Lial m et al college algebra and trigonometry 6ed global 2017

Global edition College Algebra & Trigonometry For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects and requirements, equipping students with the best possible learning tools This Global Edition preserves the cutting-edge approach and pedagogy of the original, but also features alterations, customization, and adaptation from the North American version Global edition sixth edition Lial Hornsby Schneider Daniels College Algebra & Trigonometry Sixth edition Margaret L Lial • John Hornsby • David I Schneider • Callie J Daniels G LOBa l edition This is a special edition of an established title widely used by colleges and universities throughout the world Pearson published this exclusive edition for the benefit of students outside the United States and Canada If you purchased this book within the United States or Canada, you should be aware that it has been imported without the approval of the Publisher or Author Pearson Global Edition Lial_06_1292151951_Final.indd 23/09/16 11:45 AM    Get the most out of MyMathLab® MyMathLab, Pearson’s online learning management system, creates personalized experiences for students and provides powerful tools for instructors With a wealth of tested and proven resources, each course can be tailored to fit your specific needs Talk to your Pearson Representative about ways to integrate MyMathLab into your course for the best results Data-Driven Reporting for Instructors MyMathLab’s comprehensive online gradebook automatically tracks students’ results to tests, quizzes, homework, and work in the study plan • The Reporting Dashboard, found under More Gradebook Tools, makes it easier than ever to identify topics where students are struggling, or specific students who may need extra help • Learning in Any Environment Because classroom formats and student needs continually change and evolve, MyMathLab has built-in flexibility to accommodate various course designs and formats • With a new, streamlined, mobile-friendly design, students and instructors can access courses from most mobile devices to work on exercises and review completed assignments • www.mymathlab.com A00_LIAL1953_06_GE_FEP.indd 07/09/16 4:35 pm CHAPTER 6  Analytic Geometry Available in MyMathLab® for Your Course Achieve Your Potential Success in math can make a difference in your life MyMathLab is a learning experience with resources to help you achieve your potential in this course and beyond MyMathLab will help you learn the new skills required, and also help you learn the concepts and make connections for future courses and careers Visualization and Conceptual Understanding These MyMathLab resources will help you think visually and connect the concepts NEW! Guided Visualizations These engaging interactive figures bring mathematical concepts to life, helping students visualize the concepts through directed explorations and purposeful manipulation Guided Visualizations are assignable in MyMathLab and encourage active learning, critical thinking, and conceptual learning Video Assessment Exercises Assignable in MyMathLab, Example Solution Videos present the detailed solution process for every example in the text Additional Quick Reviews cover definitions and procedures for each section Assessment exercises check conceptual understanding of the mathematics www.mymathlab.com A00_LIAL1953_06_GE_FEP.indd 07/09/16 4:35 pm    Preparedness and Study Skills MyMathLab® gives access to many learning resources which refresh knowledge of topics previously learned Getting Ready material and Skills for Success Modules are some of the tools available Getting Ready Students refresh prerequisite topics through skill review quizzes and personalized homework integrated in MyMathLab With Getting Ready content in MyMathLab students get just the help they need to be prepared to learn the new material Skills for Success Modules Skills for Success Modules help foster success in collegiate courses and prepare students for future professions Topics such as “Time Management,” “Stress Management” and “Financial Literacy” are available within the MyMathLab course www.mymathlab.com A00_LIAL1953_06_GE_FEP.indd 07/09/16 4:35 pm This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM College Algebra & Trigonometry SIXTH EDITION GLOBAL EDITION Margaret L Lial American River College John Hornsby University of New Orleans David I Schneider University of Maryland Callie J Daniels St Charles Community College Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo A01_LIAL1953_06_GE_FM.indd 15/09/16 10:40 am Editorial Director: Chris Hoag MathXL Content Manager: Kristina Evans Editor in Chief: Anne Kelly Marketing Manager: Claire Kozar Editorial Assistant: Ashley Gordon Marketing Assistant: Fiona Murray Program Manager: Danielle Simbajon Senior Author Support/Technology Specialist: Joe Vetere Project Manager: Christine O’Brien Rights and Permissions Project Manager: Gina Cheselka Program Management Team Lead: Karen Wernholm Procurement Specialist: Carol Melville Project Management Team Lead: Peter Silvia Associate Director of Design: Andrea Nix Assistant Acquisitions Editor, Global Edition: Murchana Borthakur Program Design Lead: Beth Paquin Senior Project Editor, Global Edition: Amrita Naskar Text Design: Cenveo® Publisher Services Manager, Media Production, Global Edition: Vikram Kumar Illustrations: Cenveo® Publisher Services Senior Manufacturing Controller, Global Edition: Jerry Kataria Cover Design: Lumina Datamatics Ltd Media Producer: Jonathan Wooding Cover Image: gusenych/Shutterstock TestGen Content Manager: John Flanagan Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2017 The rights of Margaret L Lial, John Hornsby, David I Schneider, and Callie J Daniels to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled College Algebra & Trigonometry, 6th edition, ISBN 978-0-13-411252-7, by Margaret L Lial, John Hornsby, David I Schneider, and Callie J Daniels, published by Pearson Education © 2017 Acknowledgments of third-party content appear on page 1181, which constitutes an extension of this copyright page All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC 1N 8TS All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners TI-84 Plus C screenshots from Texas Instruments Courtesy of Texas Instruments MYMATHLAB, MYMATHLAB PLUS, MATHXL, LEARNING CATALYTICS, and TESTGEN are exclusive trademarks owned by Pearson Education, Inc or its affiliates in the U.S and/or other countries Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners and any references to third-party trademarks, logos, or other trade dress are for demonstrative or descriptive purposes only Such references are not intended to imply any sponsorship, endorsement, authorization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc or its affiliates, authors, licensees, or distributors ISBN 10: 1-292-15195-1 ISBN 13: 978-1-292-15195-3 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library 10 Typeset by Cenveo® Publisher Services Printed and bound by Vivar in Malaysia A01_LIAL1953_06_GE_FM.indd 22/09/16 5:34 pm To Rhonda, Sandy, and Betty Johnny To my MS & T professors, Gus Garver, Troy Hicks, and Jagdish Patel C.J.D A01_LIAL1953_06_GE_FM.indd 15/09/16 10:40 am Contents Preface 17 Resources for Success 22 R Review of Basic Conceptsâ•… 27 R.1 Setsâ•… 28 Basic Definitions╇ ■╇ Operations on Sets R.2 Real Numbers and Their Propertiesâ•… 35 Sets of Numbers and the Number Linꕇ ■╇Exponents╇■╇ Order of Operations╇ ■ Properties of Real Numbers╇ ■╇ Order on the Number Linꕇ ■╇ Absolute Value R.3 Polynomialsâ•… 50 Rules for Exponents╇ ■╇Polynomials╇■╇ Addition and Subtraction╇ ■╇ Multiplication╇ ■╇Division R.4 Factoring Polynomialsâ•… 62 Factoring Out the Greatest Common Factor╇ ■╇ Factoring by Grouping╇ ■╇ Factoring Trinomials╇ ■╇ Factoring Binomials╇ ■╇ Factoring by Substitution R.5 Rational Expressionsâ•… 72 Rational Expressions╇ ■╇ Lowest Terms of a Rational Expression╇ ■╇ Multiplication and Division╇ ■╇ Addition and Subtraction╇ ■╇ Complex Fractions R.6 Rational Exponentsâ•… 81 Negative Exponents and the Quotient Rulꕇ ■╇ Rational Exponents╇ ■╇ Complex Fractions Revisited R.7 Radical Expressionsâ•… 92 Radical Notation╇ ■╇ Simplified Radicals╇ ■╇ Operations with Radicals╇ ■╇ Rationalizing Denominators Test Prepâ•… 103╇ ■╇ Review Exercisesâ•… 107╇ ■╇Testâ•… 111 1.1 Equations and Inequalitiesâ•… 113 Linear Equationsâ•… 114 Basic Terminology of Equations╇ ■╇ Linear Equations╇ ■╇ Identities, Conditional Equations, and Contradictions╇ ■╇ Solving for a Specified Variable (Literal Equations) 1.2 Applications and Modeling with Linear Equationsâ•… 120 Solving Applied Problems╇ ■╇ Geometry Problems╇ ■╇ Motion Problems╇ ■╇ Mixture Problems╇ ■╇ Modeling with Linear Equations 1.3 Complex Numbersâ•… 131 Basic Concepts of Complex Numbers╇ ■╇ Operations on Complex Numbers A01_LIAL1953_06_GE_FM.indd 15/09/16 10:40 am CONTENTS 1.4 Quadratic Equations  139 The Zero-Factor Property  ■  The Square Root Property  ■  Completing the Square  ■  The Quadratic Formula  ■  Solving for a Specified Variable  ■  The Discriminant Chapter Quiz (Sections 1.1–1.4)  149 1.5 Applications and Modeling with Quadratic Equations  150 Geometry Problems  ■  The Pythagorean Theorem  ■  Height of a Projected Object  ■  Modeling with Quadratic Equations 1.6 Other Types of Equations and Applications  162 Rational Equations  ■  Work Rate Problems  ■  Equations with Radicals  ■  Equations with Rational Exponents  ■  Equations Quadratic in Form Summary Exercises on Solving Equations  175 1.7 Inequalities  176 Linear Inequalities  ■  Three-Part Inequalities  ■  Quadratic Inequalities  ■  Rational Inequalities 1.8 Absolute Value Equations and Inequalities  188 Basic Concepts  ■  Absolute Value Equations  ■  Absolute Value Inequalities  ■  Special Cases  ■  Absolute Value Models for Distance and Tolerance Test Prep  196  ■  Review Exercises  201  ■ Test  207 Graphs and Functions  209 2.1 Rectangular Coordinates and Graphs  210 Ordered Pairs  ■  The Rectangular Coordinate System  ■  The Distance Formula  ■  The Midpoint Formula  ■  Equations in Two Variables 2.2 Circles  221 Center-Radius Form  ■  General Form  ■  An Application 2.3 Functions  229 Relations and Functions  ■  Domain and Range  ■  Determining Whether Relations Are Functions  ■  Function Notation  ■  Increasing, Decreasing, and Constant Functions 2.4 Linear Functions  245 Basic Concepts of Linear Functions  ■  Standard Form Ax + By = C  ■  Slope  ■  Average Rate of Change  ■  Linear Models Chapter Quiz (Sections 2.1–2.4)  259 2.5 Equations of Lines and Linear Models  260 Point-Slope Form  ■  Slope-Intercept Form  ■  Vertical and Horizontal Lines  ■  Parallel and Perpendicular Lines  ■  Modeling Data  ■  Graphical Solution of Linear Equations in One Variable Summary Exercises on Graphs, Circles, Functions, and Equations  273 A01_LIAL1953_06_GE_FM.indd 15/09/16 10:40 am 1184 INDEX Cartesian equations, 840 Celsius-Fahrenheit relationship, 120, 272 Center of an ellipse, 996 of a circle, 221 of a hyperbola, 1008 Center-radius form of the equation of a circle, 221 Certain events, 1092 Chain rule, 311 Change in x, 248 Change in y, 248 Change-of-base theorem for logarithms, 480 Chords, table of, 563 Circle(s) applications of, 224–225 arc length of, 595 circumference of, 592 definition of, 221 eccentricity of, 1001, 1013 equations of, 221–223 general form of the equation of, 222–223 geometric interpretation of, 986 graphing by calculator, 222 graph of, 222 polar form of, 840–841, 845 radius of, 221 sector of, 597 tangent line to, 227 unit, 221, 604 Circular functions definition of, 604 domains of, 606 evaluating, 606 finding numbers with a given circular function value, 608–609 unit circle and, 221 Circumference of a circle, 592 Cis u, 820 Cissoid graphing, 847 with a loop, 847 Clinometer, 576 Closed interval, 177 Closure properties, 40 Cloud ceiling, 576 Coefficient binomial, 1067 correlation, 375 definition of, 52 leading, 330 Cofactor of an element of a matrix, 903–904 Cofunction identities, 548, 697–698 Cofunctions of trigonometric functions, 548 Collinear points, 213 Column matrix, 946 Column of a matrix, 890, 904 Combinations definition of, 1083 distinguishing from permutations, 1085–1086 formula for, 1084 Combined variation, 411 Z04_LIAL1953_06_GE_IDX.indd 1184 Common difference of an arithmetic sequence, 1043 Common factors of a rational expression, 72, 73 Common logarithms applications of, 476 definition of, 475 modeling with, 476 Common ratio of a geometric sequence, 1053 Commutative properties, 40, 41, 952 Complement of an event, 1092 of a set, 30, 32 Complementary angles, 525 Completing the square, 141, 332 Complex conjugates definition of, 135 property of, 135, 358 Complex fractions definition of, 77 simplifying, 77–78, 87–88 Complex numbers absolute value of, 820 argument of, 820 conjugates of, 135 definition of, 131 De Moivre’s theorem for, 830 graph of, 819 imaginary part of, 131 modulus of, 820 nth root of, 831 nth root theorem for, 832 operations on, 133–137 polar form of, 820 powers of, 830 product theorem for, 824 quotient theorem for, 825 real part of, 131 rectangular form of, 819 roots of, 831–834 standard form of, 131, 819 trigonometric form of, 820 Complex plane imaginary axis of, 819 real axis of, 819 Components of a vector horizontal, 809 vertical, 809 Composite function definition of, 308 derivative of, 311 Composition of functions, 308 Compound amount, 453 Compound events, 1094–1095 Compounding continuously, 454 Compound interest definition of, 452 general formula for, 452 Concours d’elegance, 428 Conditional equation, 116, 117 Conditional statement converse of, 212 definition of, 212 Conditional trigonometric equations with half-angles, 742 linear methods for solving, 738 with multiple angles, 743–744 satisfying, 738 solving with the quadratic formula, 740 steps to solve algebraically, 741 using trigonometric identities to solve, 740–741 zero-factor property method for solving, 739 Congruence axioms, 770 Congruent triangles, 770–771 Conic definition of, 1001 eccentricity of, 1001 Conic sections definition of, 986 equations of, 1017, 1119 geometric definition of, 1021 geometric interpretation of, 986 identifying from equations, 1019 polar form of, 1113–1115 summary of, 1018–1019 Conjugate(s) complex, 135, 358 properties of, 358 of a radical, 99 Conjugate zeros theorem, 358 Consistent systems, 872 Constant function, 238, 246 Constant of variation, 409 Constant velocity problems, 121–122 Constraints, 938 Continuity of a function, 274 Continuous compounding, 454 Continuous function, 274, 1032 Contradiction equations, 116 Convergent sequence, 1033 Converse of a conditional statement, 212 Coordinate plane, 210 Coordinates, photographic, 784 Coordinates of a point on a line, 35, 210 Coordinate system definition of, 35 polar, 618, 837 rectangular, 837 Corner point of a region of feasible solutions, 939 Correlation coefficient, 375 Cosecant function characteristics of, 653 definition of, 533 domain of, 653 graph of, 653 inverse of, 728 period of, 653 range of, 540, 653 steps to graph, 654 Cosine function amplitude of, 621 characteristics of, 620 definition of, 533 difference identity for, 695–697 domain of, 620 double-angle identity for, 709–710 graph of, 620 half-angle identity for, 715, 716 horizontal translation of, 633 inverse of, 726–727 07/09/16 3:34 pm INDEX period of, 620, 623 range of, 540, 620 steps to graph, 624, 635 sum identity for, 695–697 translating graphs of, 633, 634, 635 vertical translation of, 634 Cosines, law of applications of, 787–788 applying to solve a triangle, 785 derivation and statement of, 785–786 Heron’s formula and, 790–792 Cost fixed, 252 per item, 252 Cost-benefit model, 81 Cost function, 252 Cotangent function characteristics of, 644 definition of, 533 domain of, 644 graph of, 644 horizontal translation of, 647–648 inverse of, 728 period of, 644 range of, 540, 644 steps to graph, 645 vertical translation of, 647–648 Coterminal angles, 527 Counting numbers, 28 Counting theory, 1080–1086 Cramer’s rule for solving linear systems derivation of, 907 general form of, 908 Critical angle, 765 Cryptography, 440 Cube root function, 277 Cubes difference of, 66 sum of, 66 Cubic equation, 144 Cubing function, 276 Curvature, degree of, 603 Cycloid calculator graphing of, 852 definition of, 852 D Damped oscillatory motion, 661 Dantzig, George, 937 Decibels, 477 Decimal degrees, 526 Decimals, 35 Decimal window, 388 Decomposition into partial fractions, 915, 920 of rational expressions, 915 Decreasing function, 238 De Fermat, Pierre, 210 Definite integral, 121, 335, 1037 Degree measure converting to radian measure, 592–593 definition of, 524 Degree mode, calculators in, 535, 554 Degree of a polynomial, 52 Z04_LIAL1953_06_GE_IDX.indd 1185 Degree of a term, 52 Degree of curvature, 603 Degree/radian relationship converting measures, 592–593 table of, 594 Delta x, 248 Delta y, 248 De Moivre, Abraham, 830 De Moivre’s theorem, 830 Denominator least common, 75 rationalizing, 98, 534, 538, 541 Dependent equations, 872 Dependent variable, 229, 231 Depression, angle of, 567 Derivatives of a function, 249, 308, 311, 354 of parametric equations, 566 power rule for, 93 of trigonometric functions, 687 Descartes, René, 210, 211 Descartes’ rule of signs, 360 Descending order, 53, 56 Determinant(s) definition of, 902 evaluating, 904 expansion of, 904 of a * matrix, 902 of a * matrix, 903 theorems for, 906 Determinant equation, 913 Diagonal form of a matrix, 891 Diastolic pressure, 628 Difference of cubes, 66 of squares, 66 Difference identity application of, 700–704 for cosine, 695–697 for sine, 698–699 for tangent, 699–700 Difference quotient, 307 Difference tones, 750 Differential equations, 499 Differentiation, 311 Digits, significant, 564–565 Dimensions of a matrix, 890 Direction angle of a vector, 809 Directly proportional definition of, 409 as nth power, 410 Directrix of a parabola, 988 Direct variation definition of, 409 as nth power, 410 Discontinuity of the graph of a function, 275, 387, 397 Discriminant, 145–146 Disjoint interval, 177 Disjoint sets, 31 Distance, rate, and time problems, 121–122 Distance formula, 211–212 Distributive property, 40, 41, 952 Divergent sequence, 1033 Division of complex numbers, 136 of functions, 304 1185 order of operations for, 37 of polynomials, 56, 57, 346 of rational expressions, 74–75 synthetic, 347–348 Division algorithm for polynomials, 346 special case of, 349 Domain(s) agreement on, 232 of circular functions, 606 of a function, 234, 304 of inverse circular functions, 729 of a rational expression, 72 of a relation, 231 Dominating term, 330, 368 Dot product of vectors definition of, 812 geometric interpretation of, 813 properties of, 812 Double-angle identities definition of, 709 simplifying expressions using, 712 Double solution, 141 Doubling time, 499 E e continuous compounding and, 454 value of, 454 Eccentricity definition of, 1001 of a circle, 1001, 1013 of a conic, 1001 of an ellipse, 1001–1002, 1013 of a hyperbola, 1012, 1013 of a parabola, 1001, 1013 symbol e , 1001 Element cofactor of, 904 of a matrix, 890 minor of, 903 of a set, 28 Elevation, angle of, 567, 571 Elimination method definition of, 873 for solving linear systems, 873, 878 for solving nonlinear systems, 924 Ellipse(s) applications of, 1002 center of, 996 definition of, 851, 995 eccentricity of, 1001–1002 equations of, 997 foci of, 995 geometric definition of, 996 geometric interpretation of, 986 major axis of, 996 minor axis of, 996 reflective property of, 1003 translations of, 999–1000 vertices of, 996 Empty set definition of, 29, 116 symbol for, 29, 116 End behavior of graphs of polynomial functions, 368, 369 Endpoint of a ray, 524 07/09/16 3:34 pm 1186 INDEX Envelopes, 662, 994 Epitrochoids, 852 Equality addition property of, 114 multiplication property of, 114 Equal matrices, 946 Equal sets, 30 Equation(s) absolute value, 189–190 Cartesian, 840 of a circle, 221–223 for compound interest, 452 conditional, 116, 117 conditional trigonometric See Conditional trigonometric equations of conic sections, 1017, 1119 connecting graphs with, 625, 648, 656 contradiction, 116, 117 cubic, 144 definition of, 114 dependent, 872 differential, 499 of an ellipse, 997 equivalent, 114 exponential See Exponential equations first-degree, 115, 872 graphing of, 216 graph of, 216 of a horizontal line, 263 of a hyperbola, 1010 identity, 116, 117 independent, 872 of an inverse function, 436–437 of a line, 260 linear See Linear equations literal, 117 logarithmic See Logarithmic equations nonlinear systems of See Nonlinear systems of equations of a parabola, 986 parametric, 566, 631, 850 polar, 840–844, 845 power property of, 165–166 quadratic in form, 169 quadratic in one variable See Quadratic equations in one variable radical, 165–166 rational, 162 with rational exponents, 168 rectangular, 840, 844 roots of, 114, 350 rotation, 1117 second-degree, 139 solution set of, 114 solutions of, 114 steps to graph, 216 trigonometric See Trigonometric equations types of, 116 of a vertical line, 263 Equilibrant vector, 801–802 Equilibrium demand, 889 Equilibrium price, 889 Equilibrium supply, 889 Z04_LIAL1953_06_GE_IDX.indd 1186 Equivalent equations, 114 Equivalent systems of linear equations, 873 Even-degree polynomial functions, 367, 369 Even function, 292, 620, 680 Even-odd identities, 680–681 Event(s) certain, 1092 complement of, 1092 compound, 1094–1095 definition of, 1091 impossible, 1092 independent, 1080 mutually exclusive, 1095 odds in favor of, 1093–1094 probability of, 1091 union of two, 1094–1095 Eves, Howard, 211 Exact number, 564 Expansion of binomials, 56, 1065 of determinants, 904 by a given row or column, 904 Experiment binomial, 1097 outcomes of, 1091 Explicitly defined sequence, 1033 Exponent(s) base of, 36 definition of, 36, 445 integer, 50 negative, 81–82 notation for, 37, 85 properties of, 445–446 rational, 81–88, 168 rules for, 50–52, 82, 86 zero, 51 Exponential decay function definition of, 499 modeling with, 501–502 Exponential equations applications of, 493 definition of, 450 solving, 450, 487–489 steps to solve, 492 Exponential expressions definition of, 36 evaluating, 37 Exponential function(s) characteristics of graphs of, 449 evaluating, 447 graph of, 448–449 modeling with, 455 standard form of, 447 Exponential growth function definition of, 499 modeling with, 499–501 Expressions algebraic, 36 classifying as polynomials, 53 exponential, 36–37 rational, 72–78, 915 roots of, 37 F Factored form of a polynomial, 62 Factorial notation, 1066, 1081 Factoring binomials, 66–68 definition of, 62 difference of cubes, 66 difference of squares, 66 greatest common factor, 62 by grouping, 63 perfect square trinomial, 65 polynomials, 62 to solve quadratic equations, 139 substitution method for, 67–68 sum of cubes, 66 trinomials, 64–65 Factors of a number, 36 of a polynomial function, 372 Factor theorem, 353 Fahrenheit-Celsius relationship, 120, 272 Fan Cost Index (FCI), 887 Fibonacci, 1034 Fibonacci sequence, 1031, 1034 Finite sequence, 1032 Finite series, 1035–1036 Finite set, 28 First-degree equations definition of, 115, 872 in n unknowns, 872 Fixed cost, 252 Foci (Focus) of an ellipse, 995 of a hyperbola, 1007 of a parabola, 988 FOIL, 54 Four-leaved rose, 842 Fractals, 822 Fraction bar, 37 Fractions complex, 77–78, 87 decimal form of, 35 finding least common denominator, 75 fundamental principle of, 72 partial, 915–920 unit, 612 Frequency definition of, 660 fundamental, 745 Frustum of a pyramid, formula for volume of, 60 Function(s) absolute value, 277–278 alternative definition of, 235 argument of, 631, 683 circular See Circular functions composite, 308, 311 composition of, 308 constant, 238, 246 continuity of, 274 continuous, 274, 1032 cosecant See Cosecant function cosine See Cosine function cost, 252 cotangent See Cotangent function cube root, 277 cubing, 276 decreasing, 238 definition of, 229 derivative of, 249, 308, 311, 354 discontinuity of, 275, 387, 397 07/09/16 3:34 pm INDEX domain of, 234, 304 even, 292, 620, 680 exponential See Exponential function(s) exponential decay, 499, 501–502 exponential growth, 499–501 greatest integer, 279 identity, 275 increasing, 238 input-output machine illustration, 231 integrals of, 713 inverse See Inverse functions inverse cosecant, 728 inverse cosine, 726–727 inverse cotangent, 728 inverse secant, 728 inverse sine, 723–725 inverse tangent, 727–728 limit of, 43, 235 linear See Linear functions linear cost, 252 logarithmic See Logarithmic function(s) logistic, 510 notation, 235, 435 objective, 938 odd, 292, 619, 680 one-to-one, 432–434, 722 operations on, 304 periodic, 618 piecewise-defined, 278 polynomial See Polynomial function(s) profit, 252 quadratic See Quadratic function range of, 234 rational See Rational functions reciprocal, 387–388 revenue, 252 secant See Secant function sine See Sine function sinusoidal, 636 square root, 277 squaring, 276 step, 280 tangent See Tangent function tests for one-to-one, 434 trigonometric See Trigonometric functions vertical line test for, 233 Fundamental frequency, 745 Fundamental identities, 680–683 Fundamental principle of counting, 1080 Fundamental principle of fractions, 72 Fundamental rectangle of a hyperbola, 1009 Fundamental theorem of algebra, 356 Fundamental theorem of linear programming, 938 Future value, 117, 453, 1058–1059 ƒ1x2 notation, 235 G Galilei, Galileo, 152 Gauss, Carl Friedrich, 356 Gauss-Jordan method for solving linear systems, 891 Z04_LIAL1953_06_GE_IDX.indd 1187 General form of the equation of a circle, 222–223 Generalized principle of mathematical induction, 1075–1076 General term of a sequence, 1032 Geometric interpretation of circles, 986 Geometric progression, 1053 Geometric sequence common ratio of, 1053 definition of, 1053 infinite, 1058 nth term of, 1053 sum of infinitely many terms of, 1058 sum of terms of, 1055–1056 Geometric series definition of, 1055 infinite, 1056–1058 Geometry formulas, 1121 Geometry problems, 121, 150 Grade, 248, 405 Grade resistance, 556 Graph(s) of a circle, 222 of complex numbers, 819 compressed, 621 connecting with equations, 625, 648, 656 of cosecant function, 653 of cosine function, 620 of cotangent function, 644 of equations, 216 of exponential functions, 448–449 of a horizontal line, 246 horizontal translation of, 294, 366 of inequalities, 936 intercepts of, 216 of inverse cosecant function, 728 of inverse cosine function, 726–727 of inverse cotangent function, 728 of inverse functions, 439–440, 723 of inverse secant function, 728 of inverse sine function, 724–725 of inverse tangent function, 727–728 of linear inequalities in two variables, 934–937 of logarithmic functions, 463–467 of a parabola, 276, 987, 988 parametric, 850 of polar coordinates, 838 of polar equations, 840–844 of polynomial functions, 365–366 of quadratic functions, 330–332 of rational functions, 389–390 of the reciprocal function, 388 reflecting across a line, 288 reflecting across an axis, 288 of a relation, 230 of secant function, 652 shrinking of, 286–287 of sine function, 619 stretched, 621 stretching of, 286–287 summary of polar, 845 symmetry of, 289 of systems of inequalities, 936–937 of tangent function, 643 translations of, 293, 366 of a vertical line, 247 vertical translation of, 293, 366 1187 Graphing calculator method analyzing the path of a projectile, 854 for circles, 222 converting from trigonometric form to rectangular form, 820 decimal window on, 388 degree mode in, 535, 554 finding magnitude and direction angle of vectors, 809 finding the determinant of a matrix, 902 finding trigonometric function values with, 554 for fitting exponential curves to scatter diagrams, 456 graphing a cycloid, 852 graphing a plane curve defined parametrically, 850 graphing of polar equations, 841–844 performing row operations, 894 for rational functions, 389, 398 for the reciprocal function, 387 sine regression feature, 636–637 for solving linear equations in one variable, 267–268 for solving nonlinear systems, 923 solving problems involving angles of elevation, 571 for solving systems of equations using matrix inverses, 965 for solving systems of linear equations in two variables, 873, 875 solving trigonometric equations by linear methods, 738 square viewing window on, 222 standard viewing window on, 217 ZTrig viewing window, 621 Graphing techniques even and odd functions, 292 reflecting, 288 stretching and shrinking, 286–287 summary of, 297 symmetry, 289–291 translations, 297 Greatest common factor, 62 Greatest integer function, 279 Groundspeed, 803 Grouping, factoring by, 63 H Half-angle identities application of, 716–718 definition of, 715 general statement about, 716 simplifying expressions using, 717–718 trigonometric equations with, 742 verifying, 718 Half-life, 501 Half-plane boundary of, 934 definition of, 934 Harmonic motion, 660 Height of a projected object, 152, 158 Heron of Alexandria, 789 07/09/16 3:34 pm 1188 INDEX Heron’s formula application of, 790 derivation of, 790–792 statement of, 789 Heron triangle, 797 Hipparchus, 563 Holding time, 90 Hooke’s Law, 415 Horizontal asymptote definition of, 387, 391 determining, 391 Horizontal axis, 819 Horizontal component of a vector, 809 Horizontal line equation of, 263, 266 graph of, 246 slope of, 249 Horizontal line test for one-to-one functions, 433 Horizontal parabola definition of, 986 graph of, 987 Horizontal shrinking of the graph of a function, 287 Horizontal stretching of the graph of a function, 287 Horizontal translation(s) of cosine function, 633 of cotangent function, 647–648 of a graph, 294 graphing, 631–633 of a graph of a polynomial function, 366 of sine function, 632 of tangent function, 647 Hubble constant, 271 Hyperbola(s) asymptotes of, 1008 center of, 1008 definition of, 721, 1007 eccentricity of, 1012, 1013 equations of, 1010 foci of, 1007 fundamental rectangle of, 1009 geometric definition of, 1007 geometric interpretation of, 986 translations of, 1011 transverse axis of, 1008 vertices of, 1008 Hyperbolic spiral, 857 Hypotenuse, 547 Hypotenuse of a right triangle, 151 Hypotrochoids, 852 I i definition of, 131 powers of, 136–137 i, j unit vectors, 812 Identities, trigonometric See Trigonometric identities Identity equations, 116, 117 Identity function, 275 Identity matrix, 960 Identity property for addition, 40 for multiplication, 40 Imaginary axis, 819 Z04_LIAL1953_06_GE_IDX.indd 1188 Imaginary numbers, 131 Imaginary part of a complex number, 131 Imaginary unit, 131 Impedance, 829 Impossible events, 1092 Inclination, angle of, 802 Inconsistent systems, 872, 894 Increasing function, 238 Independent equations, 872 Independent events, 1080 Independent variable, 229, 231 Index of a radical, 92, 94 of summation, 1036 Induction, mathematical, 1072–1077 Inequalities absolute value, 190 definition of, 42, 176 graphing, 934–937 linear, 176, 178, 934–937 nonstrict, 181, 935 properties of, 176 quadratic, 179 rational, 182 strict, 181, 935 three-part, 178 Infinite discontinuity, 397 Infinite geometric sequence definition of sum of terms, 1056–1058 formula for sum of terms of, 1058 Infinite geometric series, 1056–1058 Infinite sequence, 1032 Infinite series, 1035–1036 Infinite set, 28 Infinity, limits at, 396 Inflection point, 276 Initial point of a vector, 799 Initial side of an angle, 524 Inner product of vectors, 812 Input-output (function) machine, 231 Integer exponents, 50 Integers definition of, 35 relatively prime, 247 Integral, definite, 121, 335, 1037 Integrals of functions, 713 Intercepts of a graph, 216 Interest compound, 452 simple, 117 Intermediate value theorem, 373 Intersection of sets, 31, 32, 1092 Interval closed, 177 definition of, 177 disjoint, 177 notation, 177 open, 177 Inverse additive, 40 multiplicative, 40, 961 Inverse cosecant function definition of, 728 graph of, 728 Inverse cosine function definition of, 726 graph of, 726–727 Inverse cotangent function definition of, 728 graph of, 728 Inverse functions definition of, 435, 723 equation of, 436–437 facts about, 440 graphs of, 439–440, 723 notation for, 435, 723 review of concepts, 723 theorem on, 470 Inverse properties, 40 Inverse secant function definition of, 728 graph of, 728 Inverse sine function definition of, 724 graph of, 724–725 Inverse tangent function definition of, 727 graph of, 727–728 Inverse trigonometric equations, 751–754 Inverse trigonometric functions domains and ranges of, 729 equations with, 751–754 graphs of, 724–728 notation for, 723 summary of, 729 Inverse variation definition of, 411 as nth power, 411 Irrational numbers, 35 Is greater than, 42 Is less than, 42 J Joint variation, 411 Julia set, 822 Jump discontinuity, 397 K kth term of the binomial expansion, 1070 L Laffer curve, 406 Lambert’s law, 694 Latitude, 596 Law of cosines applications of, 787–788 applying to solve a triangle, 785 derivation and statement of, 785–786 Heron’s formula and, 790–792 Law of sines ambiguous case of, 774 applications of, 772–773 derivation and statement of, 771 Leading coefficient, 330 Least common denominator in fractions, 75 steps to find, 75 Legs of a right triangle, 151 Leibniz notation, 478 Lemniscates, 843, 845 Leonardo of Pisa, 1034 07/09/16 3:34 pm INDEX Life table, 1101 Like radicals, 96 Like terms, 52 Limaỗons, 844, 845 Limit(s) definition of, 43 of a function, 43, 235 at infinity, 396 notation, 43, 235 one-sided, 391 for the sum of the terms of an infinite geometric sequence, 1057 Line(s) See also Line segment(s) definition of, 524 equation of, 260 horizontal, 246, 249, 263, 266 parallel, 263 perpendicular, 263 polar form of, 840–841 secant, 307 slope of, 248 vertical, 247, 249, 263, 266 Linear cost function, 252 Linear equations definition of, 114, 872 in n unknowns, 872 standard form of, 247, 266 Linear equations in one variable definition of, 114 graphing calculator method for solving, 267–268 modeling with, 124–125 solution of, 114 solving by graphing, 267–268 Linear equations in two variables graphing calculator method for solving, 873, 875 modeling with, 266–267 point-slope form of, 260 slope-intercept form of, 261 standard form of, 247 summary of, 266 Linear functions definition of, 245 graphing, 245 point-slope form of, 260 slope-intercept form of, 261 x-intercept of graph of, 246 y-intercept of graph of, 246 Linear inequalities in one variable applications of, 178 solutions of, 176–177 standard form of, 176 Linear inequalities in two variables graphs of, 934–937 standard form of, 934 Linear methods for solving trigonometric equations, 738 Linear models, 124, 252 Linear programming constraints for, 938 definition of, 937 fundamental theorem of, 938 steps to solve problems with, 937 Linear regression, 267 Linear speed applications of, 611–613 definition of, 610 formula for, 610 Z04_LIAL1953_06_GE_IDX.indd 1189 Linear systems, 872 See also Systems of linear equations Line segment(s) definition of, 524 finding lengths of, 610 function values as lengths of, 609–610 Lissajous figure, 856 Literal equation, 117 Logarithmic differentiation, 468 Logarithmic equations definition of, 462 solving, 462–463, 490–492 steps to solve, 492 Logarithmic function(s) characteristics of graphs of, 465 graphs of, 463–467 modeling with, 476, 478, 493 standard form of, 463 Logarithms argument of, 461 base of, 461 change-of-base theorem for, 480 common, 475 definition of, 461 natural, 478 properties of, 467, 487 Logistic function, 510 LORAN, 1015 Lowest terms of a rational expression, 72 M Mach number, 721 Magnitude of a vector, 799 Major axis of an ellipse, 996 Mandelbrot set, 835 Mapping, 230 Marginal propensity to consume, 272 Mathematical induction generalized principle of, 1075–1076 principle of, 1072–1073 proof by, 1074 Mathematical model, 124, 252 Matrix (Matrices) addition of, 946 additive inverse of, 948 augmented, 890 cofactor of an element of, 903–904 column, 946 column of, 890 definition of, 890 determinant of, 902–903 diagonal form of, 891 dimensions of, 890 element of, 890 equal, 946 expansion by a given row or column, 904 identity, 960 minor of an element of, 903 modeling with, 953 multiplication of See Matrix multiplication multiplicative inverse of, 961 negative of, 948 1189 order of, 890 properties of, 946–954 reduced-row echelon form of, 891 row, 946 row of, 890 row transformations of, 890 scalar multiplication of, 949 sign array for, 905 singular, 965 size of, 890 square, 946 steps to find inverse of, 963 subtraction of, 948 zero, 947 Matrix inverses method for solving linear systems, 965 Matrix method for solving linear systems, 891 Matrix multiplication applications of, 953–954 definition of, 950 properties of, 949, 952 by scalars, 949 Maturity value, 117 Maximum point of a parabola, 334 Measure of an angle, 524, 554, 555 Members of a set, 28 Midpoint formula, 213 Minimum point of a parabola, 333 Minor axis of an ellipse, 996 Minor of a matrix element, 903 Minute, 526 Mixture problems, 122–123 Models absolute value, 192 cost-benefit, 81 exponential, 455 linear, 124, 252 linear systems, 879 logarithmic, 476, 478, 493 mathematical, 124, 252 matrix, 953 polynomial, 375–376 quadratic, 153, 336 rational, 398 Modulus of a complex number, 820 Monomial, 52 Motion problems, 121–122 Multiple-angle identity, deriving, 712 Multiple angles, trigonometric equations with, 743–744 Multiplication of binomials, 54–56 of complex numbers, 134 of functions, 304 identity property for, 40 of matrices See Matrix multiplication order of operations for, 37 of polynomials, 54–56 of radicals, 98 of rational expressions, 74–75 Multiplication property of equality, 114 Multiplication property of zero, 40 Multiplicative inverse definition of, 961 of a matrix, 961 of a real number, 40, 961 Multiplicity of zeros, 357 Mutually exclusive events, 1095 07/09/16 3:34 pm 1190 INDEX N Napier, John, 469 Natural logarithms applications of, 478 definition of, 478 modeling with, 478 Natural numbers, 28, 35 Nautical mile, 603 Negation, order of operations for, 37 Negative of a matrix, 948 of a real number, 40 Negative angle, 524 Negative exponents, 81–82, 86 Negative reciprocals, 263 Negative slope, 251 Newton, 801 Newton’s law of cooling, 503 n-factorial (n!), 1066, 1081 Nonlinear systems of equations application of, 927–928 with complex solutions, 927 definition of, 923 elimination method for solving, 924 graphing calculator method for solving, 923 with nonreal solutions, 927 Nonstrict inequalities, 181, 935 Notation exponential, 37, 85 factorial, 1066, 1081 function, 235 interval, 177 inverse function, 435, 723 Leibniz, 478 limit, 43, 235 radical, 92 set-builder, 28 for sets of numbers, 28 sigma, 1035 summation, 1035 nth root of a complex number, 831 nth root theorem for complex numbers, 832 nth term of an arithmetic sequence, 1044 of a geometric sequence, 1053 of a sequence, 1032 Null set definition of, 29, 116 symbol for, 29, 116 Number line definition of, 35 distance between points on, 44 order on, 42 Number of zeros theorem, 356–357 Numbers complex See Complex numbers conjugate of, 99 counting, 28 decimals, 35 exact, 564 factors of, 36 imaginary, 131 integers, 35 irrational, 35 natural, 28, 35 negative reciprocals, 263 Z04_LIAL1953_06_GE_IDX.indd 1190 rational, 35 real, 35, 40 sets of, 28, 35 whole, 35 Numerator, rationalizing, 99 Numerical coefficient, 52 O Objective function, 938 Oblique asymptote definition of, 391, 393 determining, 391 Oblique triangle congruency and, 770–771 data required for solving, 770 definition of, 770 finding area of, 778–779 solving procedures for, 770–771 Obtuse angle, 524 Obtuse triangle, 771 Odd-degree polynomial functions, 367 Odd function, 292, 619, 680 Odds definition of, 1093 in favor of an event, 1093–1094 rules for, 1094 One-sided limits, 391 One-to-one functions definition of, 432, 722 horizontal line test for, 433 tests to determine, 434 Open interval, 177 Operations on complex numbers, 133–137 on functions, 304 order of, 37 with radicals, 96 on sets, 30–32 Opposite of a vector, 800 Opposite side to an angle, 547 Orbital period of a satellite, 848 Order descending, 53 of a matrix, 890 on a number line, 42 of operations, 37 Ordered pair, 210, 215 Ordered triple(s), 212, 878 Order symbols, 42 Ordinates, addition of, 656 Origin of rectangular coordinate system, 210 symmetry with respect to, 290 Orthogonal vectors, 814 Outcomes of an experiment, 1091 P Pair, ordered, 210, 215 Parabola(s) application of, 991 axis of, 331 definition of, 276, 330 directrix of, 988 eccentricity of, 1001 equations of, 986 focus of, 988 geometric definition of, 988 geometric interpretation of, 986 graphing by completing the square, 332 graph of, 276, 987 horizontal, 986, 987, 989 maximum point of, 334 minimum point of, 333 reflective property of, 991 translations of, 990 vertex formula for, 335 vertex of, 276, 331, 335, 986 vertical, 988 Parallel lines definition of, 263 slope of, 263 Parallelogram rule for vectors, 800 Parallelograms, properties of, 800 Parametric equations definition of, 631, 850 derivatives of, 566 of a plane curve, 850 Parentheses, 37 Partial fractions decomposition, 915 definition of, 915 Pascal, Blaise, 469, 1066 Pascal’s triangle, 1065–1066 Path of a projectile, 854 Pendulum, period of, 112 Perfect square trinomial, 65 Perfect triangle, 797 Period of cosecant function, 653 of cosine function, 620, 623 of cotangent function, 644 definition of, 623 of secant function, 652 of sine function, 619, 623 of tangent function, 643 Periodic function definition of, 618 period of, 618 Period life table, 1101 Period of a pendulum, 112 Permutations definition of, 1081 distinguishing from combinations, 1085–1086 formula for, 1081 Perpendicular lines definition of, 263 slope of, 263 pH, 476 Phase angle, 672 Phase shift, 631 Phelps, Phlash, 228 Photographic coordinates, 784 Pi (p), 103, 592 Piecewise-defined function, 278 Plane curve, 850 Plane trigonometry, 563 Plimpton 322 (tablet), 563 Point-slope form of the equation of a line, 260, 266 Poiseuille’s law, 416 Polar axis, 837 07/09/16 3:34 pm INDEX Polar coordinates of a point definition of, 618, 837 graph of, 838 Polar coordinate system definition of, 837 polar axis of, 837 pole of, 837 Polar equations calculator graphing of, 841–844 classifying, 844–845 converting to rectangular equations, 844 definition of, 840 graphs of, 840–844 Polar form of a complex number, 820 of conic sections, 1113–1115 of lines and circles, 840–841 Polar grid, 841 Pole of a polar coordinate system, 837 Polynomial(s) addition of, 53 classifying expressions as, 53 definition of, 52 degree of, 52 descending order of, 53, 56 division algorithm for, 346 division of, 56–57 factored form of, 62 factoring of, 62 multiplication of, 54 prime, 62 special products of, 55 subtraction of, 53 synthetic division for, 347 term of, 52 in x, 52 zero, 330 Polynomial function(s) approximating real zeros of, 375 boundedness theorem for, 374 comprehensive graph of, 370 conjugate zeros theorem for, 358 definition of, 330 of degree n, 330 derivative of, 354 dominating term of, 330, 368 end behavior of graph of, 368, 369 of even degree, 367 factors of, 372 factor theorem for, 353 graph of, 365–366 intermediate value theorem for, 373 leading coefficient of, 330 modeling with, 375–376 number of zeros theorem for, 356–357 of odd degree, 367 rational zeros theorem for, 354 roots of, 350 solutions of, 350, 372 steps to graph, 370 turning points of graph of, 368 zero, 330 zeros of, 350, 359, 367, 372 Polynomial interpolation, 971 Polynomial models, 375–376 Population growth, 1034 Position vector, 809 Z04_LIAL1953_06_GE_IDX.indd 1191 Positive angle, 524 Positive slope, 251 Power property of equations, 165–166 Power rule for derivatives, 93 Power rules for exponents, 50, 86 Power series, 190 Powers of complex numbers, 830 Powers of i, 136–137 Present value, 453 Prime polynomial, 62 Principal root of a radical, 92 Principle of mathematical induction, 1072–1073 Probability of an event, 1091 basics of, 1091–1098 binomial, 1097 of compound events, 1095 definition of, 1091 properties of, 1097 of union of events, 1094–1095 Product of the sum and difference of two terms, 55 Product rule for exponents, 50, 86 Product theorem for complex numbers, 824 Product-to-sum identities, 713–714 Profit function, 252 Progression arithmetic, 1043 geometric, 1053 Proportional directly, 409 as nth power, 410–411 Proposed solutions, 162 Ptolemy, 563, 784 Pyramid frustum of, 60 volume of a frustum of, 60 Pythagorean identities, 541–542, 681 Pythagorean theorem, 151 Q Quadrantal angles, 527 Quadrants of a coordinate system, 210 Quadratic equations in one variable applications of, 150 completing the square to solve, 141 discriminant of, 145 factoring to solve, 139 modeling with, 153 quadratic formula for solving, 142–143 square root method for solving, 140 standard form of, 139 types of solutions for, 146 Quadratic formula discriminant of, 145 standard form of, 143 Quadratic function definition of, 330 graph of, 330–332 maximum of, 334 minimum of, 333 steps to graph, 335 Quadratic inequalities standard form of, 179 steps to solve, 179 1191 Quadratic models, 153, 336 Quadratic regression, 338 Queuing theory, 404 Quotient identities, 542–544, 681 Quotient rule for exponents, 82, 86 Quotient theorem for complex numbers, 825 R Radian, 592 Radian/degree relationship converting measures, 592–593 table of, 594 Radian measure applications of, 596–597 converting to degree measure, 592–593 definition of, 592 Radiative forcing, 479 Radical(s) addition of, 96 conjugate of, 99 index of, 92, 94 like, 96 multiplication of, 98 notation, 92 operations with, 96 principal root of, 92 rational exponent form of, 93 rationalizing the denominator of, 98 rationalizing the numerator of, 99 simplifying, 95–96 steps to simplify, 95 subtraction of, 96–97 summary of rules for, 94 symbol for, 92 unlike, 96 Radical equations power property and, 165–166 steps to solve, 166 Radical notation, 92 Radical symbol, 92 Radicand, 92 Radius of a circle, 221 Range(s) of a function, 234 of inverse circular functions, 729 of a relation, 231 of trigonometric functions, 540 Rate of change, average, 251 Rational equations, 162 Rational exponents definition of, 84 equations with, 168 radical form of, 92 Rational expressions addition of, 75 common factors of, 72, 73 decomposition of, 915 definition of, 72 division of, 74–75 domain of, 72 least common denominator of, 75 lowest terms of, 72 multiplication of, 74–75 subtraction of, 75–77 07/09/16 3:34 pm 1192 INDEX Rational functions definition of, 386 features of, 390 graphing by calculator, 389, 398 graph of, 389–390 modeling with, 398 steps to graph, 393 Rational inequalities definition of, 182 steps to solve, 182 Rationalizing the denominator, 98 Rationalizing the numerator, 99 Rational numbers, 35 Rational zeros theorem, 354 Ray definition of, 524 endpoint of, 524 r cis u, 820 Real axis, 819 Real numbers additive inverse of, 40 definition of, 35 multiplicative inverse of, 40, 961 negative of, 40 properties of, 40 reciprocal of, 40 summary of properties of, 40 Real part of a complex number, 131 Reciprocal, 537 Reciprocal functions graphing by calculator, 387 graph of, 388 important features of, 388 Reciprocal identities, 537–538, 681 Reciprocal of a real number, 40 Rectangular coordinates, converting to polar coordinates, 838 Rectangular coordinate system, 210 Rectangular equations converting polar equations to, 844 definition of, 840 Rectangular form of a complex number converting to trigonometric form, 821 definition of, 819 Recursively defined sequence, 1033 Reduced-row echelon form of a matrix, 891 Reduction formula, 708 Reference angles definition of, 550 special angles as, 552–554 table of, 552 Reference arc, 604 Reflection of a graph across a line, 288 Reflection of a graph across an axis, 288 Reflective property of an ellipse, 1003 of a parabola, 991 Region of feasible solutions corner point of, 939 definition of, 938 vertex of, 939 Regression linear, 267 quadratic, 338 sine, 636–637 Related rates, 566 Z04_LIAL1953_06_GE_IDX.indd 1192 Relation definition of, 229 domain of, 231 graph of, 230 range of, 231 Relatively prime integers, 247 Remainder theorem, 349 Removable discontinuity, 397 Resistance, grade, 556 Resultant vectors, 800, 819 Revenue function, 252 Right angle, 524 Right triangle(s) hypotenuse of, 151, 547 legs of, 151 in Pythagorean theorem, 151 solving, 565 Right-triangle-based definitions of trigonometric functions, 547 Rise, 248 Roots of a complex number, 831–834 of an equation, 114 evaluating for radicals, 92 of an expression, 37 of polynomial functions, 350 Rose curve, 845 Rotation angles of, 1118 of axes, 1117–1119 definition of, 1117 Row matrix, 946 Row of a matrix, 890 Row transformations, 890 Rules for exponents, 50–52, 82, 86 Rules for odds, 1094 Run, 248 Rutherford, Ernest, 1015 S Sample space, 1091 Saturation constant, 1042 Scalar, 949 Scalar multiplication definition of, 949 properties of, 949 Scalars, 799 Scatter diagrams, 266, 456 Secant function characteristics of, 652 definition of, 533 domain of, 652 graph of, 652 inverse of, 728 period of, 652 range of, 540, 652 steps to graph, 654 Secant line, 307 Second, 526 Second-degree equation, 139 Sector of a circle area of, 597–598 definition of, 597 Segment of a line See Line segment(s) Semiperimeter, 789 Sequence(s) arithmetic, 1043, 1044, 1047 convergent, 1033 definition of, 1032 divergent, 1033 explicitly defined, 1033 Fibonacci, 1031, 1034 finite, 1032 general term of, 1032 geometric See Geometric sequence infinite, 1032 infinite geometric, 1056–1058 nth term of, 1032, 1044, 1053 recursively defined, 1033 terms of, 1032 Series arithmetic, 1047 definition of, 1035–1036 finite, 1035–1036 geometric, 1055 infinite, 1035–1036 infinite geometric, 1056–1058 power, 190 Set(s) complement of, 30, 32 definition of, 28 disjoint, 31 elements of, 28 empty, 29, 116 equal, 30 finite, 28 infinite, 28 intersection of, 31, 32, 1092 members of, 28 notation, 28 null, 29, 116 of numbers, 35 operations on, 30–32 subset of, 29 summary of operations on, 32 union of, 31, 32, 181, 1092 universal, 29 Set braces, 28 Set-builder notation, 28 Shrinking graphs graphing techniques for, 286–287 horizontally, 287 vertically, 287 Side adjacent to an angle, 547 Side-Angle-Angle (SAA) axiom, 770 Side-Angle-Side (SAS) axiom, 770 Side opposite an angle, 547 Side-Side-Side (SSS) axiom, 770 Sigma notation definition of, 1036 symbol for, 1036 Sign array for matrices, 905 Significant digits, 564–565 Signs of trigonometric function values, 538 Simple harmonic motion, 660 Simple interest formula, 117 Simplifying complex fractions, 77–78, 87–88 Simplifying radicals, 95–96 Sine function amplitude of, 621 characteristics of, 619 definition of, 533 difference identity for, 698–699 domain of, 619 double-angle identity for, 709–710 19/09/16 1:46 pm INDEX graph of, 619 half-angle identity for, 715, 716 horizontal translation of, 632 inverse of, 724–725 period of, 619, 623 range of, 540, 619 steps to graph, 624, 635 sum identity for, 698–699 translating graphs of, 632, 635 Sine regression, 636–637 Sines, law of ambiguous case of, 774–777 applications of, 772–773 derivation of, 771 Sine wave, 619 Singular matrix, 965 Sinusoid, 619 Sinusoidal function, 636 Size of a matrix, 890 Slope(s) of a curve at a point, 248 definition of, 248 formula for, 248 of a horizontal line, 249 of a line, 248 negative, 251 of parallel lines, 263 of perpendicular lines, 263 point-slope form, 260, 266 positive, 251 slope-intercept form, 261, 266 undefined, 249 of a vertical line, 249 zero, 249, 251 Slope-intercept form of the equation of a line, 261, 266 Snell’s law, 765–766 Solar constant, 630 Solution(s) of an equation, 114 double, 141 of a polynomial function, 350, 372 proposed, 162 Solution set, 114 Solving applied problems, 120 Solving for a specified variable, 117, 145 Solving triangles, 565, 772–773, 775–777, 787–789 Sound waves, 630, 707 Special angles finding angle measures with, 554 reference angles as, 552–554 trigonometric function values of, 549–550 Special products of polynomials, 55 Speed angular, 610–613 definition of, 121 linear, 610–613 Spherical trigonometry, 563 Spiral, hyperbolic, 857 Spiral of Archimedes, 843–844 Square(s) of a binomial, 55 completing the, 141, 332 difference of, 66 Square matrix, 946 Square root function, 277 Z04_LIAL1953_06_GE_IDX.indd 1193 Square root method for solving quadratic equations, 140 Square root property, 140 Square viewing window on graphing calculators, 222 Squaring function, 276 Standard form of a complex number, 132, 819, 820 of the equation of a line, 260 of a linear equation, 247, 266 of a quadratic equation, 139 Standard position of an angle, 527 Standard viewing window on graphing calculators, 217 Statute mile, 603 Stefan-Boltzmann law, 416 Step function, 280 Stirling’s formula, 1072 Straight angle, 524 Stretching graphs graphing techniques for, 286–287 horizontal, 287 vertical, 287 Strict inequalities, 181, 935 Subscripts, 1036 Subset, 29 Substitution method definition of, 872 for factoring, 67–68 for solving linear systems, 873 Subtend an angle, 602 Subtense bar method, 570 Subtraction of complex numbers, 134 of functions, 304 of matrices, 948 order of operations for, 37 of polynomials, 53 of radicals, 96–97 of rational expressions, 75–77 of vectors, 800 Sum identity application of, 700–704 for cosine, 695–697 for sine, 698–699 for tangent, 699–700 Summation, index of, 1036 Summation notation definition of, 1036 with subscripts, 1036 Summation properties, 1037 Summation rules, 1037 Sum of cubes, 66 Sum of terms of an arithmetic sequence, 1047 of an infinite geometric sequence, 1058 of an infinite geometric series, 1056–1058 of a geometric sequence, 1055–1056 Sum-to-product identities, 714–715 Superelevation, 561 Supplementary angles, 525 Supply and demand, 889 Swokowski, Earl, 713 Symbols of order, 42 Symmetry example of, 209 graphing techniques for, 289 1193 of graphs, 289 with respect to origin, 290 with respect to x-axis, 289–290 with respect to y-axis, 289–290 tests for, 289–291 Synthetic division for polynomials, 347–349 Systems of equations definition of, 872 solutions of, 872 Systems of inequalities definition of, 936 graphing, 936–937 Systems of linear equations in three variables applications of, 881 elimination method for solving, 878 Gauss-Jordan method for solving, 893 geometric interpretation of, 877 matrix inverses method for solving, 965–966 modeling with, 879 solving, 877–878 Systems of linear equations in two variables applications of, 876 consistent, 872 Cramer’s rule for solving, 907–908 elimination method for solving, 873 equivalent, 873 Gauss-Jordan method for solving, 891 graphing calculator method for solving, 873, 875 inconsistent, 872 matrix method for solving, 891 solution set for, 872 substitution method for solving, 872 transformations of, 873 Systolic pressure, 628 T Tangent function characteristics of, 643 definition of, 533 difference identity for, 699–700 domain of, 643 double-angle identity for, 710 graph of, 643 half-angle identity for, 715, 716 horizontal translation of, 647 inverse of, 727–728 period of, 643 range of, 540, 643 steps to graph, 645 sum identity for, 699–700 vertical translation of, 647 Tangent line to a circle, 227 to a curve, 260 Tartaglia, Niccolo, 364 Term(s) coefficient of, 52 definition of, 52 dominating, 330, 368 like, 52 of a polynomial, 52 of a sequence, 1032 07/09/16 3:34 pm 1194 INDEX Terminal point of a vector, 799 Terminal side of an angle, 524 Tests for symmetry, 289–291 Three-part inequalities, 178 Threshold sound, 477 Tolerance, 192 Traffic intensity, 398 Transformations of linear systems, 873 Transit, 574 Translation(s) combinations of, 635–636 definition of, 293 of an ellipse, 999–1000 of graphs, 293, 366 horizontal, 294, 631–633, 647 of a hyperbola, 1011 of a parabola, 990 summary of, 297 vertical, 293, 634, 647 Transverse axis of a hyperbola, 1008 Tree diagram, 1080 Trial, 1091 Triangle(s) acute, 771 area of, 778–779, 789 congruent, 770–771 Heron, 797 oblique, 770–771 obtuse, 771 Pascal’s, 1065–1066 perfect, 797 right See Right triangle(s) solving, 565–566, 772–773, 775–777, 787–789 Triangular form, 900 Triangulation method, 796 Trigonometric equations conditional, 738 with half-angles, 742 inverse, 751–754 linear methods for solving, 738 with multiple angles, 743–744 quadratic methods for solving, 739–740 solving using the quadratic formula, 740 solving by squaring, 741 solving by zero-factor property, 739 Trigonometric form of a complex number, 820 Trigonometric functions circular, 604 cofunctions of, 548, 697–698 combinations of translations of, 635–636 definitions of, 533 derivatives of, in calculus, 687 domains of, 606 inverses of, 722–729 ranges of, 540 right-triangle-based definitions of, 547–548 translations of, 631–637 unit circle and, 221 Trigonometric function values of acute angles, 547 finding with a calculator, 554–555 Z04_LIAL1953_06_GE_IDX.indd 1194 of nonquadrantal angles, 553 of quadrantal angles, 535–536 signs and ranges of, 538 of special angles, 549–550 undefined, 536 Trigonometric identities cofunction, 548, 697–698 difference, 695–697, 698–703, 700–703 double-angle, 709–712 even-odd, 680–681 fundamental, 680–683 half-angle, 715–718, 742 product-to-sum, 713–714 Pythagorean, 541–542, 681 quotient, 542–544, 681 reciprocal, 537–538, 681 solving conditional trigonometric equations using, 740–741 sum, 695–697, 698–703, 700–703 sum-to-product, 714–715 verifying, 687 Trigonometric models, 626, 636–637 Trigonometric substitution, 542 Trinomials definition of, 52 factoring, 64–65 perfect square, 65 Triples, ordered, 212, 878 Trochoid, 852 Turning points of polynomial function graphs, 368 U Undefined slope, 249 Union of sets, 31, 32, 181, 1092 Union of two events definition of, 1092 probability of, 1094–1095 Unit, imaginary, 131 Unit circle, 221, 604 Unit fractions, 612 Unit vector, 811 Universal set, 29 Unlike radicals, 96 Upper harmonics, 745, 750 V Value absolute, 42–43, 188 future, 117, 453, 1058–1059 maturity, 117 present, 453 Variable dependent, 229, 231 independent, 229, 231 solving for a specified, 117–118, 145 Variation combined, 411 constant of, 409 direct, 409 directly as the nth power, 410 inverse, 411 inversely as the nth power, 411 joint, 411 in sign, 360 steps to solve problems with, 410 Vector(s) algebraic interpretation of, 809–810 angle between, 814 applications of, 802–804 components of, 809 direction angle for, 809 dot product of, 812–814 equilibrant of, 801–802 horizontal component of, 809 i, j units, 812 initial point of, 799 inner product of, 812 magnitude of, 799 naming practices, 799 operations with, 811 opposite of, 800 orthogonal, 814 parallelogram rule for, 800 position, 809 quantities, 799 resultant of, 800, 819 symbol for, 799 terminal point of, 799 unit, 811 vertical component of, 809 zero, 800 Vector cross products, 903 Velocity constant, 121 in motion problems, 121 non-constant, 121 Venn diagram, 30, 1093 Verifying trigonometric identities, 687 Vertex (Vertices) of an angle, 524 of an ellipse, 996 of a hyperbola, 1008 of a parabola, 276, 331, 335, 986 of a polygonal region, 939 Vertex formula for a parabola, 335 Vertical asymptote behavior of graphs near, 396 definition of, 387, 391, 642 determining, 391 Vertical axis, 819 Vertical component of a vector, 809 Vertical line equation of, 263, 266 graph of, 247 slope of, 249 Vertical line test for functions, 233 Vertical parabola definition of, 988 graph of, 988 Vertical shrinking of the graph of a function, 287 Vertical stretching of the graph of a function, 287 Vertical translations of cosine function, 634 of cotangent function, 647–648 of a graph, 293, 366 graphing, 634 of tangent function, 647 07/09/16 3:34 pm INDEX W Y Waiting-line theory, 404 Wave sine, 619 sound, 630, 707 Whispering gallery, 1003 Whole numbers, 35 Windchill formula, 102 Work rate problems, 164–165 y-axis definition of, 210 symmetry with respect to, 289–290 y-intercept, 216 X x-axis definition of, 210 symmetry with respect to, 289–290 x-intercept, 216, 372 xy-plane, 210 Z04_LIAL1953_06_GE_IDX.indd 1195 1195 Zero polynomial, 330 Zero slope, 249, 251 Zero vector, 800 ZTrig viewing window, 621 Z Zero(s) multiplication property of, 40 multiplicity of, 357 of polynomial functions, 350, 359, 367, 372 as whole number, 35 Zero exponent, 51 Zero-factor property, 72, 139, 739 Zero matrix, 947 07/09/16 3:34 pm Geometry Formulas  Square Rectangle Triangle Perimeter: P = 4s Perimeter: P = 2L + 2W Perimeter: P = a + b + c Area: Ꮽ = s Area: Ꮽ = LW Area: Ꮽ = bh s s s s a W L c h b Parallelogram Trapezoid Circle Perimeter: P = 2a + 2b Perimeter: P = a + b + c + B Diameter: d = 2r Area: Ꮽ = bh Area: Ꮽ = h1B + b2 Circumference: C = 2pr = pd Area: Ꮽ = pr b Chord b a r c h a h d B Cube Rectangular Solid Sphere pr Surface area: S = 4pr Volume: V = e 3 Volume: V = LWH Volume: V = Surface area: S = 6e Surface area: S = 2HW + 2LW + 2LH e r H e e Cone L W Right Circular Cylinder Right Pyramid 1 Volume: V = pr 2h Volume: V = pr 2h Volume: V = Bh 3 Surface area: S = pr 2r + h Surface area: S = 2prh + 2pr B = area of the base (includes top and bottom) (excludes the base) h Z05_LIAL1953_06_GE_BEP.indd 1196 h h r r 07/09/16 5:03 pm Graphs of Functions 2.6 Identity Function 2.6 Squaring Function y 2.6 Cubing Function y y x –2 –1 x 2.6 Cube Root Function 2.6 Absolute Value Function 2.6 Greatest Integer Function y f (x) = [[x]] x –2 x y 1 –2 –3 –4 –1 x –4 –3 –2 x 3.5 Reciprocal Function y f (x) = | x | f (x) = √x –8 x y f (x) = √x 2 y f (x) = x3 f (x) = x f (x) = x 2.6 Square Root Function x –1 f (x) = 1x 3.4 Polynomial Functions y y y y 6 –3 –2 x x –4 –6 Degree 3; one real zero Degree 3; three real zeros  –12 Z05_LIAL1953_06_GE_BEP.indd 1197 y f(x) = a x, a > (0, 1) x Degree 4; four real zeros –2 –1 –1 x Degree 6; three real zeros –2   4.3 Logarithmic Functions y y f(x) = log a x, < a < f(x) = log a x, a > (–1, 1a) (1, a) (–1, 1a)   4.2 Exponential Functions y –2 –1 f(x) = a x, < a < (0, 1) x   (a, 1) (1, a) x (1, 0) , –1 a ( ) x   (a, 1) (1, 0) x ( 1a , –1) 07/09/16 5:03 pm Global edition College Algebra & Trigonometry For these Global Editions, the editorial team at Pearson has collaborated with educators across the world to address a wide range of subjects and requirements, equipping students with the best possible learning tools This Global Edition preserves the cutting-edge approach and pedagogy of the original, but also features alterations, customization, and adaptation from the North American version Global edition sixth edition Lial Hornsby Schneider Daniels College Algebra & Trigonometry Sixth edition Margaret L Lial • John Hornsby • David I Schneider • Callie J Daniels G LOBa l edition This is a special edition of an established title widely used by colleges and universities throughout the world Pearson published this exclusive edition for the benefit of students outside the United States and Canada If you purchased this book within the United States or Canada, you should be aware that it has been imported without the approval of the Publisher or Author Pearson Global Edition Lial_06_1292151951_Final.indd 23/09/16 11:45 AM ... understanding of college algebra and trigonometry and the analytical skills necessary for success in mathematics The MyMathLab features described here support college algebra and trigonometry students... 4:35 pm This page intentionally left blank 561590_MILL_MICRO_FM_ppi-xxvi.indd 24/11/14 5:26 PM College Algebra & Trigonometry SIXTH EDITION GLOBAL EDITION Margaret L Lial American River College. .. from most mobile devices to work on exercises and review completed assignments • www.mymathlab.com A00 _LIAL1 953_06_GE_FEP.indd 07/09/16 4:35 pm CHAPTER 6  Analytic Geometry Available in MyMathLab®