www.elsolucionario.net http://www.elsolucionario.net LIBROS UNIVERISTARIOS Y SOLUCIONARIOS DE MUCHOS DE ESTOS LIBROS LOS SOLUCIONARIOS CONTIENEN TODOS LOS EJERCICIOS DEL LIBRO RESUELTOS Y EXPLICADOS DE FORMA CLARA VISITANOS PARA DESARGALOS GRATIS www.elsolucionario.net http://www.elsolucionario.blogspot.com LIBROS UNIVERISTARIOS Y SOLUCIONARIOS DE MUCHOS DE ESTOS LIBROS LOS SOLUCIONARIOS CONTIENEN TODOS LOS EJERCICIOS DEL LIBRO RESUELTOS Y EXPLICADOS DE FORMA CLARA VISITANOS PARA DESARGALOS GRATIS www.elsolucionario.net bar51969_fm_i-xviii.qxd 31/1/08 4:47 PM Page i THIRD EDITION Precalculus GRAPHS AND MODELS Raymond A Barnett Merritt College Michael R Ziegler Marquette University Karl E Byleen Marquette University Dave Sobecki Miami University Hamilton www.elsolucionario.net bar51969_fm_i-xviii.qxd 31/1/08 6:26 PM Page ii PRECALCULUS: GRAPHS AND MODELS, THIRD EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2009 by The McGraw-Hill Companies, Inc All rights reserved Previous editions © 2005, 2000 No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper DOW/DOW ISBN 978–0–07–305196–3 MHID 0–07–305196–9 ISBN 978–0–07–334180–4 (Annotated Instructor’s Edition) MHID 0–07–334180–0 Editorial Director: Stewart K Mattson Sponsoring Editor: Dawn R Bercier Vice-President New Product Launches: Michael Lange Developmental Editor: Katie White Senior Marketing Manager: John Osgood Senior Project Manager: Sheila M Frank Senior Production Supervisor: Kara Kudronowicz Senior Media Project Manager: Sandra M Schnee Senior Designer: David W Hash Cover Illustration: John Albert Joran (USE) Cover Image: Curved glass roof, low angle view, ©Luis Veiga/Photographer's Choice/GETTY IMAGES Lead Photo Research Coordinator: Carrie K Burger Project Coordinator: Melissa M Leick Compositor: Aptara Typeface: 10/12 Times Roman Printer: R R Donnelley Willard, OH Photo Credits: CO 1: © DigitalVision/PunchStock; CO 2: © Vol 88 PhotoDisc/Getty; p 215: © Vol 77 PhotoDisc/Getty; CO 3: © BigStock Photo; CO 4: © Corbis RF; CO 5: © Corbis RF; p 478: NASA; p 535: © RF Corbis; CO 6: © Vol 64 PhotoDisc/Getty; p 584: © RF Corbis; p 603: Jacqui Hurst/Corbis; CO 7: © Vol 1/ Corbis RF; p 637: © BrandX/Punchstock; p 682: © Big Stock Photos; CO 8: © BigStock Photo; p 730: © Vol PhotoDisc/Getty; CO 9: © Vol 25 PhotoDisc/Getty; CO 10: © Corbis RF; CO 11: © BrandX RF/Superstock Library of Congress Cataloging-in-Publication Data Precalculus : graphs and models — 3rd ed / Raymond A Barnett [et al.] p cm — (Barnett, Ziegler, and Byleen’s precalculus series) Rev ed of: Precalculus : graphs and models / Raymond A Barnett, Michael R Ziegler, Karl E Byleen 2nd ed c2005 Includes index ISBN 978–0–07–305196–3 — ISBN 0–07–305196–9 (hard copy : alk paper) Functions—Textbooks Functions—Graphic methods—Textbooks I Barnett, Raymond A II Barnett, Raymond A Precalculus QA331.3.B394 2009 515—dc22 2007042225 www.mhhe.com www.elsolucionario.net bar51969_fm_i-xviii.qxd 26/1/08 12:39 AM Page iii Pinnacle 110:MHIA065:SE:FM: About the Authors Raymond A Barnett, a native of and educated in California, received his B.A in mathematical statistics from the University of California at Berkeley and his M.A in mathematics from the University of Southern California He has been a member of the Merritt College Mathematics Department and was chairman of the department for four years Associated with four different publishers, Raymond Barnett has authored or co-authored 18 textbooks in mathematics, most of which are still in use In addition to international English editions, a number of the books have been translated into Spanish Co-authors include Michael Ziegler, Marquette University; Thomas Kearns, Northern Kentucky University; Charles Burke, City College of San Francisco; John Fujii, Merritt College; and Karl Byleen, Marquette University Michael R Ziegler received his B.S from Shippensburg State College and his M.S and Ph.D from the University of Delaware After completing postdoctoral work at the University of Kentucky, he was appointed to the faculty of Marquette University where he currently holds the rank of Professor in the Department of Mathematics, Statistics, and Computer Science Dr Ziegler has published more than a dozen research articles in complex analysis and has co-authored more than a dozen undergraduate mathematics textbooks with Raymond Barnett and Karl Byleen Karl E Byleen received his B.S., M.A., and Ph.D degrees in mathematics from the University of Nebraska He is currently an Associate Professor in the Department of Mathematics, Statistics, and Computer Science of Marquette University He has published a dozen research articles on the algebraic theory of semigroups and co-authored more than a dozen undergraduate mathematics textbooks with Raymond Barnett and Michael Ziegler Dave Sobecki earned a B.A in math education from Bowling Green State University, then went on to earn an M.A and a Ph.D in mathematics from Bowling Green He is an Associate Professor in the Department of Mathematics and Statistics at Miami University in Hamilton, Ohio He has written or co-authored five journal articles, eleven books, and five interactive CD-ROMs Dave lives in Fairfield, Ohio, with his wife (Cat) and dogs (Large Coney and Macleod) His passions include Ohio State football, Cleveland Indians baseball, heavy metal music, travel, and home improvement projects www.elsolucionario.net bar51969_fm_i-xviii.qxd 26/1/08 12:39 AM Page iv Pinnacle 110:MHIA065:SE:FM: Barnett, Ziegler, Byleen and Sobecki’s Precalculus Series College Algebra, Eighth Edition This book is the same as Precalculus without the three chapters on trigonometry ISBN 0-07-286738-8, ISBN 978-0-07-286738-1 Precalculus, Sixth Edition This book is the same as College Algebra with three chapters of trigonometry added The trigonometry functions are introduced by a unit circle approach ISBN 0-07-286739-6, ISBN 978-0-07-286739-8 College Algebra with Trigonometry, Eighth Edition This book differs from Precalculus in that College Algebra with Trigonometry uses right triangle trigonometry to introduce the trigonometric functions ISBN 0-07-331264-9, ISBN 978-0-07-331264-4 College Algebra: Graphs and Models, Third Edition This book is the same as Precalculus: Graphs and Models without the three chapters on trigonometry This text assumes the use of a graphing calculator ISBN 0-07-305195-0, ISBN 978-0-07-305195-6 Precalculus: Graphs and Models, Third Edition This book is the same as College Algebra: Graphs and Models with three additional chapters on trigonometry The trigonometric functions are introduced by a unit circle approach This text assumes the use of a graphing calculator ISBN 0-07-305196-9, ISBN 978-0-07-305-196-3 College Algebra with Trigonometry: Graphs and Models This book is the same as Precalculus: Graphs and Models except that the trigonometric functions are introduced by right triangle trigonometry This text assumes the use of a graphing calculator ISBN 0-07-291699-0, ISBN 978-0-07-291699-7 www.elsolucionario.net bar51969_fm_i-xviii.qxd 26/1/08 12:39 AM Page v Pinnacle 110:MHIA065:SE:FM: Contents Preface vii Application Index xvi CHAPTER 1-1 1-2 1-3 1-4 1-5 1-6 CHAPTER Functions, Graphs, and Models Using Graphing Calculators Functions 21 Functions: Graphs and Properties 46 Functions: Graphs and Transformations 67 Operations on Functions; Composition 84 Inverse Functions 99 Chapter Review 119 Chapter Group Activity: Mathematical Modeling: Choosing a Cell Phone Provider 126 4-1 4-2 4-3 4-4 4-5 Modeling with Exponential and Logarithmic Functions 381 Exponential Functions 382 Exponential Models 399 Logarithmic Functions 416 Logarithmic Models 430 Exponential and Logarithmic Equations 440 Chapter Review 462 Chapter Group Activity: Comparing Regression Models 456 Cumulative Review Exercises Chapters 3–4 457 CHAPTER CHAPTER 2-1 2-2 2-3 2-4 2-5 2-6 2-7 Modeling with Linear and Quadratic Functions 127 Linear Functions 128 Linear Equations and Models 151 Quadratic Functions 172 Complex Numbers 190 Quadratic Equations and Models 206 Additional Equation-Solving Techniques 226 Solving Inequalities 241 Chapter Review 258 Chapter Group Activity: Mathematical Modeling in Population Studies 265 CHAPTER 3-1 3-2 3-3 3-4 3-5 3-6 5-1 5-2 5-3 5-4 5-5 5-6 Polynomial and Rational Functions 273 Polynomial Functions and Models 274 Polynomial Division 291 Real Zeros and Polynomial Inequalities 303 Complex Zeros and Rational Zeros of Polynomials 320 Rational Functions and Inequalities 336 Variation and Modeling 361 Chapter Review 371 Chapter Group Activity: Interpolating Polynomials 378 6-1 6-2 6-3 6-4 6-5 Trigonometric Identities and Conditional Equations 561 Basic Identities and Their Use 562 Sum, Difference, and Cofunction Identities 572 Double-Angle and Half-Angle Identities 584 Product-Sum and Sum-Product Identities 596 Trigonometric Equations 604 Chapter Review 618 Chapter Group Activity: From M sin Bt ϩ N cos Bt to A sin (Bt ϩ C)—A Harmonic Analysis Tool 622 CHAPTER 7-1 7-2 7-3 Trigonometric Functions 463 Angles and Their Measure 464 Trigonometric Functions: A Unit Circle Approach 478 Solving Right Triangles 493 Properties of Trigonometric Functions 501 More General Trigonometric Functions and Models 518 Inverse Trigonometric Functions 535 Chapter Review 550 Chapter Group Activity: A Predator–Prey Analysis Involving Mountain Lions and Deer 558 CHAPTER Cumulative Review Exercises Chapters 1–2 267 Additional Topics in Trigonometry 625 Law of Sines 626 Law of Cosines 639 Vectors in the Plane 650 V www.elsolucionario.net bar51969_fm_i-xviii.qxd 26/1/08 12:39 AM Page vi Pinnacle 110:MHIA065:SE:FM: 7-4 7-5 Polar Coordinates and Graphs 667 Complex Numbers in Rectangular and De Moivre’s Theorem 683 Chapter Review 696 Chapter Group Activity: Conic Sections and Planetary Orbits 704 Cumulative Review Exercises Chapters 5–7 707 CHAPTER 8-1 8-2 8-3 8-4 Modeling with Systems of Equations and Inequalities 713 Systems of Linear Equations in Two Variables 714 Systems of Linear Equations in Three Variables 732 Systems of Linear Inequalities 745 Linear Programming 760 Chapter Review 772 Chapter Group Activity: Heat Conduction 776 CHAPTER 9-1 9-2 9-3 9-4 9-5 9-6 9-7 Matrices and Determinants 777 Systems of Linear Equations: Gauss–Jordan Elimination 778 Matrix Operations 797 Inverse of a Square Matrix 815 Matrix Equations and Systems of Linear Equations 828 Determinants 838 Properties of Determinants 847 Determinants and Cramer’s Rule 854 Chapter Review 860 Chapter Group Activity: Using Matrices to Find Cost, Revenue, and Profit 866 Cumulative Review Exercises Chapters 8–9 868 CHAPTER 10-1 10-2 10-3 10-4 10-5 10 10-6 The Binomial Formula 946 Chapter 10 Review 954 Chapter 10 Group Activity: Sequences Specified by Recursion Formulas 959 CHAPTER 11-1 11-2 11-3 11-4 11-5 11 Additional Topics in Analytic Geometry 961 Conic Sections; Parabola 962 Ellipse 973 Hyperbola 985 Translation and Rotation of Axes 1001 Systems of Nonlinear Equations 1020 Chapter 11 Review 1031 Chapter 11 Group Activity: Focal Chords 1036 Cumulative Review Exercises Chapters 10–11 1037 APPENDIX A Basic Algebra Review A-1 Algebra and Real Numbers A-2 Exponents A-13 Radicals A-27 Polynomials: Basic Operations A-36 Polynomials: Factoring A-47 Rational Expressions: Basic Operations A-58 Appendix A Review * Appendix A Group Activity: Rational and Irrational Numbers * *Available online at www.mhhe.com/barnett A-1 A-2 A-3 A-4 A-5 A-6 APPENDIX B Review of Equations and Graphing A-69 B-1 Linear Equations and Inequalities A-70 B-2 Cartesian Coordinate System A-82 B-3 Basic Formulas in Analytic Geometry A-91 C Sequences, Induction, and Probability 871 Sequences and Series 872 Mathematical Induction 883 Arithmetic and Geometric Sequences 894 The Multiplication Principle, Permutations, and Combinations 909 Sample Spaces and Probability 926 APPENDIX Special Topics A-105 C-1 Significant Digits A-106 C-2 Partial Fractions A-109 C-3 Descartes’ Rule of Signs A-118 C-4 Parametric Equations * *Available online at www.mhhe.com/barnett APPENDIX D Geometric Formulas A-123 Student Answers SA-1 Index I-1 VI www.elsolucionario.net bar51969_fm_i-xviii.qxd 26/1/08 12:39 AM Page vii Pinnacle 110:MHIA065:SE:FM: Preface Enhancing a Tradition of Success We take great satisfaction from the fact that more than 100,000 students have learned college algebra or precalculus from a Barnett Series textbook Ray Barnett is one of the masters of college textbook writing His central approach is proven and remains effective for today’s students The third edition of Precalculus: Graphs and Models has benefited greatly from the numerous contributions of new coauthor Dave Sobecki of Miami University Hamilton Dave brings a fresh approach to the material and many good suggestions for improving student accessibility Every aspect of the revision focuses on making the text more relevant to students, while retaining the precise presentation of the mathematics for which the Barnett name is renowned Specifically we concentrated on the areas of writing, worked examples, exercises, technology, and design Based on numerous reviews, advice from expert consultants, and direct correspondence with many users of previous editions, we feel that this edition is more relevant than ever before We hope you will agree Writing Without sacrificing breadth or depth or coverage, we have rewritten explanations to make them clearer and more direct As in previous editions, the text emphasizes computational skills, real-world data analysis and modeling, and problem solving rather than theory Examples In the new edition, even more solved examples in the book provide graphical solutions side-by-side with algebraic solutions By seeing the same answer result from their symbol manipulations and from graphical approaches, students gain insight into the power of algebra and make important conceptual and visual connections Likewise, we added expanded color annotations to many examples, explaining the solution steps in words Each example is then followed by a similar matched problem for the student to solve Answers to the matched problems are located at the end of each section for easy reference This active involvement in learning while reading helps students develop a more thorough understanding of concepts and processes Exercises With an eye to improving student performance and to make the book more useful for instructors, we have extensively revised the exercise sets We added hundreds of new writing questions as well as exercises at the easy to moderate level and expanded the variety of problem types to ensure a gradual increase in difficulty level throughout each exercise set Technology Although technology is employed throughout, we strive to balance algebraic skill development with the use of technology as an aid to learning and problem solving We assume that students using the book will have access to one of the various graphing calculators or computer programs that are available to perform the following operations: • Simultaneously display multiple graphs in a user-selected viewing window • Explore graphs using trace and zoom VII www.elsolucionario.net bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-2 pinnacle 110:MHIA065:SE:Index: I-2 SUBJECT INDEX Bisection method, 309–311 Boundary line, for half-planes, 747 Bounded solution region, 754 Break-even point, 252 Cardano, Girolamo, 332n Cardano’s formula, 332 Cardioid, 673–674 Cartesian coordinate system See Rectangular coordinates; Rectangular coordinate system Catenaries, 443, 968 Cayley, Arthur, 804 Center of circle, A–97 of ellipse, 973 of hyperbola, 985 Change-of-base formula, 426–428 Characteristic polynomials, 847 Circles equations and graphs of, 1002, A–96–A–100 explanation of, 704, A–97 formula for, A–125 Circular points coordinates of, 480–481 explanation of, 478 Clairaut, Alexis, 41 Closed sets, A–10 Closure property, A–7 Coefficient determinant, 856 Coefficients binomial, 949 explanation of, 714 of extinction, 451 numerical, A–38 of polynomial functions, 274 real, 322–324 Cofactor of element, 840–841 Cofunction identities for cosine, 574–575 for sine, 575 for tangent, 575 Column matrix, 778–779 Combination command, 949 Combination formula, 948–949 Combinations applications of, 921 explanation of, 919–921 order and, 922 Combined inequalities, 243 Combined variation, 367–368 Common difference, 894, 895 Common logarithms, 423, 424 Common ratio, 895 Commutative property, 654, A–5, A–7 Complementary angles, 465 Completing the square explanation of, A–99–A–100 to solve quadratic equations, 209–212 to solve quadratic expressions, 177–178 Complex numbers addition and subtraction of, 195–197, 683 applications for, 203 De Moivre’s theorem and, 689–690 division of, 198, 688–689 explanation of, 193, 683–684 historical background of, 191, 192 multiplication of, 197–200, 688–689 natural number power of, 690–691 polar-rectangular relationships for, 685 properties of, 195 radicals and, 200–202 roots of, 691–693 set of, 194 solving equations involving, 202–203 types of, 193–195 zero for, 197 Complex plane, 684 Composite numbers, A–48–A–49 Composition of functions, 88–93 of inverse functions, 107 Compound events, 927 Compound fractions, A–64–A–65 Compound interest applications of, 293, 391–393, 442, 450, 460, 461, 959 continuous, 393–395 explanation of, 391–392 Conditional equations, 152, 561, 604 Congruent triangles, A–96 Conic sections See also specific conic sections eccentricity of, 682 ellipses as, 963, 973–982 equations for, 705, 706, 1002, 1003, 1007 explanation of, 704, 961–963 focal chords of, 1036–1037 hyperbolas as, 963, 985–996 identification of, 1015–1016 parabolas as, 172, 175, 178–182, 277, 963–970 in polar form, 704–706 Conjecture counterexamples and, 884 explanation of, 883 Goldbach’s, 891 induction to prove, 886 Conjugate, of a ϩ bi, 193 Conjugate axis, 987 Conjugate hyperbolas, 991 Connected mode, on graphing calculators, 60, 348 Consistent systems, 717 Constant, A–3 Constant functions, 51, 52, 129 Constant of proportionality, 361, 362, 364–367 Constant-profit line, 764 Constant terms, in systems of equations, 714 Continuous compound interest, 393–395 Continuous compound interest formula, 394, 395 Continuous graphs, 59, 60 Contradiction, 152 Coordinate axis explanation of, A–82 rotation of, 1009–1011 Coordinate-free definition, 963 Coordinates of circular points, 480–481 of point, A–4 www.elsolucionario.net bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-3 pinnacle 110:MHIA065:SE:Index: SUBJECT INDEX polar, 667–671 rectangular, 668 Corner point, of solution region, 752 Correspondence, 21, 22 Cosecant function See also Trigonometric functions domain of, 484–485 explanation of, 482 inverse, 546 properties of, 513 Cosine function See also Trigonometric functions domain and range of, 484 double-angle identity for, 588 explanation of, 482 half-angle identity for, 588 inverse, 540–543 as periodic function, 509 sum and difference identities for, 572–574 Cosine-inverse cosine identities, 541 Cosines, law of See Law of cosines Cotangent function See also Trigonometric functions domain of, 485 explanation of, 482 inverse, 546 properties of, 512 Coterminal angles, 464 Counterexamples, 786, 885 Counting techniques combinations, 919–922 explanation of, 909–910 factorial, 913–915 multiplication principle, 910–913 permutations, 916–918, 922 Cramer, Gabriel, 855 Cramer’s rule, 855–858 Cube functions, 68 Cube root functions, 68 Cube roots, 68, A–18 Cubic equations, 692–693 Cursor, Curve fitting, 161, 162 Curves equation of, 1002, 1003 explanation of, 161 family of, 83 polar, 677–678 Dantzig, George B., 760 Data analysis explanation of, A–85 exponential regression and, 410–411 logarithmic models and, 436–437 sinusoidal regression and, 528–529 Data analysis and linear regression applications break-even, profit, and loss, 251–253 diamond prices, 161–167 optimal speed, 219–220 solid waste disposal, 216–218 Decibel, 431 Decimal degrees (DD), 466 Decimal expansions, A–6 Decision variables, 761, 764 Decoding matrix, 824, 825 Decomposition, of partial fractions, A–110, A–112–A–116 Decreasing functions, 50–53 Degenerate conic, 963 Degree-radian conversion formulas, 469 Degrees of angles, 465–468 conversions between radians and, 468–471 explanation of, 465 Degrees-minutes-seconds (DMS), 466 De Moivre, Abraham, 689–690 De Moivre’s theorem, 689–690 Denominators explanation of, A–10 least common, A–62 rationalizing, A–32–A–33 Dependent systems in three variables, 735–736 in two variables, 717 Dependent variables, 25, 29 Descartes, René, 192, A–13, A–119 Descartes’ rule of signs explanation of, A–119–A–120 use of, A–118, A–120 DET command, 839, 842, 851, 856, 858 Determinants coefficient, 856 Cramer’s rule and, 855–859 defining first-order, 838 evaluating higher-order, 843–845 evaluating second-order, 839 evaluating third-order, 840–843 explanation of, 838 of order n, 838 properties of, 847–852 Diagnostics, 162 Diagonal matrix, 812 Difference function, 85 Difference identities for cosine, 572–574 explanation of, 577 on graphing calculators, 578 for sine, 575 use of, 577 Difference of cubes formula, A–53, A–54 Difference of squares formula, A–53, A–54 Difference quotients, 36–38 Dimensions, of matrix, 778 Directrix, 704, 963 Dirichlet, Lejeune, 41 Discontinuous graphs, 59, 60 Discriminant, 215, 1015, 1016 Distance between two points, A–91–A–96 Distance-between-two-points formula, 640, A–93, A–96 Distributive property explanation of, A–5, A–7, A–38 factoring using, A–50 multiplication and, A–40 simplifying radicals using, A–31 for vectors, 654 Dividend, 293 Divisibility property, 889–890 Division of complex numbers, 198, 688–689 polynomial, 291–300 www.elsolucionario.net I-3 bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-4 pinnacle 110:MHIA065:SE:Index: I-4 SUBJECT INDEX Division (continued ) of rational expressions, A–60–A–61 of real numbers, A–8–A–9 synthetic, 294–298, 300 Division algorithm, 293–294, 298 Division property of equality, A–71 of inequality, A–78 Divisor, 293, 296 Domain of composition of functions, 90–93 explanation of, 22 of functions, 24, 25, 29–31, 33–36, 48–50 of polynomials, 33, 275 of quadratic function, 173 of trigonometric functions, 484–485, 509, 537 of variable, A–70–A–71 Dot mode, on graphing calculators, 60, 348 Double-angle identities, 585–586, 588 Double root, 208 Double subscript notation, 779 Double zero, 321 Doubling time, 400 Doubling time growth model, 400–402 Eccentricity, 682, 704, 705, 999 Eigenvalues, 847 Element cofactor of, 840 of matrix, 778 minor of, 840 Elimination by addition, 721–724, 733–738 to solve nonlinear systems, 1025–1026 Ellipses applications of, 980–982 equations of, 974–979 explanation of, 704, 963, 973 graphs of, 974–979 method to draw, 973–974 Empirical probability application of, 942–943 explanation of, 930, 938 finding or approximating, 938–942 to test theoretical probability, 943 Empty sets, A–2 Encoding matrix, 824 Equality, properties of, A–71 Equally likely assumptions, 934–937 Equal polynomials, A–110–A–111 Equations See also Systems of linear equations; Systems of nonlinear equations of circles, 1002, A–96–A–100 with complex numbers, 202–203 conditional, 152, 561, 604 of conic sections, 705, 706, 1002, 1003, 1007 cubic, 692–693 of curves, 1002, 1003 defining functions by, 25–29 of ellipses, 974–979 equivalent, A–71 explanation of, A–70 exponential, 440–444 fractions in, 152–155 on graphing calculators, 3–4, 12–14 graphs of, 27, 46, A–83–A–84 of hyperbolas, 986–994, 1012 inconsistent, 735 of lines, 141 logarithmic, 440, 445–448 matrix, 828–830 Mollweide’s, 637 of parabola, 178–182, 965–968 polar, 669–678, 705–706 power operations on, 227–228 quadratic, 206–220 of quadratic type, 231–234 with radicals, 226–231 root of, A–71 solution of, A–71–A–73 trigonometric, 489, 604–614 with two or more variables, 155 Equilibrium, static, 661–663 Equilibrium point, 166 Equilibrium price, 166 Equilibrium quantity, 166 Equivalent equations, A–71 Equivalent systems, 721–724 Eratosthenes, A–48 Euclid’s theorem, 223n Euler, Leonhard, 41, 191, 389, 693, 884 Even functions explanation of, 76 testing for, 77–78 Events arbitrary, 932, 935 compound, 927 explanation of, 926, 927, 930 probability of, 930–937 simple, 927, 931–932, 935 Exact values graphing calculators to find, 539–540 method for finding, 578–581, 587–588, 590–592, 597–600 Expansion, binomial, 946 Expected frequency, 939 Experiments explanation of, 926–927 outcome of, 930, 931 Exponential decay, 412 Exponential equations applications of, 442, 443 explanation of, 440 method to solve, 440–442, 444 Exponential form, converting between logarithmic form and, 419–421 Exponential functions with base e, 389–391 compound interest and, 391–395 explanation of, 382–383, 416 graphs of, 383–387, 390 properties of, 384, 387–389 Exponential growth models examples of, 400–402 explanation of, 399–400 limited, 407–408 negative, 403–404 Exponential models Carbon-14 dating, 405–406 comparison of, 412 www.elsolucionario.net bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-5 pinnacle 110:MHIA065:SE:Index: SUBJECT INDEX data analysis and regression, 410–411 exponential growth, 399–402 limited growth, 407–409 negative exponential growth, 403–405 Exponents integer, A–14–A–18 logarithms as, 418 notation for, A–13 proof of law of, 888–889 rational, A–19–A–21, A–24 Extended principle of mathematical induction, 891 Extraneous roots, 1022 Extraneous solution, 228 Extrema, local, 53–56 Face cards, 922 Factored form, polynomials in, 277 Factorials evaluation of, 914 explanation of, 913–914 reducing fractions involving, 915 zero, 913 Factoring composite numbers, A–48–A–49 explanation of, A–47 by grouping, A–51 of polynomials, 299, 322–324, A–47–A–56 to solve nonlinear systems, 1026–1027 to solve quadratic equations, 207–209 to solve trigonometric equations, 606–609 Factoring formulas, A–53 Factor theorem, 299, 300 Family of curves, 83 Feasible region, 755, 756, 765, 766 Feasible solutions, 755, 756 Fermat, Pierre de, 891 Fermat’s last theorem, 891 Fibonacci, Leonardo, 874 Fibonacci sequences, 874–875 Finite sequences, 873 Finite series arithmetic, 898–900 explanation of, 877 geometric, 901–902 Finite sets, A–2 First-order determinants, 838 Fixed cost, 156 Focal chords, 971, 1036–1037 Focus of ellipse, 973 explanation of, 704 of hyperbola, 985 of parabola, 963 Force vectors addition of, 659–660 application for, 662–663 explanation of, 659 resolved into components, 661 Four-leafed rose, 675 Fourth-order determinants, 843–845 Fractional expression, A–58 FRACTION command, 199 Fractions compound, A–64–A–65 in equations, 152–155 explanation of, A–10 involving factorials, 915 partial, A–109–A–116 proper, A–110 properties of, A–10, A–11 reduced to lowest terms, A–59–A–60 significant digits in decimal, A–107 simple, A–64 Frequency, 938, 939 Function notation, 30–32, 693 Functions absolute value, 68 basic concepts of, 46–50 composition of, 88–91 constant, 51, 52, 129 cube, 68 cube root, 68 decreasing, 50–53, 104 defined by equations, 25–29 definition of, 23, 100 difference, 85 domain of, 24, 25, 29–31, 33–36, 48–50 evaluation of, 32 even, 76–78 explanation of, 23 exponential, 382–395, 416 finding quotient of two, 87 finding sum of two, 86–87 finding zeros of, 304 graphs of, 46–62, 67 greatest integer, 59–61 historical background of, 41 hyperbolic, 450 identity, 68 increasing, 50–53, 104 inverse, 99–115 inverse hyperbolic, 450 linear, 128–144 logarithmic, 384, 416–428 notation for, 30–32 odd, 76–78 one-to-one, 100–104 operations on, 84–87 periodic, 509–514 piecewise-defined, 57–59 polynomial, 274–284 probability, 934 product, 85 quadratic, 172–185 quotient, 85 range of, 24, 25, 49–50 rate of change of, 142–143 set form of definition of, 24 square, 68 square root, 68 sum, 85 transcendental, 381 transformations of, 68 vertical line test for, 27–29 Fundamental period of f, 509 Fundamental sample spaces, 928 Fundamental theorem of analytic geometry, A–83 www.elsolucionario.net I-5 bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-6 pinnacle 110:MHIA065:SE:Index: I-6 SUBJECT INDEX Fundamental theorem of arithmetic, A–49 Fundamental theorem of linear programming, 765 f(x), 30–32 Galileo, 184 Gauss, Carl Friedrich, 320, 785 Gauss-Jordan elimination applications of, 824–825, 833–835 determinants and, 838–845 determinants and Cramer’s Rule and, 854–859 explanation of, 778 on graphing calculators, 778, 788–791 identity matrix for multiplication and, 815–817 inverse of square matrix and, 817–823 inverses to solve systems of equations and, 830–833 matrix equations and, 828–830 matrix operations and, 797–811 modeling and, 792–794 performing row operations on matrix and, 778–782 properties of determinants and, 847–852 recognizing reduced matrix and, 782–785 solving systems by, 785–791 General form of polynomial, 274, 379 of quadratic equation, 206 of quadratic function, 172 General term, of sequence, 872, 876–877 Geometric sequences explanation of, 895 finding terms in, 897 method to recognize, 895–896 nth term of, 897 Geometric series sum formulas for finite, 901–902 sum formulas for infinite, 903–904 Geostationary orbit, 649 Goldbach, Christian, 891 Goldbach’s conjecture, 891 Graphing calculator commands combination, 949 DET, 839, 842, 851, 856, 858 FRACTION, 199 INTERSECT, 9–13, 47, 163, 166, 226, 231, 393, 407, 426, 612, 720, 1028 MAXIMUM, 284, 311 MINIMUM, 311 randInt, 941 ROOT, 47 RREF, 787–791 SEQ, 899, 900 SUM, 899, 900 TABLE, 949 TRACE, 8–12, 676, 683, 941 ZERO, 47, 284, 304, 311 ZOOM, 8–12 Graphing calculators as aid to hand graphing, 6–7 angle conversions using, 466 changing graphing mode on, 60 combinations on, 921 complex numbers on, 196–201 compound interest on, 392, 393, 395 connected mode on, 60, 348 conversions between rectangular and polar form using, 669–670, 685–687 to determine if equation defines a function, 28 domain of functions on, 34–36 dot mode on, 60, 348 entering composition of two functions in, 92, 93 exponential equations on, 440–444 exponential functions on, 382–383, 385–386, 388 exponential models on, 401, 402, 404, 406, 407, 409 exponential regression on, 410, 411 finding appropriate viewing window on, finding equation of simple harmonic graph on, 526–527 finding exact values with, 539–540, 543 fractions on, 154 Gauss-Jordan elimination on, 778, 788–791 graphics of conics on, 1006 graphing equations on, 3–4 harmonic analysis using, 622–623 histograms on, 941 linear inequalities on, 747–749 linear regression on, 161–166 logarithm functions on, 424, 425 logarithmic equations on, 446, 447 matrices on, 779, 780, 784n, 798–804, 806, 807, 819, 823 maximum and minimum on, 54, 55, 311 modeling on, 15–17, 56–57 overview of, 2–4 parabolas on, 967 partial fraction decomposition on, A–114 permutations on, 918, 919 polar equations on, 675–676, 705 polynomial functions on, 284–286, 304, 306, 307, 309, 312 polynomial inequalities on, 313, 314 rational functions on, 348–349, 351 rational inequalities on, 354 rounding to significant digits on, 626 scatter plots on, 39–40, 161–163 scientific notation on, A–22–A–23 screen coordinates on, 7–8 sequences on, 873–875, 877 series on, 880, 900, 902 SINDIV program for, 307 slope-intercept form on, 138 to solve equations, 12–15 to solve linear equations, 151–152 to solve nonlinear systems, 1021, 1023–1026, 1028 to solve trigonometric equations, 604, 611–614 storing functions on, 68 synthetic division on, 296–297, 307 systems of linear equations on, 715, 720, 725–728, 740–743, 831 systems of linear inequalities on, 751–754 testing identities using, 567–568 trigonometric functions on, 485, 486, 488, 489, 495, 496, 511 viewing windows on, 3, 5, 26 zeros of polynomials on, 325, 326 Graphs of arbitrary rational functions, 352–353 of circles, 26, A–96–A–100 of conics, 1004–1007 continuous, 59, 60 discontinuous, 59, 60 of ellipses, 974–979 of equations, 27, 46, A–89–A–91 www.elsolucionario.net bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-7 pinnacle 110:MHIA065:SE:Index: SUBJECT INDEX of exponential functions, 383–387, 390 finding domain and range from, 49–50 four-leafed rose, 675 of functions, 46–62, 67 horizontal and vertical shifts in, 69–70 of hyperbolas, 986–994 of intervals and inequalities, A–75, A–76 of inverse functions, 112–115 of linear equations in two variables, 132 of linear functions, 128, 129, 131–134 of linear inequalities, 746–750 of lines, 132–133 of logarithmic functions, 417–419 multiplicities from, 325 of parabolas, 965–968, A–84 point-by-point plotting and, 671–672, A–83–A–85 of polar equations, 671–676 of polynomials, 276–280, 283–284, 308, 309, 324–325 prepared by hand, 6–7 of quadratic functions, 173–175 of rational functions, 336–353 reflections of, 73, 74 rotation used in, 1011–1015 of simple harmonic, 526–527 to solve inequalities, 242–246 to solve systems of linear equations in two variables, 714–718, 720–721 to solve systems of linear inequalities, 750–754 stretching and shrinking in, 71–72 translation use in, 1004–1008 of trigonometric functions, 486–488, 510, 511, 519–525 types of basic, 67–68 vertically and horizontally translating, 70–71 Greatest integer, of real number, 59 Greatest integer function explanation of, 59–60 use of, 60–61 Grouping, factoring by, A–51 Half-angle identities, 588–590 Half-life, 403, 405 Half-life decay model, 403 Half-planes, 746 Hand graphing, 6–7 Harmonic analysis, 519, 622–623 Heron of Alexandria, 645 Histograms, 941 Horizontal asymptotes explanation of, 342–344 of rational functions, 344–347, 350 Horizontal axis, A–82 Horizontal lines, 141 Horizontal line test, 102–104 Horizontal shifts, 69–70 Horizontal shrinks, 72 Horizontal stretches, 72 Horizontal translations, 70–71, 74 Hyperbolas applications of, 994–996 conjugate, 991 equations of, 986–994, 1012 explanation of, 704, 963, 985 graphs of, 986–994 method to draw, 985–986 Hyperbolic functions, 450 Hyperbolic paraboloids, 994 Hyperboloids, 994, 1000 Hypotenuse, 494 Identities See also Trigonometric equations additive, A–5, A–7 basic trigonometric, 503–504, 562 cofunction, 574–575, 577 cosine-inverse cosine, 541 double-angle, 585–586, 588 explanation of, 561, 604 half-angle, 588–590 multiplicative, A–5, A–7 for negatives, 503, 562 product-sum, 596–598 Pythagorean, 503, 562 quotient, 503, 562 reciprocal, 485, 562 sine-inverse sine, 538 sum and difference for cosine, 572–574, 577 sum and difference for sine and tangent, 575–577 sum-product, 598–601 tangent-inverse tangent, 544 transformation, 622 using graphing calculator to test, 567–568 verification of, 563–567, 581, 592 Identity functions explanation of, 128 graph of, 68 Identity matrix, for multiplication, 815–817 Imaginary axis, 684 Imaginary numbers explanation of, 191–194 finding powers of pure, 192 Imaginary part, of a ϩ bi, 193 Imaginary unit, 191–193, 693 Imaginary zeros, 330–332, A–111, A–120, A–121 Incidence matrix, 814 Inconsistent systems in three variables, 735–736 in two variables, 717, 718 Increasing functions, 50–53 Independent systems, 717 Independent variables, 25, 128 Index, A–27 Inequalities combined, 243 involving absolute value, 244–246 linear, A–73–A–75, A–77–A–78 polynomial, 312–314 properties of, A–78 quadratic, 247–249 rational, 354–356 solution set for, 241, A–77 symbols for, A–73–A–75 systems of linear, 745, 750–757 trigonometric, 613–614 Infinite sequences, 873 www.elsolucionario.net I-7 bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-8 pinnacle 110:MHIA065:SE:Index: I-8 SUBJECT INDEX Infinite series explanation of, 877 geometric, 903–904 Infinite sets, A–2 Infinity symbol, A–75 Initial side, of angle, 464 Integer exponents explanation of, A–14 properties of, A–15–A–17 simplification of, A–18 Integers, A–3 Interest, 391 Interest rate, 391 Intermediate Value Theorem, 247n Interpolating polynomials, 378–379 INTERSECT command, 9–13, 47, 163, 166, 226, 231, 393, 407, 426, 612, 720, 1028 Intersection, A–76 Interval notation, A–74–A–77 Intervals explanation of, A–74 graphs of, A–75 union and intersection of, A–786 Inverse additive, A–5, A–7 of function, 104–106 matrix, 817–825, 833–835 multiplicative, A–5, A–7 property of, 110–111 to solve systems of equations and Gauss-Jordan elimination, 830–833 of square matrix, 817–823 Inverse functions explanation of, 104–106, 536 on graphing calculators, 114, 115 graphs of, 112–114 hyperbolic, 450 method for finding, 106–110, 114–115 modeling with, 111–112 properties of, 107 Inverse proportionality, 364 Inverse trigonometric functions cosecant, 546 cosine, 540–543, 546 cotangent, 546 explanation of, 535–537 secant, 546 sine, 495–496, 537–540, 546 tangent, 543–546 Irrational numbers explanation of, A–3, A–6 historical background of, 190 Irrational zeros, 330 Isoprofit line, 764 Joint variation, 366–367 Jordan, Wilhelm, 785 Kepler, Johannes, 683, 980 Kirchhoff’s laws, 837 Knots, 473 Kung, Sidney H., 601 Lagrange, Joseph Louis, 891 Lagrange’s four square theorem, 891 Law of cosines applications for, 645–646 derivation of, 639–640 explanation of, 639 solving SAS case with, 640–642 solving SSS case with, 642–645 use of, 639, 658 Law of sines applications for, 633–635 derivation of, 627–628 explanation of, 628 solving AAS case with, 629 solving ASA case with, 628–629 solving SSA case with, 629–633 use of, 626, 628, 641, 658 Learning curves, 407–408 Least common denominator (LCD), A–62 Least-squares line, 457 Left and right behavior, 278, 280–283 Left endpoint of interval, A–74 Leibniz, Gottfried, 41 Like terms, combining, A–38–A–39 Limited growth, 412 Limited-growth models, 407–409 Limits, 278n Linear and quadratic factors theorem, 323, A–111 Linear equations modeling with, 156–160 in one variable, A–71 solution to, A–71–A–73 in two variables, 132 Linear factors nonrepeating, A–112–A–113, A–115 repeating, A–114–A–1115 Linear functions explanation of, 128–130 finding x and y intercepts and, 130 graphs of, 128, 129, 131–134 modeling and, 142–144 parallel and perpendicular lines and, 141–142 slope of line and, 134–136 using special forms of, 137–140 Linear inequalities combined, 243 graphs of, 746–750 involving absolute value, 244–246 in one variable, 241–242 solution to, A–77–A–79 Linear programming explanation of, 760, 764–765 fundamental theorem of, 765 modeling and, 767–768 Linear programming problems example of, 760–764 explanation of, 760, 764–765 method to solve, 765–767 Linear regression, 161–167 Linear speed, 471–474 Lines equations of, 141 graphs of, 132–133 www.elsolucionario.net bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-9 pinnacle 110:MHIA065:SE:Index: SUBJECT INDEX parallel, 141–142 perpendicular, 141–142 secant, 148, 188 slope of, 134–137 tangent, 148 Local maxima, 54–56 Local minima, 54–56 Location theorem, 247, 308 Logarithmic equations explanation of, 440 method to solve, 445–448 Logarithmic-exponential conversions, 419–421 Logarithmic-exponential relationships, 425–426 Logarithmic functions common, 423, 424 explanation of, 384, 416–418 graphs of, 417–419 natural, 423, 424 properties of, 418, 421–422 Logarithmic models data analysis and regression, 436–437 logarithmic scales, 431–436 Logarithmic scales, 431–436 Logarithms calculator evaluation of, 424–426 change of base and, 426–428 common, 423, 424 natural, 423, 424 Logistic equation, 408 Logistic growth, 412 Long division, 292–294 Lower bound, 305 Magnitude of earthquake, 433 vector, 650–652, 655 Major axis, 973 Marginal cost, 157 Mathematical induction extended principle of, 891 principle of, 885, A–15 proof of binomial formula using, 952–953 proof of conjecture using, 886 proof of divisibility property using, 889–890 proof of law of exponents using, 888–889 proof of summation formula using, 887–888 Matrices addition and subtraction of, 798–801 augmented coefficient, 780–783 column, 778–779, 804 decoding, 824, 825 diagonal, 812 element of, 778 encoding, 824, 825 explanation of, 778–780 Gauss-Jordan elimination and, 797–811 on graphing calculators, 779, 780, 784n, 798–804, 806, 807, 819, 823 identity, 815–817 incidence, 814 inverse of square, 817–823 multiplication of, 801–810 m x n, 778 negative of, 799 principal diagonal of, 779 properties of, 829 reduced, 782–785 row, 804 row-equivalent, 781 singular, 817 square, 778 upper triangular, 812, 846 Matrix equations, 828–830 Matrix inverses applications of, 824–825, 833–835 explanation of, 817–820 methods to find, 820–823 to solve systems of equations, 830–833 Maxima, local, 54–56 Maximization problems, 764 MAXIMUM command, 284, 311 Member of set, A–2 Midpoint formula, A–93–A–95 Midpoint of line segment, A–93–A–96 Minima, local, 54–56 Minimization problems, 764 MINIMUM command, 311 Minor axis, 973 Minor of element, 840 Minutes, 466 Modeling to construct model in economics, 94–95 with direct variation, 361–363 explanation of, 15, 792 on graphing calculators, 15–17, 56–57 with inequalities, 250–251 with inverse variation, 364–365 with joint and combined variation, 366–368 with linear equations, 156–157 with linear programming, 767–768 with polynomial regression, 285–286 with polynomials, 314–315 with quadratic equations, 215–216 with quadratic functions, 183–185 with radicals, 234–235 with sinusoidal regression, 528–529 solutions obtained from, 57 with systems in three variables, 738–742 with systems of linear inequalities, 754–757 Modeling applications agriculture, 767–768 construction, 314–315 cost analysis, 156–157 data analysis, 38–41, 740–742 design, 215–216 distance-rate-time, 157–159 economics, 94–95 manufacturing, 15–17 maximum area, 183–184 mixture, 159–160 production scheduling, 738–740, 754–757 projectile motion, 184–185, 250–251 purchasing, 792–793 rate of change, 142–143 revenue, 111–112 weight estimates, 285–286 well depth, 234–235 www.elsolucionario.net I-9 bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-10 pinnacle 110:MHIA065:SE:Index: I-10 SUBJECT INDEX Modulus, 685 Mollweide’s equation, 637 Monomials, A–38 Moore’s law, 414 Multiplication of binomials, A–41–A–42 of complex numbers, 197–200, 688–689 identity matrix, 815–817 of matrices, 801–810 of polynomials, A–40–A–43 with radical forms, A–32 of rational expressions, A–60–A–61 of real numbers, A–4–A–7 Multiplication principle applications of, 910–913 explanation of, 911–912 Multiplication property of equality, A–71 of inequality, A–78 Multiplicative identity for complex numbers, 198 explanation of, A–5, A–7 for vectors, 654 Multiplicative inverse, A–5, A–7 Multiplicities explanation of, 321 from graphs, 325 of zeros, 321, 322 Multiplier doctrine, 905 m x n matrix, 778 n!, 915–916 Natural logarithmic base, 693 Natural logarithms, 423, 424 Natural numbers, A–3 Nautical miles, 473 Navigational compass, 567, 702 Negative angles, 464 Negative growth, 403 Negative real numbers explanation of, A–4, A–9, A–10 principal square root of, 201 Negatives, identities for, 503 Negative zeros, A–119 Newton, Isaac, 689 n linear factors theorem, 321 Nonlinear systems See Systems of nonlinear equations Nonnegative constraints, 762, 764 Nonnegative restrictions, 753 Norm, of vector, 652 Notation/symbols absolute value, 838n double subscript, 779 exponent, A–13 factorial, 915n function, 30–32, 693 imaginary unit, 693 inequality, A–73–A–75 interval, A–74–A–75 n!, 915–916 natural logarithmic base, 693 radical, A–27 scientific, A–21–A–24 set builder, A–2 for set of real numbers, A–3 summation, 878 nth root, A–18–A–19 nth-root radicals, A–27 nth-term formulas, 896–897 nth-term theorem, 691, 692 Null sets, A–2 Numbers complex, 191–203 composite, A–48–A–49 factor of, A–47 imaginary, 191–194 irrational, 190, A–3, A–6 natural, A–3 prime, A–48 rational, 382, A–3, A–4–A–7 real, 59, 201, A–2–A–11, A–4 Numerators, A–10 Numerical coefficients, A–38 Objective function, 761 Oblique asymptotes, 351, 352 Oblique triangles, 626 Obtuse angles, 465 Obtuse triangles, 626 Odd functions, 77–78 One-to-one correspondence, 684, A–4, A–82–A–83 One-to-one functions explanation of, 100–101 identification of, 101–104 Optimal solution, 763, 765, 766 Optimal value, 764, 767 Ordered pairs, 23, 24 Ordering, 916, 922 Order of operations, A–43 Ordinate, 46 Origin explanation of, A–4, A–82 in polar coordinate system, 667 reflection in, 73, 74 Parabolas applications of, 968–970 axis of symmetry of, 175, 963, 968 equations of, 178–182, 704, 965–968 focal chord of, 971 graphs of, 965–968, A–84 method to draw, 964 turning point on, 277 vertex of, 178–181, 963, 965, 966 Paraboloids, 969, 994 Parallel lines, 141–142 Parallelogram formula, A–124 Parallelogram rule, 653, 659–660 Partial fractions basic theorems and, A–110–A–111 decomposition of, A–112–A–116 explanation of, A–109–A–110 Pascal’s triangle, 947–948 Perfect square formula, A–53, A–54 Perihelion, 683, 703 Periodic functions www.elsolucionario.net bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-11 pinnacle 110:MHIA065:SE:Index: SUBJECT INDEX explanation of, 509–514 trigonometric functions as, 501, 536 Periods of f, 509 formula for, 521–522 of trigonometric functions, 520–522, 525, 526 Permutations application of, 918–919 explanation of, 916–918 order in, 922 Perpendicular lines, 141–142 Phase shift, 523, 525, 526 Photic zone, 414, 451 ph scale, 438 Piecewise-defined functions applications of, 57–59 evaluation of, 58 explanation of, 57 greatest integer function as, 59–61 sharp corners in, 57 Pitiscus, 493 Pixels explanation of, 2, screen coordinates of, 7–8 Planetary orbits, 705–706 Point-by-point plotting, 671–672, A–83–A–85 Point-slope form, 139–141 Polar axis, 667 Polar coordinate system conic sections and, 705 explanation of, 667–668 Polar curves, 677–678 Polar equations of conics, 705, 706 conversions between rectangular and, 669–671 graphs of, 671–678 Polar form conic sections in, 704–706 conversions between rectangular and, 669–671, 685–687 explanation of, 685–688 multiplication and division in, 688–689 product of complex numbers in, 690 Polar sketching, rapid, 673–675 Pole, 667 Polygons, finding side of regular, 644 Polynomial division factor theorem and, 299, 300 remainder theorem and, 297–298, 300 using long division, 291–294 using synthetic division, 294–297 Polynomial functions explanation of, 274 graphs of, 276–280, 283–284 identification of, 275–276 left and right behavior of, 278, 280–283 Polynomial inequalities on graphing calculators, 313, 314 methods to solve, 312–314 Polynomial regression models, 285–286 Polynomials addition and subtraction of, A–39–A–42 application of, A–44 bisection method and, 309–311 characteristic, 847 classification of, A–37–A–38 combining like terms in, A–38–A–39 degree of, A–37 domain of, 33, 275 equal, A–110–A–111 evaluation of, 298 explanation of, A–36 in factored form, 277 factors of, 299, 322–324, A–47–A–56 fundamental theorem of algebra and, 320–322 general form, 274, 379 graphs of, 276–280, 283–284, 308, 309, 324–325 interpolating, 378–379 multiplication of, A–40–A–43 of odd degree, 324 in one variable, A–36 prime, A–49 rational zeros of, 325–332 with real coefficients, 322–325 real zeros of, 303–312, 324 reduced, 329, 330 second-degree, A–52–A–53 simplification of, A–39 Taylor, 398 in two variables, A–36 zeros of, 276–277, 299–300, 325–332 Position, of element in matrix, 779 Positive angles, 464 Positive real numbers, A–4 Positive zeros, A–119 Power operations, on equations, 227–228 Prime numbers, A–48, A–57 Prime polynomials, A–49 Principal, 391, 392 Principal diagonal, of matrix, 779, 839 Principal nth root, A–19, A–27 Principle of mathematical induction, 885 Principle square root of negative real number, 201 Probability empirical, 930, 938–943 of events, 931–937 explanation of, 930 theoretical, 930, 939–940, 943 Probability function, 934 Problem constraints, 761, 764 Product function, 85 Product-sum identities, 596–598 Proper fractions, A–110 Pure imaginary numbers, 192, 193 Pythagorean identity, 503 Pythagoreans, 190 Pythagorean theorem, 497, 974, 993, A–124 Quadrantal angles, 464 Quadratic equations completing the square to solve, 209–212 data analysis and regression and, 216–220 explanation of, 206–207 factoring to solve, 207–209 general form of, 206 modeling with, 215–216 quadratic formula to solve, 212–215 www.elsolucionario.net I-11 bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-12 pinnacle 110:MHIA065:SE:Index: I-12 SUBJECT INDEX Quadratic factors nonrepeating, A–115 repeating, A–116 Quadratic formula explanation of, 213 imaginary zeros and, A–111 to solve quadratic equations, 212–215, 1016, 1028 to solve trigonometric equations, 610–611 Quadratic functions analysis of, 176 completing the square and, 176–178 explanation of, 172–173 general form of, 172 graphs of, 173–175 modeling with, 183–185 properties of, 175 vertex form of, 173–176 Quadratic inequalities, 247–249 Quadratic-type equations, 231–234 Quotient function, 85 Quotient identities, 503 Quotients difference, 36–38 explanation of, 293 of functions, 86–87 Radian-degree conversion formulas, 469 Radians conversions between degrees and, 468–471 explanation of, 467–468 linear and angular speed and, 472 Radicals complex numbers and, 200–202 conversions between rational exponents and, A–28 explanation of, A–27 modeling using, 234–235 multiplication with, A–32 nth-root, A–27 properties of, A–29–A–30 in simplified form, A–30–A–31 solving equations involving, 226–231 Radicand, A–27 Radius, of circle, A–97 randInt command, 941 Random experiments, 926 Range explanation of, 22 of functions, 24, 25, 49–50 Rapid polar sketching, 673–675 Rate of change, 142–143 Rational exponents evaluation of, A–24 explanation of, A–19–A–20 properties of, A–20–A–21 radical conversions and, A–28 use of, A–21 Rational expressions addition and subtraction of, A–62–A–64 compound fractions and, A–64–A–65 explanation of, A–58–A–59 expressed as partial fractions, A–109–A–110 multiplication and division of, A–60–A–61 reduced to lowest terms, A–59–A–60 Rational functions domain and x intercepts of, 337–338 explanation of, 336–337 graphs of, 336–353 method for graphing, 345–353 properties of, 339 vertical and horizontal asymptotes of, 340–345 zeros of, 337–338 Rational inequalities explanation of, 354 on graphing calculators, 354 method to solve, 355–356 Rationalizing factor, A–32, A–33 Rationalizing the denominator, A–32–A–33 Rational numbers addition and multiplication of, A–4–A–7 explanation of, A–3, A–4 use of calculator for, 382 Rational zeros explanation of, 325–327 method for finding, 327–332 Rational zero theorem, 326–328 Real axis, 684 Real coefficients, 322–324 Real number line, 684 Real numbers addition and multiplication of, A–4–A–7 greatest integer of, 59 negative, A–4, A–9, A–10 positive, A–4 properties of, A–7, A–8 set of, A–3, A–4, A–7 subtraction and division of, A–8–A–9 zero properties of, A–9, A–10 Real part, of a ϩ bi, 193 Real plan, A–82 Real roots approximation of, 206 explanation of, 47, 215 Real values, 29 Real zeros approximation of, 307, 311–312 bisection method to locate, 309–311 complications caused by, 337 explanation of, 47, a–119 method to find, 303 polynomials of odd degree and, 324 upper and lower bound for, 304–308 Réaumur, René, 369 Reciprocal identities, 485, 503 Reciprocals, 198, 199 Rectangles area of, 645 formulas for, A–124 Rectangular coordinates conversions between polar and, 669–671 explanation of, 668 graphs of, A–83–A–85 Rectangular coordinate system, A–82–A–83 Rectangular form conversions between polar and, 669–671, 685–687 explanation of, 683–684 www.elsolucionario.net bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-13 pinnacle 110:MHIA065:SE:Index: SUBJECT INDEX Recursion formulas explanation of, 874 sequences specified by, 959–960 Recursive definition of an, 888 Reduced augmented coefficient matrix, 790 Reduced matrix, 782–785 Reduced polynomials, 329, 330 Reduced system, 785 Reference angles, 506 Reference triangles, 505–507, 580 Reflection, in x and y axes, 73, 74 Regression exponential, 410–411 polynomial, 285–286 sinusoidal, 528–529 Regression analysis applications for, 216–220 explanation of, 161 Regression models comparison of, 456–457 linear, 457 polynomial, 285–286 Relations, 22, 23 Relative frequency, 938–939 Relative growth rate, 402 Remainder, 293 Remainder theorem, 297–298, 300 Replacement set, A–3, A–70 Resultant, of vectors, 653 Resultant force, 659–660 Resultant velocity, 657–659 Richter, Charles, 433 Richter scale, 433 Right angles, 465 Right circular cones, 962n, A–125 Right circular cylinders, A–125 Right endpoint of interval, A–74 Right triangle ratios, 494 Right triangles explanation of, 493–494 solution to, 494–498, 626 ROOT command, 47 Roots cube, 68, A–18 double, 208 extraneous, 1022 nth, 691, A–18–A–19 real, 47, 206, 215, A–18–A–19 Rotation angle of, 1013 of axis, 1009–1011 of coordinate axis, 1009–1011 used in graphing, 1011–1015 Rotation formulas, 1010, 1011 Rounding convention, A–108 Row-equivalent matrix, 781 Row operations, on matrices, 778–782 Rule, A–2 Sample spaces example of, 929–930 explanation of, 927–929 fundamental, 928 method to choose, 927–928 SAS case, solution to, 639–642 Scalar components, of vector, 652 Scalar multiplication, 653–654 Scalar product, 653, 655 Scalar quantities, 650 Scatter plots explanation of, 39, 161 on graphing calculators, 39–40, 161–163 Scientific notation, A–21–A–24, A–106–A–107 Screen coordinates, of pixel, 7–8 Secant function See also Trigonometric functions domain of, 485 explanation of, 482 inverse, 546 properties of, 514 Secant lines, 148, 188 Secondary diagonal, 839 Second-degree polynomials, A–52–A–53 Second-order determinants, 839 Seconds, 466 Semiperimeter, 645 SEQ command, 899, 900 Sequences arithmetic, 894–897 explanation of, 871–874 Fibonacci, 874–875 finite, 873 general term of, 872, 876–877 geometric, 895–897 on graphing calculators, 874 infinite, 873 recursion formula and, 874, 875 specified by recursion formulas, 959–960 terms of, 872 Series alternating, 879 arithmetic, 898–900 explanation of, 877–878 finite, 877 infinite, 877 in summation notation, 879 terms of, 878–880 Set-builder notation, A–2 Sets closed, A–10 explanation of, A–2–A–3 of real numbers, A–3, A–4, A–7 replacement, A–3, A–70 Sharp corners, in piecewise-defined functions, 57 Shrink, vertical and hozontal, 71–72 Side adjacent to angle , 494 Side opposite angle , 494 Sieve of Eratosthenes, A–48 Significant digits angles and, 626 in decimal fractions, A–107 explanation of, A–106–A–107 rounding numbers to, A–108 Sign properties, in trigonometric functions, 505 Similar triangles, A–124 Simple events, 927, 931–932, 935 www.elsolucionario.net I-13 bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-14 pinnacle 110:MHIA065:SE:Index: I-14 SUBJECT INDEX Simple fractions, A–64 Simple harmonic motion, 518 Simple harmonics explanation of, 518–519 finding equation from graph of, 526–528 Simplex method, 760 SINDIV, 307 Sine function See also Trigonometric functions cofunction identities for, 575 domain and range of, 484, 537 explanation of, 482 inverse, 495–496, 537–540 as periodic function, 509 sum and difference identities for, 575–576 Sine-inverse sine identities, 538 Sines, law of See Law of sines Singular matrix, 817 Sinusoidal regression, use of, 528–529 Slope explanation of, 134–135, 141 geometric interpretation of, 135 method for finding, 135–137 as rate of change, 142–143 Slope-intercept form explanation of, 137–138 use of, 138–139 Solution extraneous, 228 feasible, 755, 756 graphical approximations of real, 1027–1028 of inequality, A–77 of linear systems, 714, 716–717, 733 optimal, 763, 765, 766 unique, 717 Solution region bounded and unbounded, 754 explanation of, 750–752 Solution set for equations, 606, A–71 for inequality, A–77 of linear systems, 714 Solving the right triangle, 493 Solving the triangle, 626 Special product formulas, A–42 Speed average, 471 linear and angular, 471–474 Sphere, A–126 Squared viewing window, on graphing calculators, 26 Square functions, 68 Square matrix inverse of, 817–823 of order n, 778, 815 Square root functions, 68 Square root property explanation of, 209 to solve quadratic equations, 210–211 Square roots explanation of, A–18 principle, 201 SSA case solution to, 629–633 variations in, 630 SSS case, solution to, 639, 642–644 Standard deck, 922 Standard form, of equations, 141 Standard position, angle in, 464 Standard vectors, 651–652 Standard viewing window, on graphing calculators, Standard window variables, on graphing calculators, Stant asymptote, 351 Static equilibrium, 661–663 Statute miles, 473 Straight angles, 465 Stretch, vertical and hozontal, 71–72 Subsets, of real numbers, A–3, A–4 Substitution to solve linear systems, 718–721, 723–724 to solve nonlinear systems, 1020–1027 to solve quadratic-type equations, 232 to solve trigonometric equations, 608–609 Substitution property, of equality, A–71 Subtraction of complex numbers, 195–197, 683 of matrices, 800–801 of polynomials, A–39–A–42 of rational expressions, A–62–A–64 of real numbers, A–8–A–9 Subtraction property of equality, A–77 of inequality, A–78 Sum of functions, 86–87 of infinite geometric series, 904 SUM command, 899, 900 Sum formulas for finite arithmetic series, 898–900 for finite geometric series, 901–903 for infinite geometric series, 903–904 Sum function, 85 Sum identities for cosine, 574 explanation of, 577 on graphing calculators, 578 for sine, 575 Summation formula, proof of, 887–888 Summation notation, 878, 879 Summing index, 878 Sum of cubes formula, A–53, A–54 Sum of the squares of the residuals (SSR), 457 Sum-product identities, 598–601 Supplementary angles, 465 Symmetry in even and odd functions, 76 trigonometric functions and, 510–514 Synthetic division examples using, 296, 298, 300, 306 explanation of, 294–296 on graphing calculators, 296–297, 307 Synthetic division table, 306 Systems of linear equations applications of, 724–728 basic terms of, 717–718 determinants and, 854–858 elimination by addition to solve, 721–724, 733–738 equivalent, 721–724 www.elsolucionario.net bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-15 pinnacle 110:MHIA065:SE:Index: SUBJECT INDEX explanation of, 714 Gauss-Jordan elimination to solve, 778, 785–791 graphing to solve, 714–718, 720–721 matrix inverse methods to solve, 830–833 modeling with, 738–742 substitution to solve, 718–721, 723–724 in three variables, 732–742 in two variables, 714–728 Systems of linear inequalities graphs of, 745 modeling with, 754–757 solutions to, 750–754 Systems of nonlinear equations elimination to solve, 1025–1026 explanation of, 1020 factoring to solve, 1026–1027 graphical approximations of real solutions and, 1027–1028 substitution to solve, 1020–1027 TABLE command, 949 Tail-to-tip rule, 653, 658 Tangent function See also Trigonometric functions cofunction identities for, 575 domain of, 485 explanation of, 482 half-angle identity for, 589 inverse, 543–545 properties of, 511–512 sum and difference identities for, 575–576 Tangent identities difference identity for, 576 tangent-inverse, 544 Tangent lines, 148 Taylor polynomials, 398 Terminal side, of angle, 464 Terms of sequence, 872, 876–877 of series, 878–880 Theorems, 883 Theoretical probability empirical probability to test, 943 explanation of, 930 method to find, 939–940 Third-order determinants, 840–843 TRACE command, 8–12, 676, 683, 941 Transcendental functions, 381 Transformation identities, 622 Transformations combining, 75 explanation of, 68, 173 reflections and, 73, 74 stretching and shrinking and, 71–72 vertical and horizontal shifts and, 69–70 Transitive property, A–78 Translation of coordinate axis, 1001–1003 in graphing, 1004–1008 vertical and horizontal, 70–71, 74 Translation formulas, 1001 Transverse axis, 985 Trapezoid formula, A–125 Triangles area of, 645–646 congruent, A–96 formulas for, A–124 oblique, 626 obtuse, 626 Pascal’s, 947–948 reference, 505–507, 580 right, 493–498, 626 significant digits and, 626 similar, A–124 Trichotomy property, A–74 Trigonometric equations See also Identities explanation of, 604 solution to, 489 use of algebraic approach to solve, 605–611 use of graphing calculator to solve, 604, 611–614 Trigonometric functions See also specific functions angles and, 464–474 applications for, 493, 561 definitions for, 482–485, 502–503 evaluation of, 483–484, 486 finding values of, 507–508 on graphing calculators, 485–486, 488, 489, 495, 496, 511 graphs of, 486–489, 510, 519–525 graphs of simple harmonic, 526–528 inverse, 495–496, 535–546 mathematical modeling and data analysis and, 528–529 properties of, 501–514 sign changes in, 505 to solve right triangles, 493–498, 626 unit circle approach to, 478–489 wrapping function and, 478–481 Trigonometric inequalities, solution to, 613–614 Trigonometric ratios explanation of, 483 geometric interpretation of, 499–500 used for right triangles, 494–498 Trigonometry, 493 Trinomials, A–38 Turning points approximating real zeros at, 311–312 explanation of, 277, 278, 488 of trigonometric functions, 488–489, 519–520, 522 Twin primes, A–57 Unbounded solution region, 754 Union, A–76 Unique solution, 717 Unit circle circumference of, 508 finding exact values by using, 580 trigonometric functions in terms of coordinates of points on, 478–489 Unit vectors explanation of, 655–656 expressed in terms of i and j vectors, 656–657 Unlimited growth, 412 Upper and lower bound theorem, 305–308 Upper bound, 305 Upper triangular matrix, 812, 846 www.elsolucionario.net I-15 bar51969_sndx_001-016.qxd 01/28/2008 06:45 PM Page I-16 pinnacle 110:MHIA065:SE:Index: I-16 SUBJECT INDEX Variable cost, 156 Variables decision, 761 dependent, 25 domain of, A–70–A–71 explanation of, A–3 independent, 25, 128 Variation combined, 367–368 direct, 361–363 explanation of, 361 inverse, 364–365 joint, 366–367 Variation in sign, A–118–A–119 Vector addition, 653–654, 659–660 Vector components, 653, 660–661 Vector quantities, 650 Vectors algebraic properties of, 654 applications of, 650 explanation of, 650–651 force, 659–663 magnitude of, 652 scalar multiplication and, 653–654 standard, 651–652 unit, 655–657 velocity, 657–659 zero, 650, 661 Velocity, apparent and actual, 657–659 Velocity vectors, 657–659 Vertex of angle, 464 of cone, 962n of ellipse, 973 of hyperbola, 985 of parabola, 178–181, 963, 965, 966 Vertex form, of quadratic functions, 173–176 Vertical asymptotes explanation of, 340–342 of rational functions, 342, 346, 347, 350 Vertical axis, A–82 Vertical lines, 141 Vertical line test, for functions, 27–29 Vertical shifts, 69–70 Vertical shrink, 72, 74 Vertical stretch, 72, 74 Vertical translation, 70–71, 74 Viewing windows, on graphing calculators, 3, 5, 26 Wiles, Andrew, 891 Window variables, on graphing calculators, Wrapping function, 478–481 x axis, reflection in, 73, 74 x intercepts explanation of, 47 of functions, 47–48 of linear functions, 130 of polynomial functions, 276–277 Xmas, Xmin, Xres, Xscl, y axis, reflection in, 73, 74 y intercepts explanation of, 47 of functions, 47–48 of linear functions, 130 Ymax, Yscl, ZERO command, 47, 284, 304, 311 Zero product property, 207, 209, 298, 808 Zero properties, of real numbers, A–9, A–10 Zeros of complex numbers, 197 double, 321 explanation of, 193 of functions, 206 imaginary, A–111, A–120, A–121 irrational, 330 multiplicities of, 321, 322 negative, A–119, A–121 of polynomials, 276–277, 299–300, 303–312, 325–332 positive, A–119, A–121 of rational functions, 337–338 of rational inequities, 354–356 real, 47, 303–312, 324, 337, A–119 of trigonometric functions, 488–489, 519–520 Zero vectors, 650, 661 ZOOM command, 9–11 www.elsolucionario.net ... Data Precalculus : graphs and models — 3rd ed / Raymond A Barnett [et al.] p cm — (Barnett, Ziegler, and Byleen’s precalculus series) Rev ed of: Precalculus : graphs and models / Raymond A Barnett, ... calculator ISBN 0-07-305195-0, ISBN 978-0-07-305195-6 Precalculus: Graphs and Models, Third Edition This book is the same as College Algebra: Graphs and Models with three additional chapters on trigonometry... 0-07-305196-9, ISBN 978-0-07-305-196-3 College Algebra with Trigonometry: Graphs and Models This book is the same as Precalculus: Graphs and Models except that the trigonometric functions are introduced