485–86, 505 of matrices, 605–6, 611–12 of polynomials, T30 of rational expressions, T41–T42 Aerodynamic lift, 503 Age and height, 5–6 Air pressure and elevation, 122–23 Alcohol, surge function and, 563, 565–66 Alcohol consumption, P1–P4, 11, 25, 52, 88 absorption, P1, P3 collecting data on, P1 metabolism, P1, P3 modeling absorption and metabolism, P3–P4, 261, 272, 295, 376 Algebraic expression(s) adding or subtracting, T26–T27 average rate of change of function defined by, 147–48 combining, T25–T32 defined, T25 factoring, T33–T38 form and value of, T25–T26 multiplying, T27–T29 polynomials, combining, T29–T31 Algebraic method of solving equation, T85, T86 Ancient objects, finding the age of, 371–72 Anglerfish, depth and pressure experienced by, 35 Annual percentage yield (APY), 266 Archimedes, 209, 312, 568 crown problem, 568, 573 Arctic sea ice, extent of, 197–98 Area(s) proportionality to square of “length” measurement, 557 of similar objects, 555–56 Arithmetic expression, evaluating, T2 Arrow notation, 509, 528 Asbestos and cancer, links between, 194, 195 Aspirin metabolism, 308 Assets, fair division of, 242–45 Associative properties, T2 Astronaut, net change in weight of, 55–56 Asymptote(s) horizontal, 287, 288, 528, 540, 541 of rational functions, 540, 541 of reciprocal functions, 528 vertical, 329, 528, 540, 541 Atlanta, population of, 150–51 Attenuated growth, 398 Augmented matrix of linear system, 590–91, 593, 594 Average age of preschoolers, Average income, 3–4 Average (mean), 2–3 definition of, formula for, 104–5 Average rate of change, 142–53, 272–75 average speed, 145–46, 148 definition of, 143 for function defined by algebraic expression, 147–48 of linear function, 155–56 See also Rate of change recognizing changes in, 229–33 Average speed of moving object, 145–46, 148 Bach, Johann Sebastian, 411 Back-substitution, solving triangular system using, 581, 582 Bacteria count, in colony-forming units per milliliter, 356 Bacterial growth, 248–49 comparing growth rates, 264 models for different time periods, 263 Bankruptcy, fair division of assets in, 242–45 Barometer, invention of, 476 Barton, Otis, 30 Base exponential function with base a, 286, 288–90 exponential function with base e, 351 exponential notation, T14 of logarithm, 324, 326–27, 337–39 Bat cave, species-area relation in, 499, 517–18 Bathysphere, 30 Beebe, William, 30 Beer-Lambert Law, 363–64 Belgium, population of, 302–3 Bias in presenting data, 128–33, 130–33 Bicycle race, average speed in, 145–46 Billionaire ruler, 403 Binomials, T26 multiplying, using FOIL, T28 Biodiesel fuels from seeds of Jatropha curcsa, 391–92 Biodiversity in Pasoh Forest Reserve of Malaysia, 526 Bird flight, weight and wingspan and, 525 Blood alcohol concentration (BAC), P1, P3, 11, 52, 88, 189, 376 using surge functions to model data on, 563, 565–66 Body Mass Index (BMI), 51, 189–90 Boiling point, elevation and, 33–34 Bonds, coupon, 309 Boyle’s Law, 111, 529, 534 Braking distance, finding maximum carrying capacity of road with, 562–63 Bridge science, 237–39 Bungee jumping, 148 California, population of, 282 Camera lenses, focusing distance for, 536, 537–39 Canceling, simplifying rational expressions by, T39 Cancer and asbestos, links between, 194, 195 I1 I2 INDEX Carbon dioxide levels in atmosphere, 197 Carrying capacity in logistic growth model, 277, 298 of road, 80–81, 560–63 of road, safe following distance and, 560, 561–62 of road, using braking distance to find maximum, 562–63 Car sales, hybrid, 22, 302 Cartesian plane See Coordinate plane Catching up with leader on bike ride, 213–14 Categorical data, 602–11 collecting, 635 getting information from, 615–16 organizing, in matrix, 603–5, 636–37 proportions and, 604–5 tabulating, 604 Catfish, stocking pond with, 298 Causation correlation and, 239–42 direction of, 242 Cell phone plan, 57 Cell phone usage in India, 363 Central tendency, measures of, 2, 3, Change, describing See Function(s) Change of base formula, 337–39 China, Internet usage in, 363 Chirping rate of crickets, temperature and, 198, 208 Chocolate-powered car, 34 Chord, 412 Christmas bird count, 22 Circle(s), T76–T77 equation of, identifying, T77 equation of, standard form of, T76–T77 graphing, T76 graphing, on graphing calculator, T83–T84 Closed interval, T9 Coefficient(s), 177–82 constant, 177–80 correlation, 195 heat transfer, 370 of x, 177, 180–82 Coefficient matrix, 621 Colony-forming units, 356 Columbia-Ecuador earthquake (1906), 346 Columns (matrix), 590 Combining “like terms,” T26, T30 Combining logarithmic expressions, 336 Common factors canceling, T39 factoring out, T33–T34 Common logarithms, 324–26 Common redpoll, wintering habits of, 25 Commutative properties, T2 Completing the square, T57–T59 defined, T58 to express quadratic function in standard form, 431 for general quadratic function, 440 solving quadratic equations by, T58–T59 Composite functions See Composition of functions Composition of functions, 377–79 Compound fraction form of rational function, 539 Compound interest, 265–66 calculating annual percentage yield, 266 comparing yields for different compounding periods, 266 continuously compounded, 354–55 formula, 265 investment growth and, 369 Concentration, mixtures and, 204–5 Conjugate radical, T42 Constant of proportionality, 183, 496, 529, 557 spring, 188–89 universal gravitational, 104 Constant coefficient, 177 varying, 177–80 Constant function, 65 Constant rate of change, 153–65 linear model of, 156–57 Continuously compounded interest, 354–55 Cooling, Newton’s Law of, 370–71, 375, 399, 427 Cooling coffee, 41 Coordinate, T7 Coordinate line See Real number line (real line) Coordinate plane, T67–T68, T71 distance between points on, T68–T69 graphing points and sets in, T68 graphing two-variable data in, 14–16 midpoint of line segment on, T69–T70 Correlation, causation and, 239–42 Correlation coefficient, 195 Cost fixed, 26 model for, 89–90 unit, 26, 27 Cost comparison of gas-powered and hybrid-electric cars, 212–13 Coughing, mathematical model of, 88 Coupon bonds, 309 Crime scene investigation, 364 Newton’s Law of Cooling and, 371 Crop yield density (plants/acre) and, 465 rainfall and, 465 Cross-tab matrix, 603, 636 Crude oil imports in U.S., 493 Cube root function, 498 Data, 2–11 See also Modeling analyzing, 136–37, 239 bias in presenting, 128–33 categorical See Categorical data collecting, 134–36, 238, 476–77 describing relationships in, 25–35 exponential, recognizing, 316–17 fitting exponential curves to, 274–75, 295–303 fitting polynomial curves to, 520–22 fitting power curves to, 517–18, 519–20 linear, recognizing, 233–37 linearizing, 408–9 log-log plot of, 518, 519 numerical, 602 one-variable, 2–5, 136–37 quadratic, recognizing, 478–79 scatter plot of, 14, 518, 519 semi-log plot of, 407–9, 518, 519 two-variable, 5–7, 13, 14–16, 37–38, 137 visualizing relationships in, 12–25 Data mining, 233 Dating, radiocarbon, 371–72 Dead Sea Scrolls, 376 Death in United States, leading causes of, 130–32 Decay factor, 253, 264 Decay rate, 253 instantaneous, 355–64 Decibel scale, 344–45 Decreasing functions, 78–79 Deforestation, paper usage and, 90–91 Degree of polynomial, 504 Demand, linear model for 170–71 Demand equation, 171, 214, 218 Demographics, life expectancy and, 23 Denominator least common (LCD), T41 rationalizing the, T42 Dependent system, 571, 582–84, 596–98 Dependent variable, 37–38, 54 net change in, 38 Depreciation, straight-line, 176 Depth and pressure, 6–7, 14–15, 26 model for, 29–30 INDEX Difference(s) first, 28–30 of functions, 484–87 second, 478–79, 480, 481 of squares, factoring, T35–T36 Dimension of matrix, 590 Dimensions of lot, quadratic function modeling, 453–54 Direct proportionality, 182–84 definition of, 183 to power function, 496–97 Discriminant, 452–53 Dissonance, 411–12 Distance, T11 maximum distance seen from a height, 61 between points on real line, T12 between two points in coordinate plane, T68–T69 Distance Formula, T68–T69 Distance function, inverse function of, 498 Distance-speed-time problem, 574–75 Distributive property, T3–T5 expanding using, T3–T4 factoring using, T4–T5 multiplying expressions using, T27–T28 multiplying polynomials using, T30 solving equations with, T48 Division See also Quotient(s) of assets, fair, 242–45 of exponents, T16 graphical, 536 long, T42–T43 of rational expressions, T40–T41 Domain of function, 56–57, 77–78 of relation, 13, 14 Doppler effect, 111, 545 Drip irrigation system for garden, 91–92 Eames, Ray and Charles, 402 Earthquakes intensity of, 346–47 magnitude of, 346, 349 Einstein, Albert, 101 Electrical capacity of solar panels, 182–84 Electrical resistance, 111 Elementary row operations, 591–92 Elements of set, T8 Elevation air pressure and, 122–23 boiling point and, 33–34 temperature and, 28–29 Elimination method, 570–71 Gaussian, 581–82 Empty set, T8 End behavior defined, 509 of polynomial function, 508–10 of rational function, 540 Energy, world consumption of, 48–49 Energy and mass formula, Einstein’s, 101 Entries of matrix, 590 Environmental management, species survival and, 639 Enzymes in expectant mothers, levels of, 15–16 Equation(s), 25–35 See also System of equations defined, T47 demand, 171, 214, 218 exponential, 365–66, 401–2 false, 582 in function form, 39 graphing, 27 of horizontal line, 171–72 inequality compared to, T62 of lines, 165–77 See also Linear equation(s) logarithmic, 367–68 logistic growth, 277 matrix, 621–25 operations on, T47 power, T51–T53 quadratic See Quadratic equation(s) reading, T48 satisfying, T72 solutions (or roots) of, T47 solving the See Solving the equation that represent functions, 38–40, 42 two-variable, T71–T80 of vertical line, 171–72 Equilibrium point, 214 Equivalent system, operations leading to, 581 e (the number), 351 exponential models in terms of, 357–58 Ethiopia, population of, 374–75 Expanding expression distributive property used in, T3–T4 logarithmic expressions, 335–36 Expectant mothers, levels of enzymes in, 15–16 Exponent(s), T14 integer, T14–T19 negative, T15–T16 rational, T20–T24 I3 rules for working with, T16–T18, T21, T22 zero, T15–T16 Exponential decay, 252–55, 356 of “iceman,” radiocarbon dating and, 372 modeling 252–55, 252–55, 264–65, 290, 356–57 Exponential equation(s), 365–66 logarithms used to solve, 401–2 steps in solving, 365 Exponential form, 327 Exponential function(s), 247–61 with base a, 286 with base a, effect of varying a or C, 288–90 with base e, 351 comparing linear functions and, 275–76 comparing power functions and, 494–95 for compound interest, 265–66 definition of, 249 finding, from a graph, 290–91 fitting exponential curves to data, 295–303 graphs of, 286–95, 406–9 inverse functions of, 385–86 modeling with, 249–55, 261–66, 274–75, 290, 295–98, 356–58 modeling with, appropriateness of, 518, 519 musical scale and, 409–12 natural, 351–52 rates of change of, 273–74 semi-log graph of, 407–9 transformations of, 420–21 Exponential growth, 248–85, 356 example of, 249–50 of investment, 265–66 linear growth versus, 275–76 modeling, 249–52, 249–52, 263, 265–66, 356, 357–58 of savings, 315–16 Exponential model(s), 276 appropriateness of, 296–97 changing time period in, 262–65, 266 for data, finding, 295–96 of decay, 252–55, 264–65, 290, 356–57 fitting model to data, 274–75, 295–97 getting information from, 368–71 of growth, 249–52, 263, 265–66, 356, 357–58 of investment growth, 265–66, 369 in terms of e, 357–58 I4 INDEX Exponential model(s) (continued) in terms of instantaneous growth rate, 358 of world population, 289, 296, 369–70 Exponential notation, T14–T15 for nth root of a, T20 for nth root of am, T21 Exponential patterns, 316–19 finding exponential functions fitting data, 317–19 recognizing exponential data, 316–17 Extraneous lines in graph, avoiding, T82–T83 Extraneous solutions, T50 Extrapolation, 191–92 Extreme numbers, 312–14 Factor(s), T1 canceling common, T39 factoring out common, T33–T34 Factored form of rational function, 539 Factoring algebraic expressions, T33–T38 completely, T36 distributive property used in, T4–T5 factoring out common factors, T33–T34 graphing polynomial functions by, 505–8 by grouping, T37 solving quadratic equations by, 448–50, T56–T57 special factoring formulas, T35–T36 trinomials, T34–T35 Fair division of assets, 242–45 Falling sky diver, 61 False equation, 582 Family of linear equations, graphing, 177–78, 180–81 Family of logarithmic functions, graphing, 329–30, 338–39 Farms in United States, average rate of change in number of, 144–45 Fechner, Gustav, 344 Femur length, height and, 197 Fencing a garden, 94–96, 443 Financial problem, using linear system to model, 585–86 First differences, 28–30 Fish, length-at-age data for, 521–22 Fish population, limited, 277–78 Fitt’s Law, 350 Fixed cost, 26 Focusing distance for camera lenses, 536, 537–39 FOIL, multiplying binomials using, T28 Force/mass/acceleration, Newton’s formula of, 107 “Forgetting” curves, experimenting with, 525 Formula(s), 101–12 for average (mean), 104–5 compound interest, 265 definition of, 101 distance, T68–T69 equivalent, 105 finding, 102–3 midpoint, T69–T70 quadratic, 451, 454, 455, T59–T61 reading and using, 105–8 simple interest, 203 for surface area, 103 variables with subscripts in, 104–5 Fractional positive powers, power function with, 497–98 Fractions properties of, T40 solving an equation involving, T50 Frequencies of musical notes, 410–12 Fudging the data, 128 Function(s), 35–52 absolute value, 70 average rate of change of, 142–53 composition of, 377–79 constant, 65 cube root, 498 decreasing, 78–79 definition of, 36–37 dependent variable in, 37–38, 54 difference of, 484–87 domain of, 56–57, 77–78 equations that represent, 38–40, 42 evaluating, 54–56 exponential See Exponential function(s) four ways to represent, 43 of functions, 377–79 graphical addition and subtraction of, 485–87, 505 graph of, 41–43, 64–88 graph of, finding values of its inverse from, 380 graph of, reading, 74–76 graph of, using graphing calculator, 67–68 identity, 66 increasing, 78–79 independent variable in, 37–38, 54 inverse, 379–86, 498 linear See Linear function(s) local maximum and minimum values of, 79–81 logarithmic See Logarithmic function(s) logistic, 297 modeling with, 89–100 natural exponential, 351–52 natural logarithm, 353 one-to-one, 383–85 piecewise defined, 57–58, 68–70 polynomial, 504–16, 520–22 power See Power functions product of, 488–89 quadratic See Quadratic function(s) quotient of, 488 range of, 77–78 rational, 536–45, 560–63 reciprocal, 527–30 as relation, 36–37 root, 497–98 as rule, 52–53 square root, 66–67, 497–98 squaring, 428–29, 432 sum of, 484–87 surge, 563–66 value of f at x, 53, 55–56, 75–76 vertical shifts, 414–15, 417–18 Function form, 39 Function notation, 52–54 Galileo Galilei, 153 Galileo’s Law, 493, 498 Garden fencing a, 94–96, 443 irrigating, 91–92 Gas mileage formula for, 102 maximum, for car, 442 Gasoline price, year and, 38 Gas-powered and hybrid-electric cars, cost comparison of, 212–13 Gaussian elimination, 581–82 General form of equation of line, 172–73 General form of quadratic function, 429–30 completing the square for, 440 General Social Survey (GSS), 635 Geometry, constructing model involving, 205–6 Geometry of space, inverse square laws and, 531–32 Germany, population of, 375 Global warming, 141 Grade of road, 157, 158 Graph(s) INDEX of absolute value function, 70 of circle, T76, T83–T84 of exponential functions, 286–95, 406–9 of family of linear equations, 177–78, 180–81 finding domain and range of function from, 77–78 finding exponential function from, 290–91 finding linear functions from, 160–61 finding local maximum and minimum values from, 79–81 finding values of function from, 75–76 of function, 41–43, 64–88 of function, finding values of its inverse from, 380 of function, reading, 74–76 of general linear equation, 172–73 horizontal shifts of, 416–18 of increasing and decreasing functions, 78–79 of inequalities, T8 intercepts, finding, T75 See also x-intercepts; y-intercepts intersection points on, 68, 76 of intervals, T9, T10 of inverse square function, 530 of linear function, 155 of logarithmic functions, 328–30, 338–39 misleading, 128–30 of model, getting information from, 93–96 of natural exponential functions, 352 of parallel lines, 179, 180 of piecewise defined functions, 68–70 of polynomial function, 505–8 of power functions, 495 of quadratic functions, 432–34, 449–50 rates of change and shapes of, 230–33 of rational functions, 536–37, 539–41 reading, 16–17, 138–39, 473–75 of reciprocal function, 527–28 reflecting, 419–22, 495 of root functions, 497, 498 scatter plot, 14, 519 semi-log, 407–9, 518, 519 shifting, 414–18, 495 solving a polynomial equation graphically, 511 of squaring function, 432 stretching and shrinking, vertically, 418–19, 495 of system of two linear equations in two variables, 571–72 transformations of function and, 414–27, 495 two-variable data in coordinate plane, 14–16, T67–T68 of two-variable equations, T72–T75 verbal description from, 75 vertical shifts, 414–15, 417–18 Graphical addition of functions, 485–86, 505 Graphical division, 536 Graphical method of finding intersection points of linear functions, 211 Graphical method of solving equations, T85–T88 equation in an interval, T88 inequalities, T88–T89 quadratic equations, T87 Graphical subtraction of functions, 486–87 Graphing calculator, T80–T85 avoiding extraneous lines, T82–T83 choosing viewing rectangle, T80–T82 CubicReg command on, 521 exponential function, 296 family of exponential functions, 289 graphing circle on, T83–T84 graphing functions with, 67–68 logistic growth function, 278, 297–98 LOG key, 338 multiplying matrices on, 614–15 PwrReg command, 517, 519 QuadReg command on, 462, 463 two graphs on same screen, T82 Xmin and Xmax command, T81 Ymin and Ymax command, T81 Gravitation, Newton’s Law of, 535 Gravitational force between moon and astronaut in space ship, 73 Newton’s formula for, 104, 107 Greater than symbol (>), T7 Grimshaw, John, 34 Grouping, factoring by, T37 Growth attenuated, 398 exponential See Exponential growth linear model of, 156–57 logistic, 272, 276–78, 297–98 Growth factor, 249–51, 252 changing time period and, 262–65 Growth rate, 251–52 annual percentage yield, 266 changing time period and, 262–65 I5 comparing, 264 instantaneous, 355–64 percentage rate of change, 272–75, 297 Half-life, 254, 264 Hanselman sextuplets, weights of, Hare and tortoise race, modeling, 210–11 Health-care expenditures, U.S., 301 Heat transfer coefficient, 370 Height age and, 5–6 of box, 106–7 femur length and, 197 Hidden variable, 239–42 Highway engineering, 80–81 Home sales in United States, 492 Hooker, Steven, 194 Hooke’s Law, 188–89 Horizontal asymptote, 287, 288, 528, 540, 541 Horizontal lines, 171–72 Horizontal Line Test, 384–85 Horizontal shifts of graphs, 416–18 combining vertical shifts and, 417–18 House, median price of, 4–5 “Housing bubble,” bursting of U.S., 259–60 Howler monkeys, modeling population of, 637–39 Hybrid car sales, 22, 302 Hydrogen ion concentration, pH scale and, 344 “Iceman” of Neolithic Age, radiocarbon dating of, 372 Identity function, 66 Identity matrix, 619 Image of x under f, 53 Income, average and median, 3–4 Inconsistent system, 571, 572, 582–84, 596–98 Increasing functions, 78–79 Independent variable, 37–38, 54 India cell phone usage in, 363 population of, 270, 392 Inequalities, T62–T66 graphing, T8 linear, solving, T63 nonlinear, solving, T63–T65 operations on, T62 order on real line and, T7–T8 reversing direction of, T62 solving, graphically, T88–T89 I6 INDEX Infant mortality, U.S., 190–91, 192, 195 Infinite intervals, T9 Initial population, 249 Initial value, 26, 156, 250, 356 Inner product, 612 Input(s), 12–14 evenly spaced, 28 in function, 36 Installation of flooring, average rate of, 143 Instantaneous rates of growth or decay, 355–64 expressing exponential models in terms of, 358 Integer exponents, T14–T19 Integers, T1 Intelligibility, noise and, 199 Intensity of earthquakes, 346–47 sound, decibel scale of, 344–45 Intercepts See x-intercepts; y-intercepts Interest compound, 265–66, 369 continuously compounded, 354–55 simple, 203–4 Internet usage in China, 363 Interpolation, 192 Intersection of intervals, T10–T11 of sets, T8–T9 Intersection points, 68, 76 equilibrium point, 214 of linear functions, 211–15 Interval(s), T9–T11 closed, T9 frequency, on musical scale, 410–12 graphing, T9, T10 infinite, T9 open, T9 solving equation in, T88 unions and intersections of, finding, T10–T11 verbal description of, T10 Inverse function(s), 379–86 definition of, 380 of distance function, 498 of exponential and logarithmic functions, 385–86 finding values of, graphically, 380 properties of, 382 steps in finding, 380 Inverse of matrix, 620–25 Inverse proportionality, 528–30 Inverse square function, graph of, 530 Inverse square laws, 530–32 laws of nature as, 531–32 Investment, growth of, 265–66, 369 Irrational numbers, T1 e as, 351 Jet takeoff, sound intensity of, 345 Kepler’s Law for Periods of Planets, 524 Kepler’s Third Law, 493 King Hiero’s crown, amount of gold in, 568, 573 Kirchhoff’s Laws, 589 Labor force, U.S., 492 La Condamine, Charles-Marie de, 134 Lang Lang, 410 Law of Laminar Flow, 50 Law of the Lever, 209, 568 Law of the Pendulum, 503 Laws of Logarithms, 334–41 defined, 335 expanding and combining logarithmic expressions using, 335–36 Lead emissions, U.S., 524 Leading entry of matrix row, 592 Leading term of polynomial, 504 end behavior of polynomial function and, 509–10 Leading variable in linear system, 595 Least common denominator (LCD), T41 Length-at-age data for fish, 521–22 Less than symbol (
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Xem thêm: College algebra , College algebra , 3 Equations: Describing Relationships in Data, 5 Function Notation: The Concept of Function as a Rule, 6 Working with Functions: Graphs and Graphing Calculators, 7 Working with Functions: Getting Information from the Graph, 1 Working with Functions: Average Rate of Change, 2 Linear Functions: Constant Rate of Change, 5 Linear Regression: Fitting Lines to Data, 6 Linear Equations: Getting Information from a Model, 5 Exponential Equations: Getting Information from a Model, 3 Maxima and Minima: Getting Information from a Model, 4 Quadratic Equations: Getting Information from a Model, 5 Matrix Operations: Getting Information from Data, 6 Matrix Equations: Solving a Linear System

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