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 Algebra and Trigonometry Judith A Beecher Indiana University Purdue University Indianapolis Judith A Penna Indiana University Purdue University Indianapolis Marvin L Bittinger Indiana University Purdue University Indianapolis Addison-Wesley Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo th EDITION Editor-in-Chief Executive Content Editor Senior Content Editor Editorial Assistant Senior Managing Editor Senior Production Project Manager Design Manager Senior Design Specialist Digital Assets Manager Supplements Production Coordinator Media Producer Software Development Executive Marketing 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designations have been printed in initial caps or all caps Library of Congress Cataloging-in-Publication Data Beecher, Judith A Algebra and trigonometry / Judith A Beecher, Judith A Penna, Marvin L Bittinger.—4th ed p cm ISBN-13: 978-0-321-69398-3 ISBN-10: 0-321-69398-1 Algebra—Textbooks Trigonometry—Textbooks I Penna, Judith A II Bittinger, Marvin L III Title QA152.3.B42 2012 512’.13—dc22 2010034358 Copyright © 2012, 2008, 2005, 2002 Pearson Education, Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher Printed in the United States of America For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA 02116, fax your request to 617-671-3447, or e-mail at http://www.pearsoned.com/legal/permissions.htm 10—DOW—14 13 12 11 10 ISBN-13: 978-0-321-69398-3 ISBN-10: 0-321-69398-1 This work is protected by United States copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning Dissemination or sale of any part of this work (including on the World Wide Web) will destroy the integrity of the work and is not permitted The work and materials from it should never be made available to students except by instructors using the accompanying text in their classes All recipients of this work are expected to abide by these restrictions and to honor the intended pedagogical purposes and the needs of other instructors who rely on these materials Contents Preface R xi Basic Concepts of Algebra R.1 The Real-Number System Real Numbers / Interval Notation / Properties of the Real Numbers / Absolute Value R.2 Integer Exponents, Scientific Notation, and Order of Operations Integers as Exponents / Scientific Notation / Order of Operations R.3 Addition, Subtraction, and Multiplication of Polynomials 17 Polynomials / Addition and Subtraction / Multiplication R.4 Factoring 22 Terms with Common Factors / Factoring by Grouping / Trinomials of the Type x + bx + c / Trinomials of the Type ax + bx + c, a Z / Special Factorizations R.5 The Basics of Equation Solving 31 Linear Equations and Quadratic Equations / Formulas R.6 Rational Expressions 36 The Domain of a Rational Expression / Simplifying, Multiplying, and Dividing Rational Expressions / Adding and Subtracting Rational Expressions / Complex Rational Expressions R.7 Radical Notation and Rational Exponents 45 Simplifying Radical Expressions / An Application / Rationalizing Denominators or Numerators / Rational Exponents Study Guide 53 Review Exercises 54 Chapter Test 56 Graphs, Functions, and Models 59 1.1 Introduction to Graphing 60 Graphs / Solutions of Equations / Graphs of Equations / The Distance Formula / Midpoints of Segments / Circles VISUALIZING THE GRAPH 71 1.2 Functions and Graphs 75 Functions / Notation for Functions / Graphs of Functions / Finding Domains of Functions / Visualizing Domain and Range / Applications of Functions CONTENTS iii iv CONTENTS 1.3 Linear Functions, Slope, and Applications 91 Linear Functions / The Linear Function f1x2 = mx + b and Slope / Applications of Slope / Slope–Intercept Equations of Lines / Graphing f1x2 = mx + b Using m and b / Applications of Linear Functions VISUALIZING THE GRAPH 102 MID-CHAPTER MIXED REVIEW 107 1.4 Equations of Lines and Modeling 109 Slope–Intercept Equations of Lines / Point–Slope Equations of Lines / Parallel Lines / Perpendicular Lines / Mathematical Models / Curve Fitting 1.5 Linear Equations, Functions, Zeros, and Applications 121 Linear Equations / Applications Using Linear Models / Zeros of Linear Functions 1.6 Solving Linear Inequalities 138 Linear Inequalities / Compound Inequalities / An Application Study Guide 144 Review Exercises 150 Chapter Test 154 More on Functions 157 2.1 Increasing, Decreasing, and Piecewise Functions; Applications 158 Increasing, Decreasing, and Constant Functions / Relative Maximum and Minimum Values / Applications of Functions / Functions Defined Piecewise 2.2 The Algebra of Functions 171 The Algebra of Functions: Sums, Differences, Products, and Quotients / Difference Quotients 2.3 The Composition of Functions 179 The Composition of Functions / Decomposing a Function as a Composition MID-CHAPTER MIXED REVIEW 187 2.4 Symmetry and Transformations 189 Symmetry / Even Functions and Odd Functions / Transformations of Functions / Vertical Translations and Horizontal Translations / Reflections / Vertical and Horizontal Stretchings and Shrinkings VISUALIZING THE GRAPH 205 2.5 Variation and Applications 211 Direct Variation / Inverse Variation / Combined Variation Study Guide 220 Review Exercises 227 Chapter Test 231 CONTENTS Quadratic Functions and Equations; Inequalities 233 3.1 The Complex Numbers 234 The Complex-Number System / Addition and Subtraction / Multiplication / Conjugates and Division 3.2 Quadratic Equations, Functions, Zeros, and Models 241 Quadratic Equations and Quadratic Functions / Completing the Square / Using the Quadratic Formula / The Discriminant / Equations Reducible to Quadratic / Applications 3.3 Analyzing Graphs of Quadratic Functions 257 Graphing Quadratic Functions of the Type f1x2 = a1x - h22 + k / Graphing Quadratic Functions of the Type f1x2 = ax + bx + c, a Z / Applications VISUALIZING THE GRAPH 266 MID-CHAPTER MIXED REVIEW 270 3.4 Solving Rational Equations and Radical Equations 272 Rational Equations / Radical Equations 3.5 Solving Equations and Inequalities with Absolute Value 280 Equations with Absolute Value / Inequalities with Absolute Value Study Guide 285 Review Exercises 289 Chapter Test 293 Polynomial Functions and Rational Functions 4.1 Polynomial Functions and Models 296 The Leading-Term Test / Finding Zeros of Factored Polynomial Functions / Polynomial Models 4.2 Graphing Polynomial Functions 309 Graphing Polynomial Functions / The Intermediate Value Theorem VISUALIZING THE GRAPH 317 4.3 Polynomial Division; The Remainder Theorem and the Factor Theorem 320 Division and Factors / The Remainder Theorem and Synthetic Division / Finding Factors of Polynomials MID-CHAPTER MIXED REVIEW 328 4.4 Theorems about Zeros of Polynomial Functions 330 The Fundamental Theorem of Algebra / Finding Polynomials with Given Zeros / Zeros of Polynomial Functions with Real Coefficients / Rational Coefficients / Integer Coefficients and the Rational Zeros Theorem / Descartes’ Rule of Signs 295 v vi CONTENTS 4.5 Rational Functions 340 The Domain of a Rational Function / Asymptotes / Applications VISUALIZING THE GRAPH 356 4.6 Polynomial Inequalities and Rational Inequalities Polynomial Inequalities / Rational Inequalities Study Guide 372 Review Exercises 381 Chapter Test 385 360 Exponential Functions and Logarithmic Functions 387 5.1 Inverse Functions 388 Inverses / Inverses and One-to-One Functions / Finding Formulas for Inverses / Inverse Functions and Composition / Restricting a Domain 5.2 Exponential Functions and Graphs 400 Graphing Exponential Functions / Applications / The Number e / Graphs of Exponential Functions, Base e 5.3 Logarithmic Functions and Graphs 413 Logarithmic Functions / Finding Certain Logarithms / Converting Between Exponential Equations and Logarithmic Equations / Finding Logarithms on a Calculator / Natural Logarithms / Changing Logarithmic Bases / Graphs of Logarithmic Functions / Applications VISUALIZING THE GRAPH 425 MID-CHAPTER MIXED REVIEW 429 5.4 Properties of Logarithmic Functions 430 Logarithms of Products / Logarithms of Powers / Logarithms of Quotients / Applying the Properties / Simplifying Expressions of the Type log a ax and aloga x 5.5 Solving Exponential Equations and Logarithmic Equations 439 Solving Exponential Equations / Solving Logarithmic Equations 5.6 Applications and Models: Growth and Decay; Compound Interest 450 Population Growth / Interest Compounded Continuously / Models of Limited Growth / Exponential Decay Study Guide 465 Review Exercises 471 Chapter Test 475 The Trigonometric Functions 6.1 Trigonometric Functions of Acute Angles 478 The Trigonometric Ratios / The Six Functions Related / Function Values of 30°, 45°, and 60° / Function Values of Any Acute Angle / Cofunctions and Complements 477 CONTENTS vii 6.2 Applications of Right Triangles 490 Solving Right Triangles / Applications 6.3 Trigonometric Functions of Any Angle 503 Angles, Rotations, and Degree Measure / Trigonometric Functions of Angles or Rotations / The Six Functions Related / Terminal Side on an Axis / Reference Angles: 30°, 45°, and 60° / Function Values for Any Angle MID-CHAPTER MIXED REVIEW 519 6.4 Radians, Arc Length, and Angular Speed 520 Distances on the Unit Circle / Radian Measure / Arc Length and Central Angles / Linear Speed and Angular Speed 6.5 Circular Functions: Graphs and Properties 536 Reflections on the Unit Circle / Finding Function Values / Graphs of the Sine and Cosine Functions / Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions 6.6 Graphs of Transformed Sine and Cosine Functions 551 Variations of Basic Graphs / Graphs of Sums: Addition of Ordinates / Damped Oscillation: Multiplication of Ordinates VISUALIZING THE GRAPH 565 Study Guide 569 Review Exercises 578 Chapter Test 581 Trigonometric Identities, Inverse Functions, and Equations 7.1 Identities: Pythagorean and Sum and Difference 584 Pythagorean Identities / Simplifying Trigonometric Expressions / Sum and Difference Identities 7.2 Identities: Cofunction, Double-Angle, and Half-Angle 597 Cofunction Identities / Double-Angle Identities / Half-Angle Identities 7.3 Proving Trigonometric Identities 605 The Logic of Proving Identities / Proving Identities / Product-to-Sum and Sum-to-Product Identities MID-CHAPTER MIXED REVIEW 613 7.4 Inverses of the Trigonometric Functions 614 Restricting Ranges to Define Inverse Functions / Composition of Trigonometric Functions and Their Inverses 7.5 Solving Trigonometric Equations 625 VISUALIZING THE GRAPH Study Guide 639 Review Exercises 645 Chapter Test 648 635 583 viii CONTENTS Applications of Trigonometry 649 8.1 The Law of Sines 650 Solving Oblique Triangles / The Law of Sines / Solving Triangles (AAS and ASA) / Solving Triangles (SSA) / The Area of a Triangle 8.2 The Law of Cosines 663 The Law of Cosines / Solving Triangles (SAS) / Solving Triangles (SSS) 8.3 Complex Numbers: Trigonometric Form 673 Graphical Representation / Trigonometric Notation for Complex Numbers / Multiplication and Division with Trigonometric Notation / Powers of Complex Numbers / Roots of Complex Numbers MID-CHAPTER MIXED REVIEW 684 8.4 Polar Coordinates and Graphs 686 Polar Coordinates / Polar Equations and Rectangular Equations / Graphing Polar Equations VISUALIZING THE GRAPH 694 8.5 Vectors and Applications 697 Vectors / Vector Addition / Applications / Components 8.6 Vector Operations 705 Position Vectors / Operations on Vectors / Unit Vectors / Direction Angles / Angle Between Vectors / Forces in Equilibrium Study Guide 720 Review Exercises 731 Chapter Test 735 Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 738 Solving Systems of Equations Graphically / The Substitution Method / The Elimination Method / Applications VISUALIZING THE GRAPH 749 9.2 Systems of Equations in Three Variables 754 Solving Systems of Equations in Three Variables / Applications / Mathematical Models and Applications 9.3 Matrices and Systems of Equations 764 Matrices and Row-Equivalent Operations / Gaussian Elimination with Matrices / Gauss–Jordan Elimination 737 INDEX Curve, plane, 891, 892, 904 Curve fitting, 114 Damped oscillation, 563 Dead Sea scrolls, 458 Decay, exponential, 456, 471 Decay rate, 456, 457, 471 Decibel, 428 Decimal degree form, 484 Decimal notation, Decomposing a function, 184, 222 Decomposition, partial fraction, 813 Decreasing function, 158, 220 Degenerate conic section, 880 Degree of a polynomial, 17, 18 of a polynomial function, 296, 372 of a term, 18 Degree measure, 484, 504 converting to radians, 525, 574 decimal form, 484 D°M¿S– form, 484 DeMoivre, Abraham, 679 DeMoivre’s theorem, 679, 723 Denominator least common, 39, 40 rationalizing, 48 Dependent equations, 739, 819 Dependent variable, 64 Depreciation, straight-line, 106 Depression, angle of, 493 Descartes, René, 335 Descartes’ rule of signs, 336, 377 Descending order, 17 Determinants, 792–795, 823 Diagonals of a polygon, 370 Difference See also Subtraction common, 920, 981 of cubes, factoring, 28, 53 of functions, 172, 221 of logarithms, 432, 436, 468 square of, 20, 53 of squares, factoring, 27 Difference identities, 589–593, 639 Difference quotient, 175, 222 Dimensions of a matrix, 764 Direct variation, 211, 226 Direction angle, 712, 729 Direction of a vector, 697 Directly proportional, 211, 226 Directrix of a conic section in polar form, 885 of a parabola, 833 Discontinuous function, 298 Discriminant, 248, 287, 880, 902 Disjunction, 141, 150 Displacement, 697 Distance See also Distance formula of a fall, 250 on the number line, projectile motion, 263 Distance formula See also Distance for motion, 127 between points, 66, 144 Distinguishable permutations, 954, 983 Distributive property for matrices, 779 for real numbers, 5, 53 Dividend, 320, 321 Division of complex numbers, 239, 285, 678, 723 of exponential expressions, 10, 53 of functions, 172, 221 of polynomials, 320–324, 375 of radical expressions, 46, 53 of rational expressions, 37 synthetic, 322, 375 Divisor, 320, 321 D°M¿S– form, 484 Domain of a function, 76, 82, 145, 174, 182, 221 polynomial, 298 rational, 341 trigonometric, 536, 543, 545, 575 of a rational expression, 36 of a relation, 78 restricting, 396 DOT mode, 341 Dot product, 714, 728 Double angle identities, 599–601, 640 Doubling time, 451, 454, 470 DRAW menu, 70 Drift angle, 700 e, 405, 467 Earthquake magnitude, 424, 468 Eccentricity of a conic in polar form, 885, 886 of an ellipse, 849, 885, 886 of a hyperbola, 860, 885, 886 Economic multiplier effect, 935 Element in a function’s domain, 76 of a set, Elements of a matrix, 764 Elevation, angle of, 493 Eliminating the parameter, 892 Eliminating the xy-term, 879, 902 Elimination Gauss–Jordan, 767 Gaussian, 754, 820 Elimination method, 740, 820 Ellipse, 842, 887, 902 center, 842 eccentricity, 849, 885, 886 equations, 843, 845, 880, 887, 899, 902 foci, 842 graphs, 843, 845 I-7 major axis, 843 minor axis, 843 vertices, 843 x-, y-intercepts, 843 Empty set, 282 End behavior, 299, 300, 373 Endpoints, Entries of a matrix, 764 Equality of matrices, 772 Equally likely outcomes, 975 Equation, 31, 121 with absolute value, 280, 289 checking the solutions, 32, 33, 273, 277 of circles, 69, 145, 840, 880, 887, 898, 902 of ellipses, 843, 845, 880, 887, 899, 902 equivalent, 31 exponential, 417, 439 graphs of, 62, 144 See also Graphs of hyperbolas, 593, 596, 617, 852, 855, 880, 887, 900, 902 of inverse relations, 388 linear, 31, 100, 121 logarithmic, 417, 445 logistic, 455 matrix, 789 in one variable, 121 of parabolas, 833, 835, 880, 887, 898, 902 parametric, 892, 904 point–slope, 110, 147 polar, 688 of conics, 887, 902 Pythagorean, 481 quadratic, 31, 241, 286 quadratic in form, 249, 287 radical, 274, 289 rational, 272, 289 rectangular, 688 reducible to quadratic, 249, 287 related, 360, 800 roots of, 241 See also Solutions, of equations slope–intercept, 100, 109, 146, 147 solution of, 61 solution set, 31 solving, 31, 53, 122, 123, 148, 241, 272, 274, 280, 286, 289, 439 solving systems of See Solving systems of equations systems of, 738, 819 in three variables, 754 translating to, 125 trigonometric, 625, 644 Equation-editor screen, 64 Equilibrium, forces in, 715, 730 Equilibrium point, 464 Equilibrium price, 464 Equivalent equations, 31 systems, 742 Equivalent expressions, 38 Equivalent inequalities, 138 I-8 INDEX Equivalent vectors, 697, 707, 726, 728 Euler, Leonhard, 405 Evaluating a determinant, 792–795 Even functions, 193, 223 Even multiplicity, 303, 373 Even roots, 46, 53 Event, 974 Experimental probability, 972, 973, 984 Exponent(s), integer, negative, properties of, 10, 53 rational, 49, 53 of zero, Exponential decay model, 456, 471 Exponential equations, 417, 439 Exponential function, 401, 466 asymptote, 402, 403, 466 base, 401 graphs, 407 inverses, 413 See also Logarithmic functions properties, 403 y-intercept, 403, 466 Exponential growth model, 450, 470 and doubling time, 451, 454, 470 growth rate, 450, 454 Exponential notation See Exponent(s) Expressions complex rational, 41 equivalent, 38 radical, 36 rational, 36 with rational exponents, 49, 53 Factor, common, 22 Factor theorem, 324, 376 Factorial notation, 949, 982 and binomial expansion, 967, 984 Factoring ac-method, 26 common factors, terms with, 22 completely, 28 differences of cubes, 28, 53 differences of squares, 27 FOIL method, 25 by grouping, 23 grouping method, 26 polynomial functions, 324, 330, 376 polynomials, 22–29 strategy, 29 sums of cubes, 28, 53 trinomials, 23–27 Factors of polynomials, 320, 324, 376 Familiarize, 125 Feasible solutions, 806, 824 Fibonacci sequence, 919 Finite sequence, 912, 981 Finite series, 915, 981 First coordinate, 61 Fitting a curve to data, 114 Fixed costs, 106 Foci See Focus Focus of a conic in polar form, 885 of an ellipse, 842 of a hyperbola, 851 of a parabola, 832 FOIL, 20 and factoring, 25 Force, 697 Forces in equilibrium, 715, 730 Formulas, 33 change-of-base, 420, 468 compound interest, 13, 53, 404, 452, 467, 470 diagonals of a polygon, 370 distance of a free fall, 250 between points, 66, 144 Heron’s, 672 for inverses of functions, 393, 465 midpoint, 68, 144 mortgage payment, 55 motion, 127 quadratic, 247, 287 recursion, 917, 981 rotation of axes, 877, 902 savings plan, 16 simple interest, 128 solving for a given variable, 33 FRAC feature, 50 Fractional equations See Rational equations Fractional exponents, 49, 53 Fractional expressions See Rational expressions Fractions clearing, 272, 288 partial, 813 Functions, 76, 145 absolute value, 194 algebra of, 172, 221 circular, 536, 537, 576 cofunctions, 486, 487, 570 composite, 179 composition of, 180, 222, 620, 643 constant, 92, 158, 220, 296 continuous, 298 as correspondences, 76 cube root, 194 cubic, 296, 297 cubing, 194 decomposing, 184, 222 decreasing, 158, 220 defined piecewise, 163, 221 difference of, 172, 221 difference quotient, 175, 222 discontinuous, 298 domain, 76, 82, 83, 145, 174, 182, 221 element, 76 even, 193, 223 exponential, 401, 466 graphs, 80 greatest integer, 165, 221 horizontal-line test, 392, 465 identity, 92, 194 increasing, 158, 220 input, 79 inverse trigonometric, 616, 619, 642 inverses of, 391, 465 library of, 194 See also inside back cover linear, 92, 296 logarithmic, 415, 467 logistic, 456 nonlinear, 92 nonpolynomial, 298 notation for, 79 objective, 546 odd, 193, 223 one-to-one, 390, 465 output, 79 periodic, 543, 544, 576 phase shift, 558, 578 piecewise, 163, 221 polynomial, 296, 372 product of, 172, 221 quadratic, 241, 286, 296, 297 quartic, 296 quotient of, 172, 221 range, 76, 83, 145 rational, 340, 378 reciprocal, 194 trigonometric, 479, 569 reflections, 190, 198, 199, 204, 224 relative maxima and minima, 159, 160, 220 sequence, 912 shrinking, 200, 201, 204, 225 square root, 194 squaring, 194 stretching, 200, 201, 204, 225 sum of, 172, 221 transformations, 194–204, 224–226, 551–561, 577 translations, 195–197, 204, 224, 552, 557, 559, 577 trigonometric, 478, 479, 507, 514, 537, 543, 545, 569 values, 79, 80, 305, 323, 375 vertical-line test, 81, 145 zeros of, 130, 131, 149, 243, 301 Fundamental counting principle, 948, 982 Fundamental theorem of algebra, 330, 376 Gauss, Karl Friedrich, 494, 507 Gauss–Jordan elimination, 507 Gaussian elimination, 494, 560 General term of a sequence, 913, 914 Geometric progressions See Geometric sequences Geometric sequences, 929, 982 common ratio, 929, 982 infinite, 932 INDEX nth term, 930, 982 sums, 931, 933, 982 Geometric series, 931 infinite, 932, 933, 982 Grade, 96 Graph styles, 801 Graphing calculator and absolute value, 6, 674 ANGLE menu, 484, 525 and circles, 70, 841, 842 coefficient of linear correlation, 117 and combinations, 959 and complex numbers, 236, 237, 239, 674, 675, 676, 677, 679, 680, 682 and composition of functions, 180, 181 CONNECTED mode, 341 converting to/from degrees and radians, 526 converting to/from D°M¿S– and decimal degree form, 484 and cube roots, 46 DEGREE mode, 484, 514 and determinants, 796 DIAGNOSTIC, 117 DMS feature, 484 D°M¿S– notation, 484 DOT mode, 341, 547 DRAW menu, 70 and ellipses, 844 equation-editor screen, 64 and factorial notation, 950 FRAC feature, 50 and function values, 80, 305 graph styles, 587, 599, 801 graphing equations, 64 graphing logarithmic functions, 421 graphing rational functions, 341 graphing trigonometric functions, 542, 547 and greatest integer function, 166 and hyperbolas, 854, 856 and identities, 587, 599, 602 and inequalities, 139, 140, 141, 801, 802, 804, 805 INEQUALZ, 801, 802, 804, 805 INTERSECT feature, 123 and inverse trigonometric functions, 485, 617 and inverses of functions, 394, 421, 485, 617 linear regression, 116, 117 and logarithms, 418, 419 LOGIC menu, 140 and matrices, 766, 768, 773, 775, 787, 788, 789, 795 MAXIMUM feature, 160 MINIMUM feature, 160 and nth roots, 46 and order of operations, 13 and parabolas, 262, 834, 836, 837 and parametric equations, 541, 543, 682 and permutations, 949, 952 and polar equations, 691, 886, 888, 889 and polynomial inequalities, 365 PolySmlt, 768 and probability, 977 quadratic regression, 759 RADIAN mode, 526, 540 and reduced row-echelon form, 768 regression, 116, 117, 759 and relative maxima and minima, 160 row-equivalent operations, 766 scatterplot, 114 and scientific notation, 12 and sequences, 913, 915, 916, 917 SEQUENTIAL mode, 587 and solving equations, 123, 250, 265, 275, 277, 442, 444, 447, 452, 630, 633 and square roots, 46 and systems of equations, 738, 864, 866 and systems of inequalities, 868 TABLE feature, 65, 124 TEST menu, 140 and trigonometric function values, 485, 511, 514, 540 TVM solver, 13 VALUE feature, 80 viewing window, 64 squaring, 70 standard, 64 and xth roots, 46 ZERO feature, 132 and zeros of functions, 132, 245, 303, 447, 454 ZOOM menu, 64, 70 Graphs, 60 asymptotes See Asymptotes bar, 60 circle, 60 of circles, 69, 70, 841, 898 of complex numbers, 673, 722 of conics, polar equations, 887 cosecant function, 548 cosine function, 541, 543 cotangent function, 547 ellipses, 843, 845 equations, 62, 64, 101, 144 exponential functions, 407 functions, 80 hole in, 353 horizontal lines, 92 hyperbolas, 852, 855 identity function, 92, 194 inequalities, 800, 823 intercepts, 62, 144 See also x-intercept; y-intercept of intervals, inverse function, 394 trigonometric, 617 inverse relations, 388 inverses of exponential functions, 413 I-9 leading-term test, 300, 372 line, 60 linear inequalities, 139 logarithmic functions, 421, 467 and multiplicity of zeros, 303 of ordered pairs, 61 parabolas, 257, 258, 287, 833, 835, 898 parametric equations, 891 of points, 61 of polar equations, 690, 724 of polynomial functions, 309–316, 373 quadratic functions, 257–262, 287 rational functions, 340, 350, 378 reflection, 190, 198, 199, 204, 224 rotation of axes, 875–880, 902 scatterplot, 114 secant function, 548 of sequences, 914 shrinking, 200, 201, 204, 225 sine function, 541, 543 solving equations using, 123 stretching, 200, 201, 204, 225 of sums, 562 symmetry, 189–194, 223 systems of equations, 738 systems of inequalities, 804, 824 tangent function, 547 transformations, 194–204, 224–226, 551–561, 577 translation, 195–197, 204, 224, 551, 556, 557, 559, 578 turning point, 309, 373 vertical lines, 92 x-intercept, 62, 144 y-intercept, 62, 99, 144 Gravity, acceleration due to, 604, 638 Greater than 17 2, Greatest integer function, 165, 221 Gross domestic product, 115 Grouping, factoring by, 23 Growth, limited, 454 Growth model, exponential, 450, 470 Growth rate, exponential, 450, 454 Half-angle identities, 601, 602, 641 Half-life, 456, 457, 471 Half-open interval, Half-plane, 800 Heron’s formula, 672 Hole in a graph, 350 Hooke’s law, 218 Horizontal asymptotes, 345–348, 378, 402, 466 Horizontal component of a vector, 701, 727 scalar, 705, 728 Horizontal line, 92, 113, 146 slope, 95, 146 Horizontal-line test, 392, 465 Horizontal stretching and shrinking, 201, 204, 225, 555, 559, 577 Horizontal translation, 196, 204, 224, 556, 557, 559, 578 I-10 INDEX Hyperbola, 851 asymptotes, 852 center, 851 conjugate axis, 852 eccentricity, 860, 885, 886 equations, 852, 855, 880, 887, 900, 902 foci, 851 graphs, 852, 855 transverse axis, 851 vertices, 851 Hypotenuse, 47, 478 i, 234, 285 powers of, 237 Identities additive matrix, 774, 775 real numbers, 5, 53 cofunction, 486, 489, 570, 597, 598, 640 multiplicative matrix, 785, 822 real numbers, 5, 53 Pythagorean, 584–586, 639 trigonometric, 487, 570, 584–596, 597–602, 609, 610, 639–642 proving, 606, 641 Identity, 584, 639 See also Identities Identity function, 92, 194 Identity matrix, 785, 822 Imaginary axis, 673 Imaginary numbers, 235 Imaginary part, 235, 285 Inconsistent system of equations, 739, 819 Increasing function, 158, 220 Independent equations, 739, 819 Independent variable, 64 Index of a radical, 45 of summation, 916 Induction, mathematical, 940, 982 Induction step, 940, 982 Inequalities, 138 with absolute value, 282, 289 compound, 140, 150 conjunction, 140, 150 disjunction, 141, 150 equivalent, 138 graphing, 800 linear, 139, 800 polynomial, 360, 380 quadratic, 360 rational, 365, 381 solutions, 138, 800 solving, 138–141, 282, 289 systems of, 804, 823, 867, 901 INEQUALZ, 801, 802, 804, 805 Infinite sequence, 912, 981 Infinite series, 915, 932 Initial point of a vector, 698 Initial side of an angle, 504 Input, 79 Integers, as exponents, Intercepts, 62, 144 See also x-intercept; y-intercept Interest compound, 13, 53, 404, 467 continuously, 452, 470 simple, 128 Intermediate value theorem, 315, 374 INTERSECT feature, 123 Intersect method, 123, 133 Interval notation, 3, Inverse of a function See Inverses Inverse relation, 388, 465 Inverse variation, 213, 226 Inversely proportional, 213, 226 Inverses additive of matrices, 774 of real numbers, 5, 53 function, 391, 465 and composition, 395, 466 exponential, 413 formula for, 393, 465 and reflection across y = x, 394, 465 trigonometric, 616, 619, 642 multiplicative of matrices, 786, 822 of real numbers, 5, 53 relation, 388, 465 Invertible matrix, 788 Irrational numbers, Irrational zeros, 332, 376 Irreducible quadratic factor, 813 Isosceles triangle, 482 Joint variation, 215, 227 Jordan, Wilhelm, 767 1k + 12st term, binomial expansion, 968, 984 Latitude, 604 Law of cosines, 663, 664, 721 Law of sines, 652, 720 Leading coefficient, 17 Leading 1, 767 Leading term, 296, 372 Leading-term test, 300 Least common denominator (LCD), 39, 40 Legs of a right triangle, 47 Length of a vector, 697, 698, 706, 728 Less than 2, Less than or equal to … 2, Libby, Willard E., 456 Library of functions, 194 See also inside back cover Light-year, 15 Like terms, 18 Limit, infinite geometric series, 933, 982 Limited population growth, 454 Limiting value, 455 Line graph, 60 Line of symmetry See Axis, of symmetry Linear combination of unit vectors, 711, 729 Linear correlation, coefficient of, 117 Linear equations, 31, 100, 121 in three variables, 574 Linear functions, 92, 296 applications, 101 slope, 93, 146 zero of, 130 Linear inequalities, 139, 800 systems of, 804, 823 Linear programming, 806 Linear regression, 116, 117 Linear speed, 529, 575 Lines angle between, 595 horizontal, 92, 113, 146 parallel, 111, 147 perpendicular, 111, 112, 148 point–slope equation, 110, 113, 147 slope of, 93, 113, 146 slope–intercept equation, 100, 109, 113, 146, 147 vertical, 92, 113, 146 Lithotripter, 847 ln, 419 log, 418 See also Logarithm Logarithm, 417, 467 See also Logarithmic functions base e, 419, 468 base–exponent property, 439, 469 base-10, 418, 468 a base to a logarithmic power, 435, 469 of a base to a power, 435, 469 on a calculator, 418 change-of-base, 420, 468 common, 418, 468 natural, 419, 468 power rule, 431, 436, 468 product rule, 430, 436, 468 properties, 430–432, 436, 468 quotient rule, 432, 436, 468 Logarithm function See Logarithmic functions Logarithmic equality, property of, 440, 469 Logarithmic equation, 417, 445 Logarithmic functions, 413–415, 467 See also Logarithm asymptote, 416, 467 domain, 416, 467 graphs, 421, 467 properties, 416, 467 range, 416, 467 x-intercept, 416, 467 Logistic function, 455 LORAN, 858 Loudness of sound, 428 INDEX Magnitude of a vector, 697, 698, 706, 728 Magnitude of an earthquake, 424, 468 Main diagonal, 764 Major axis of an ellipse, 843 Mathematical induction, 940, 982 Mathematical models, 91, 114, 304, 758, 867 See also Regression Matrices, 764, 821 addition of, 772, 821 additive identity, 774, 775 additive inverses of, 774, 775 augmented, 764, 821 coefficient, 764, 821 cofactor, 793 column, 779 columns of, 764 determinant of, 792–795, 823 dimensions, 764 elements, 764, 821 entries, 764, 821 equal, 772 and Gauss–Jordan elimination, 767 and Gaussian elimination, 764 identity, multiplicative, 785, 822 inverses of, 786, 822 invertible, 788 main diagonal, 764 minor, 792 multiplicative inverse, 786, 822 multiplying, 776, 822 by a scalar, 775 nonsingular, 788 opposite of, 774 order, 764 reduced row-echelon form, 767 row, 779 row-echelon form, 766, 767, 821 row-equivalent, 764 row-equivalent operations, 765, 821 rows of, 764 and scalar multiplication, 774, 822 singular, 788 square, 764 subtraction, 772 and systems of equations, 754, 764, 767, 789, 820, 822 zero, 774 Matrix See Matrices Matrix equation, 789 Maximum linear programming, 806, 824 quadratic function, 258, 287 relative, 159, 160, 220 MAXIMUM feature, 160 Mean, arithmetic, 928 Midpoint formula, 68, 144 Mil, 535 Minimum linear programming, 806, 824 quadratic function, 258, 287 relative, 159, 160, 220 MINIMUM feature, 160 Minor, 792 Minor axis of an ellipse, 843 Models, mathematical, 91, 114, 304, 758, 867 See also Regression Monomial, 18 Mortgage payment formula, 55 Motion, projectile, 263 Motion formula, 127 Multiple, scalar, 707 Multiplication associative property, 5, 53, 779 commutative property, 5, 53 of complex numbers, 236, 285, 677, 723 of exponential expressions, 10, 53 of exponents, 10, 53 of functions, 172, 221 of matrices, 776, 822 and scalars, 774, 822 of ordinates, 562 of polynomials, 19–21, 53 of radical expressions, 46, 53 of rational expressions, 37 scalar for matrices, 774 for vectors, 707, 710, 728 Multiplication principle for equations, 31, 53, 121, 148 for inequalities, 139, 149 Multiplicative identity matrices, 785, 822 real numbers, 5, 53 Multiplicative inverse matrices, 786, 822 real numbers, 5, 53 Multiplicity of zeros, 303, 373 nth partial sum, 915, 981 nth root, 45 nth roots of unity, 681 nth term arithmetic sequence, 921, 981 geometric sequence, 930, 982 Nanometer, 15 Nappes of a cone, 832 Natural logarithms, 419, 468 Natural numbers, Nautical mile, 604, 638 Negative exponents, Negative rotation, 504 Newton’s law of cooling, 463 Nondistinguishable objects, permutations of, 954, 983 Nonlinear function, 92 Nonlinear systems of equations, 861, 901 Nonlinear systems of inequalities, 867, 901 Nonnegative root, 45 Nonpolynomial functions, 298 Nonreal zeros, 331, 376 Nonsingular matrix, 788 Notation binomial coefficient, 958, 983 combination, 957 decimal, exponential, factorial, 949, 982 function, 79 interval, inverse function, 390, 465 trigonometric, 616 for matrices, 772 permutation, 949, 981 polar, 674 radical, 45 scientific, 11 set, 4, 37 sigma, 916 standard, for complex numbers, 675 summation, 916 trigonometric, for complex numbers, 674, 722 Number line, distance between points, Numbers complex, 235, 236, 285 imaginary, 235 integers, irrational, natural, pure imaginary, 235 rational, real, whole, Numerator, rationalizing, 48 Objective function, 806 Oblique asymptotes, 348, 349, 378 Oblique triangle, 650 Obtuse angle, 505 Odd functions, 193, 223 Odd multiplicity, 303, 373 Odd roots, 46, 53 One-to-one functions, 390, 465 and inverses, 391, 465 Open interval, 3, Operations order of, 12 row-equivalent, 794 Opposite of a matrix, 774 Order descending, 17 of a matrix, 764 of operations, 12 of real numbers, Ordered pair, 61 Ordered triple, 754 Ordinate, 61 Ordinates addition of, 562 multiplication of, 563 I-11 I-12 INDEX Origin, 61 symmetry with respect to, 190, 191, 223 Orthogonal vectors, 719 Oscillation, damped, 563 Oscilloscope, 562 Outcomes, 974 equally likely, 975 Output, 79 Pair, ordered, 61 Parabola, 257, 287, 832 axis of symmetry, 257, 287, 832 directrix, 832 equations, 833, 835, 898 focus, 832 graphs, 257, 258, 287, 833, 835, 898 vertex, 257, 261, 287, 832 Parallel lines, 111, 113, 147 Parallelogram law, 699 Parameter, 892, 904 Parametric equations, 892, 904 converting to/from rectangular equations, 892–894 Parsec, 15 Partial fractions, 813 Partial sum, 915, 981 Pascal’s triangle, 964 Perimeter, 129 Periodic function, 543, 544, 555, 559, 576, 578 Permutations, 948 of n objects, k at a time, 951, 983 allowing repetition, 953, 983 of n objects, total number, 949, 982 of nondistinguishable objects, 954, 983 Perpendicular lines, 111, 112, 113, 148 pH, 427 Phase shift, 558, 578 Pi 1p2, Piecewise-defined function, 163, 221 Plane curve, 891, 892, 904 Plotting a point, 61, 686, 724 Point coordinates of, 61 plotting, 61, 686, 724 Point–slope equation, 110, 113, 147 Poker hands, 963, 979 Polar axis, 686 Polar coordinates, 686 converting to/from rectangular coordinates, 687, 724, 725 Polar equation of a conic, 887, 890, 902 converting to a rectangular equation, 689, 725, 889, 903 graphing, 690, 724 Polar notation, 674 Pole, 686 Polygon, number of diagonals, 370 Polynomial function, 296, 372 See also Polynomials coefficients, 296, 372 degree, 296, 372 domain, 298 factoring, 324, 330, 376 graph of, 297, 298, 309–316, 373 intermediate value theorem, 315, 374 leading coefficient, 296, 372 leading term, 296, 372 leading-term test, 300, 372 as models, 304 values, 322, 375 zeros of, 301–304, 330–337, 373, 376, 377 Polynomial inequalities, 360, 380 Polynomials See also Polynomial function addition, 18, 19 binomial, 18 coefficients, 17 constant term, 17 degree of, 17, 18 degree of a term, 18 descending order, 17 division of, 320–324, 375 factoring, 22–29 factors of, 320, 324, 376 leading coefficient, 17 monomial, 18 multiplication of, 19–21, 53 in one variable, 17 prime, 29 in several variables, 18 subtraction of, 18, 19 term of, 17 trinomial, 18 PolySmlt, 768 Population growth, 450, 470 limited, 454 Position vector, 705 Positive rotation, 504 Power, raising to a power, 10, 53 Power rule for exponents, 10, 53 for logarithms, 431, 468 Powers of complex numbers, 679, 723 of i, 237 principle of, 275, 289 Present value, 464 Prime polynomial, 29 Principal root, 45 Principles(s) addition, 31, 53, 122, 139, 148, 149 fundamental counting, 948, 982 of mathematical induction, 940, 982 multiplication, 31, 53, 122, 139, 148, 149 P (experimental probability), 973, 984 P (theoretical probability), 975, 984 of powers, 275, 289 of square roots, 31, 53, 241, 286 of zero products, 31, 53, 241, 286 Probability, 971 experimental, 972, 973, 984 properties, 976 theoretical, 972, 974, 975, 984 Problem-solving steps, 125 Product See also Multiplication dot, 714, 728 of functions, 172, 221 logarithm of, 430, 468 of matrices, 776, 822 raised to a power, 10, 53 scalar for matrices, 774 for vectors, 707, 710, 728 of vectors, 714, 728 Product rule for exponents, 10, 53 for logarithms, 430, 468 Product-to-sum identities, 609, 642 Profit, total, 269 Programming, linear, 806 Progressions See Sequences Projectile motion, 263 Properties of addition of real numbers, 5, 53 associative, 5, 53, 779 base–exponent, 439, 469 commutative, 5, 53, 775 of cosine function, 545 distributive, 5, 53, 779 of exponential functions, 403 of exponents, 10, 53 of logarithmic equality, 440, 469 of logarithmic functions, 416, 467 of logarithms, 430–433, 436, 468 of matrix addition, 775 of matrix multiplication, 779 of multiplication of real numbers, 5, 53 of probability, 976 of radicals, 46, 53 of real numbers, 5, 53 of scalar multiplication for matrices, 775 for vectors, 710 of sine function, 545 of vector addition, 710 Proportional See Variation Proving trigonometric identities, 606, 641 Pure imaginary numbers, 235 Pythagorean equation, 481 Pythagorean identities, 584–586, 639 Pythagorean theorem, 48, 53 Quadrantal angle, 510, 572 Quadrants, 61 Quadratic equations, 31, 241, 286 See also Quadratic function discriminant, 248, 287 solving, 31, 53, 241, 286 standard form, 241 Quadratic form, equation in, 249, 287 INDEX Quadratic formula, 247, 287 Quadratic function, 241, 286, 296, 297 See also Quadratic equations applications, 262 graph, 257, 262, 287 See also Parabola maximum value, 258, 287 minimum value, 258, 287 zeros of, 243 Quadratic inequalities, 360, 380 Quadratic regression, 759 Quartic function, 296 Quotient difference, 175, 222 of functions, 172, 222 logarithm of, 432, 468 of polynomials, 320, 321 raised to a power, 10, 53 Quotient rule for exponents, 10, 53 for logarithms, 432, 468 Radian measure, 523, 528, 574, 575 converting to degree measure, 525, 574 Radical, 45 See also Radical expressions Radical equation, 274, 289 Radical expressions, 45 See also Roots addition, 47 conjugate, 49 converting to exponential notation, 49, 53 division, 46, 53 multiplication, 46, 53 properties of, 46, 53 rationalizing denominators or numerators, 48 simplifying, 46 subtraction, 47 Radicand, 45 Radius, 69, 145, 840 Raising a power to a power, 10, 53 Raising a product or quotient to a power, 10, 53 Range of a function, 76, 83, 145 of a relation, 78 Rate of change, 93, 98, 146, 175 Ratio, common, 929, 982 Rational equations, 272, 288 Rational exponents, 49, 53 Rational expressions, 36 addition, 39 complex, 41 decomposing, 793 division, 37 domain, 36 multiplication, 37 simplifying, 37 subtraction, 39 Rational function, 340, 378 asymptotes, 342–349, 378 domain, 341 graph of, 340, 350, 378 Rational inequalities, 365, 381 Rational numbers, Rational zeros theorem, 333, 377 Rationalizing denominators or numerators, 48 Real axis, 673 Real numbers, 2, order, properties, 5, 53 Real part, 235, 285 Reciprocal, in division, 38 Reciprocal function, 194 trigonometric, 479, 569 Rectangular coordinates, converting to/from polar coordinates, 687, 724, 725 Rectangular equations converting to/from parametric equations, 892–894 converting to/from polar equations, 689, 725, 889, 903 Recursion formula, 917, 981 Recursively defined sequence, 917, 981 Reduced row-echelon form, 767 Reducible to quadratic equation, 249, 287 Reference angle, 512, 572 Reference triangle, 506 Reflection, 190, 198, 199, 204, 224 and inverse relations and functions, 388, 394 on unit circle, 537, 576 Regression correlation coefficient, 117 linear, 116, 117 quadratic, 759 Related equation, 360, 800 Relation, 78 domain, 78 inverse, 388, 465 range, 78 Relative maxima and minima, 159, 160, 220 Remainder, 320, 321 Remainder theorem, 322, 375 Repeating decimals, changing to fraction notation, 934 Representing a vector, 701, 727 Resolving a vector into its components, 701, 727 Restricting a domain, 396 Resultant, 699, 727 Revenue, total, 269 Richter scale, 424, 468 Right angle, 478, 505 Right triangle, 478 solving, 490 Rise, 93, 146 Roots See also Radical expressions of complex numbers, 680, 724 cube, 45 of equations, 241 even, 46, 53 nth, 45 nonnegative, 45 odd, 46, 53 principal, 45 square, 45 Rotation of a ray, 504 Rotation of axes, 875–880, 902 Row-echelon form, 766, 767, 821 Row-equivalent matrices, 764 Row-equivalent operations, 765, 821 Row matrix, 779 Rows of a matrix, 764 Run, 93, 146 Salvage value, 106, 929 Sample space, 975 Savings plan, 16 Scalar, 705, 707, 774, 822 Scalar components of a vector, 705, 728 Scalar multiplication for matrices, 774, 822 for vectors, 707, 710, 728 Scalar product for matrices, 774, 822 for vectors, 707, 710, 728 Scatterplot, 114 Scientific notation, 11 Secant function, 479, 507, 537, 548, 569 graph, 548 Secant line, 175 Second coordinate, 61 Semiperimeter, 672 Sequences, 912 alternating, 913 arithmetic, 920, 982 Fibonacci, 919 finite, 912, 981 general term, 913, 914 geometric, 929, 982 graph, 914 infinite, 912, 981 partial sum, 915, 981 recursive definition, 917, 981 of statements, 940 terms, 913 Series arithmetic, 923 finite, 915, 981 geometric, 931 infinite, 915, 981 partial sum, 915, 981 Set-builder notation, 37 Sets element of, empty, 282 notation, 4, 37 solution, 31, 121, 138, 800 subsets, 3, 960 union of, 141 Shift of a function See Translation of a function I-13 I-14 INDEX Shrinkings, 200, 201, 204, 225, 555, 559, 577, 578 Sigma notation, 916 Signs of trigonometric functions, 508, 571 Similar terms, 18 Similar triangles, 480 Simple-interest formula, 128 Simplifying radical expressions, 46 Simplifying rational expressions, 37 complex, 41 Simplifying trigonometric expressions, 586 Sine function, 478, 479, 507, 537, 569 amplitude, 544, 545 domain, 543, 545 graph, 541, 545 period, 544, 545 range, 543, 545 Sines, law of, 652, 720 Singular matrix, 788 Slant asymptote, 348, 349, 378 Slope, 93, 113, 146 applications, 96 of horizontal lines, 95, 146 as rate of change, 93, 98, 146 of vertical lines, 95, 146 Slope–intercept equation, 100, 109, 113, 146, 147 Solution set, 31, 121, 138, 800 Solutions of equations, 61, 121 feasible, 806, 824 of inequalities, 138, 800 of systems of equations, 738, 819 Solving equations, 31, 53, 122, 123, 148, 241, 272, 274, 280, 286, 289, 439, 625, 644 See also Solving systems of equations Solving formulas, 33 Solving inequalities, 138–141, 282, 289, 360–368, 381 systems of, 804, 823, 867, 901 Solving systems of equations with Cramer’s rule, 796, 797 elimination method, 740, 820 Gauss–Jordan elimination, 767 Gaussian elimination, 754, 764, 820 graphically, 738 with the inverse of a matrix, 779, 822 nonlinear, 861, 901 substitution method, 739, 820 Solving triangles, 490, 570, 650, 653, 655, 677, 720, 721, 722 oblique, 650, 653, 655, 664, 667, 720, 771, 772 right, 490, 570 Sound, loudness, 428 Speed angular, 529, 575 linear, 529, 575 Square, completing, 244, 286 Square matrix, 764 determinant, 792 Square root, 45 Square root function, 194 Square roots, principle of, 31, 53, 241, 286 Squares, differences of, 27 Squares of binomials, 20, 53 factoring, 28, 53 Squaring function, 194 Squaring the window, 70 Standard form for equation of a circle, 69, 145, 840, 898 for equation of an ellipse, 843, 845, 899 for equation of a hyperbola, 852, 855, 900 for equation of a parabola, 833, 835, 898 for quadratic equation, 241 Standard notation, complex numbers, 675 Standard position for an angle, 504 for a vector, 705 Standard viewing window, 64 State the answer, 125 Straight angle, 505 Straight-line depreciation, 106 Stretchings, 200, 201, 204, 225, 555, 559, 577, 578 Study tips, 95, 160, 174, 190, 246, 277, 315, 321, 337, 415, 436, 452, 480, 531, 586, 606, 615, 634, 653, 669, 674, 747, 766, 816, 835, 844, 853, 867, 877, 896, 921, 950, 972 Subsets, 3, 960 total number, 969, 984 Substitution method, 739, 820 Subtraction See also Difference of complex numbers, 236, 285 of exponents, 10, 53 of functions, 172, 221 of matrices, 772 of polynomials, 18, 19 of radical expressions, 47 of rational expressions, 39 of vectors, 709, 728 Sum and difference, product of, 20, 53 Sum and difference identities, 589–593, 639 Sum-to-product identities, 610, 642 Summation, index of, 916 Summation notation, 916 Sums See also Addition of arithmetic sequences, 923, 981 of cubes, factoring, 28, 53 of functions, 172, 221 of geometric sequences, 931, 933, 982 graphs of, 562 of logarithms, 430, 436, 468 partial, 915, 981 squares of, 20, 53 of vectors, 699, 708, 710, 727, 728 Supplementary angles, 506, 571 Supply and demand, 747 Symmetry, 189 axis of, 257, 287 with respect to the origin, 190, 191, 223 with respect to the x-axis, 190, 191, 223 with respect to the y-axis, 190, 191, 223 tests for, 191, 223 Synthetic division, 322, 375 Systems of equations, 738, 819 consistent, 739, 819 in decomposing rational expressions, 816 dependent equations, 739, 819 equivalent, 742 inconsistent, 739, 819 independent equations, 739, 819 in matrix form, 789 nonlinear, 861, 901 and partial fractions, 816 solutions, 738, 819 solving See Solving systems of equations in three variables, 754, 820 in two variables, 738, 819 Systems of inequalities, 804, 823, 867, 901 TABLE feature, 65, 124 Tangent function, 479, 507, 547, 548, 569, 577 Term, leading, 296, 372 Terminal point of a vector, 698 Terminal side of an angle, 504 Terminating decimal, Terms of a polynomial, 17 constant, 17 degree of, 18 leading, 296, 372 like (similar), 18 Terms of a sequence, 913 TEST menu, 140 Theorem binomial, 966, 967, 983 DeMoivre’s, 679, 723 factor, 324, 376 fundamental, of algebra, 330, 376 intermediate value, 315, 374 Pythagorean, 48, 53 rational zeros, 333, 377 remainder, 322, 375 Theoretical probability, 972, 974, 975, 978 Time of a free fall, 250 Total cost, profit, revenue, 269 Tower of Hanoi problem, 954 Trade-in value, 106 Transformations of functions, 194–204, 224–226, 551–561, 577 Translate to an equation, 125 Translation of a function, 195–197, 204, 224, 551, 556, 557, 559, 578 Transverse axis of a hyperbola, 851 Tree diagram, 947 Triangle, Pascal’s, 964 Triangles area, 658, 659, 721 isosceles, 482 INDEX oblique, 650 reference, 506 right, 478 semiperimeter, 672 similar, 480 solving, 490, 570, 650, 653, 655, 664, 667, 720, 721, 722 Trigonometric equations, 625, 644 Trigonometric expressions, simplifying, 586 Trigonometric functions of acute angles, 478, 479, 484, 569 of any angle, 507, 514, 571, 572 circular, 536, 537, 576 cofunctions, 486, 487, 570, 597, 598, 640 composition, 620, 643 domain, 536, 543, 545, 575 graphs, 541, 547, 548, 551–561, 577 inverse, 616, 619, 642 of quadrantal angles, 510, 572 range, 536, 619 reciprocal, 479, 569 signs of, 508, 571 value of 30°, 60°, 45°, 481–483, 569 Trigonometric identities, 487, 570, 584–586, 589–593, 597–602, 609, 610, 639–642 proving, 606, 641 Trigonometric notation for complex numbers, 674, 722 Trinomial, 18 factoring, 23–27 Triple, ordered, 444 Turning point, 309, 373 Undefined slope, 95, 146 Union of sets, 141 Unit circle, 75, 520 reflections on, 537, 576 Unit vector, 711, 729 Unity, nth roots of, 681 VALUE feature, 80 Values, critical, 365 Values, function, 79, 80, 305 Variable, 64 Variable costs, 106 Variation combined, 215, 227 direct, 211, 226 inverse, 213, 226 joint, 215, 227 Variation constant, 211, 213, 226 Variations in sign, 335 Vectors, 697, 726 addition of, 699, 708, 710, 727, 728 additive identity, 710, 729 angle between, 715, 730 component form, 706, 727 components, 701, 727 scalar, 705, 728 direction, 697 direction angle, 712, 729 dot product, 714, 728 equivalent, 697, 707, 726, 728 and forces in equilibrium, 715, 730 initial point, 698 length, 697, 698, 706, 728 as linear combinations of unit vectors, 711, 729 magnitude, 697, 698, 706, 728 orthogonal, 719 position, 705 representing, 701, 727 resolving into components, 701, 727 resultant, 699, 727 scalar multiplication, 707, 710, 728 standard position, 705 subtraction, 709, 728 sum of, 699, 708, 710, 727, 728 terminal point, 698 unit, 711, 729 zero, 710, 729 Velocity, 697 Vertex of an angle, 504 of an ellipse, 843 of a hyperbola, 851 of a parabola, 257, 261, 287, 832 Vertical asymptotes, 342–345, 378 Vertical component of a vector, 701, 727 scalar, 705, 728 Vertical line, 92, 113, 146 slope, 95, 146 Vertical-line test, 81, 145 Vertical stretching and shrinking, 200, 204, 225, 553, 559, 578 I-15 Vertical translation, 195, 204, 224, 551, 556, 557, 559, 578 Vertices See Vertex Viewing window, 64 squaring, 70 standard, 64 Visualizing the Graph, 71, 102, 205, 266, 317, 356, 425, 489, 565, 635, 694, 869, 936 Walking speed, 423 Wedge bevel, 668 Whispering gallery, 847 Whole numbers, Window, 64 squaring, 70 standard, 64 x-axis, 61 symmetry with respect to, 190, 191, 223 x-coordinate, 61 x-intercept, 62, 144 ellipse, 843 logarithmic function, 416, 467 xy-term, eliminating, 879, 902 y-axis, 61 symmetry with respect to, 190, 191, 223 y-coordinate, 61 y-intercept, 62, 99, 144 ellipse, 843 exponential function, 403, 466 Zero, exponent, ZERO feature, 132 Zero matrix, 774 Zero method, 132, 133 Zero of a function See Zeros of functions Zero products, principle of, 31, 53, 241, 286 Zero slope, 95, 146 Zero vector, 710, 729 Zeros of functions, 130, 131, 149, 243, 301–304, 330–337, 373, 376, 377 irrational, 332, 376 multiplicity, 303, 373 nonreal, 331, 376 rational, 333, 377 ZOOM menu, 64, 70 Geometry Plane Geometry Rectangle Area: A = lw Perimeter: P = 2l + 2w Square Area: A = s2 Perimeter: P = 4s Solid Geometry Rectangular Solid Volume: V = lwh w h l l s s Triangle Area: A = 12 bh Cube Volume: V = s3 s h b Sum of Angle Measures A + B + C = 180° Right Triangle Pythagorean theorem (equation): a2 + b2 = c w s s B A C c a r Right Circular Cylinder Volume: V = pr 2h Lateral surface area: L = 2prh Total surface area: S = 2prh + 2pr h r b Parallelogram Area: A = bh h b a Trapezoid Area: A = 12 h1a + b2 h Right Circular Cone Volume: V = 13pr 2h Lateral surface area: L = prs Total surface area: S = pr + prs Slant height: s = 2r + h2 s h r b Circle Area: A = pr Circumference: C = pd = 2pr d r Sphere Volume: V = 43 pr Surface area: S = 4pr r Algebra Properties of Real Numbers Factoring Formulas a + b = b + a; ab = ba a + 1b + c2 = 1a + b2 + c; a1bc2 = 1ab2c a + = + a = a Additive Identity: - a + a = a + - a2 = Additive Inverse: Multiplicative Identity: a # = # a = a 1 Multiplicative Inverse: a # = # a = 1, a Z a a a1b + c2 = ab + ac Distributive: a2 a2 a2 a3 a3 Commutative: Associative: Exponents and Radicals am # an = am + n am = am - n an 1ab2m = ambm a-n = n a 1am2n = amn a m am a b = m b b n If n is even, 2an = ƒ a ƒ n If n is odd, 2an = a n n n 2a # 2b = 2ab , a, b Ú n 2a n a = n Ab 2b 2am = A 2a B m = am>n n n + + - b2 = 1a + b21a - b2 2ab + b2 = 1a + b22 2ab + b2 = 1a - b22 b3 = 1a + b21a2 - ab + b22 b3 = 1a - b21a2 + ab + b22 Interval Notation 1a, b2 = 5x|a x b6 3a, b4 = 5x|a … x … b6 1a, b4 = 5x|a x … b6 3a, b2 = 5x|a … x b6 1- q , a2 = 5x|x a6 1a, q = 5x|x a6 1- q , a4 = 5x|x … a6 3a, q = 5x|x Ú a6 Absolute Value ƒaƒ Ú For a 0, ƒ X ƒ = a : X = - a or X = a, ƒ X ƒ a : - a X a, ƒ X ƒ a : X - a or X a Special-Product Formulas Equation-Solving Principles 1a 1a 1a 1a 1a a = b:a + c = b + c a = b : ac = bc a = b : an = bn ab = a = or b = x = k : x = 1k or x = - 1k + + + - b21a - b2 = a2 - b2 b22 = a2 + 2ab + b2 b22 = a2 - 2ab + b2 b23 = a3 + 3a2b + 3ab2 + b3 b23 = a3 - 3a2b + 3ab2 - b3 n n 1a + b2n = a a b an-kbk, where k=0 k n! n a b = k k!1n - k2! n1n - 121n - 22 Á 3n - 1k - 124 = k! Inequality-Solving Principles a b:a + c b + c a b and c : ac bc a b and c : ac bc (Algebra continued) Algebra (continued) The Distance Formula Properties of Exponential and Logarithmic Functions The distance from 1x 1, y12 to 1x 2, y22 is given by log a x = y x = ay log a MN = log a M + log a N M = log a M - log a N log a N log a M log b M = log a b log a a = log a a x = x d = 21x - x 12 + 1y2 - y12 2 The Midpoint Formula The midpoint of the line segment from 1x 1, y12 to 1x 2, y22 is given by a x + x y1 + y2 , b 2 1x - h22 + 1y - k22 = r 1x - h22 1y - k22 Ellipse: + = 1, a2 b2 1y - k22 1x - h22 + = b2 a2 Parabola: 1x - h22 = 4p1y - k2, 1y - k22 = 4p1x - h2 1x - h22 1y - k22 Hyperbola: = 1, a2 b2 1y - k22 1x - h22 = a2 b2 Circle: The slope of the line containing points 1x 1, y12 and 1x 2, y22 is given by y2 - y1 x2 - x1 Slope–intercept equation: Horizontal line: Vertical line: Point–slope equation: log a = aloga x = x Conic Sections Formulas Involving Lines m = ax = ay x = y log a M p = p log a M y y x y = = = - f1x2 = mx + b b or f1x2 = b a y1 = m1x - x 12 The Quadratic Formula Arithmetic Sequences and Series The solutions of ax + bx + c = 0, a Z 0, are given by a1, a1 + d, a1 + 2d, an + = an + d n Sn = 1a1 + an2 x = - b ; 2b2 - 4ac 2a a1 + 3d, Á an = a1 + 1n - 12d Geometric Sequences and Series Compound Interest Formulas Compounded n times per year: A = Pa + Compounded continuously: P1t2 = P0ekt i b n nt a1 , a1r , a1r 2, a1r 3, an + = anr a111 - r n2 Sn = - r an = a1r n - a1 , ƒrƒ Sq = - r Trigonometry Trigonometric Functions Real Numbers Any Angle Acute Angles y y (x, y) hyp opp r u u x adj opp hyp , opp hyp adj hyp , sec u = , cos u = hyp adj adj opp , cot u = tan u = opp adj sin u = , csc u = (x, y) s y sin u = , r x cos u = , r y tan u = , x r csc u = , y r sec u = , x x cot u = y x sin s = y, cos s = x, tan s = y , x , y sec s = , x x cot s = y csc s = Basic Trigonometric Identities sin - x2 = - sin x, cos 1- x2 = cos x, tan 1- x2 = - tan x, , sin x , sec x ϭ cos x cot x ϭ tan x sin x , cos x cos x , cot x = sin x csc x = tan x = Pythagorean Identities Double-Angle Identities sin2 x + cos2 x = 1, + cot x = csc2 x, + tan2 x = sec2 x sin 2x = sin x cos x, cos 2x = cos2 x - sin2 x = - sin2 x = cos2 x - 1, tan x tan 2x = - tan2 x Identities Involving P/ sin 1p>2 - x2 = cos x, cos 1p>2 - x2 = sin x, sin 1x ; p>22 = ; cos x, tan 1p>2 - x2 = cot x, cos 1x ; p>22 = < sin x Sum and Difference Identities sin 1u ; v2 = sin u cos v ; cos u sin v, cos 1u ; v2 = cos u cos v < sin u sin v, tan u ; tan v tan 1u ; v2 = < tan u tan v Half-Angle Identities sin x x - cos x + cos x = ; , cos = ; , A 2 A tan - cos x sin x - cos x x = ; = = A + cos x + cos x sin x (Trigonometry continued) Trigonometry (continued) Trigonometric Function Values of Special Angles The Law of Sines y In any ^ABC, ΂Ϫq, a b c = = sin A sin B sin C , ͙2 ΂Ϫ ͙2 2 ΃ ΂ B The Law of Cosines ΃ S a c In any ^ABC, Ϫ ͙3 ,q A (0, 1) ΃ q 90Њ i 60Њ 120Њ f 135Њ ΂ a = b + c - 2bc cos A , b = a + c - 2ac cos B , c = a + b - 2ab cos C ΂q, ͙3 ΃ ͙2 u , ΃ ΂ ͙2 45Њ d 30Њ 150Њ A , q΃ ΂ ͙3 0Њ (1, 0) 360Њ 2p x (Ϫ1, 0) 180Њ p C b ͙3 330Њ 210Њ G ͙3 F 315Њ ΃ 225Њ ΂ , Ϫq΃ 300Њ j h 240Њ ͙2 ͙2 ,Ϫ ΃ ,Ϫ ΃ p ΂͙2 ΂Ϫ ͙2 o 2 w 270Њ ΂Ϫq, Ϫ͙3 ΂q, Ϫ͙3 ΃ ΃ (0, Ϫ1) Ϫ ͙3 , Ϫq Graphs of Trigonometric Functions Ϫ2p Ϫw Ϫp Ϫq y y 2 1 Ϫ1 q p w 2p x Ϫ2p Ϫw Ϫp Ϫq Ϫ1 q p w 2p x 2p x 2p x Ϫ2 The sine function: f1x2 = sin x Ϫ2p Ϫw Ϫp Ϫq The cosecant function: f1x2 = csc x y y 2 Ϫ1 q p w 2p x Ϫ2p Ϫw Ϫp Ϫq The cosine function: f1x2 = cos x Ϫw Ϫp Ϫq p w The secant function: f1x2 = sec x y y 2 1 Ϫ1 q Ϫ2 Ϫ2 Ϫ2p Ϫ1 q p w 2p x Ϫ2p Ϫw Ϫp Ϫq Ϫ1 q p w Ϫ2 The tangent function: f1x2 = tan x The cotangent function: f1x2 = cot x A Library of Functions Linear function Linear function y y Ϫ4 Ϫ2 2 Ϫ2 x Ϫ4 Ϫ2 f(x) ϭ 3x ϩ Ϫ4 Squaring function f (x) ϭ Ϫ2 x Ϫ1 x Ϫ4 y 4 2 Ϫ2 Ϫ2 Ϫ2 Ϫ4 Ϫ4 Quadratic function y Ϫ2 x Ϫ2 f(x) ϭ Ϫ3 x Ϫ6 Ϫ4 x Square-root function y f(x) ϭ ͙x Ϫ2 Ϫ4 Ϫ4 2 f(x) ϭ |x| Ϫ2 Ϫ2 Ϫ4 Ϫ2 f(x) ϭ Ϫ x Ϫ x Ϫ 2 Ϫ4 x Ϫ4 y f(x) ϭ x Quadratic function y Ϫ4 y Absolute-value function Constant function x Ϫ4 Ϫ2 Ϫ2 Ϫ2 Ϫ4 Ϫ4 x f (x) ϭ x Ϫ 2x Ϫ Cubing function Cube root function 4 Ϫ2 f(x) ϭ x3 x Ϫ4 Ϫ2 Ϫ2 Ϫ2 Ϫ4 Ϫ4 x Ϫ4 f(x) ϭ x Ϫ4 Ϫ2 2 x Ϫ2 f(x) ϭ [[x]] x Ϫ4 Ϫ2 4000 f(x) ϭ log x Ϫ2 Ϫ2 Ϫ2 Ϫ4 Ϫ4 Ϫ4 x y 4 Logistic function y f (x) ϭ Ϫ4 Logarithmic function 2Ϫx f(x) ϭ x Ϫ2 Ϫ4 ex y Ϫ2 y Ϫ2 Exponential function y f (x) ϭ ͙x Exponential function Ϫ4 y y y Ϫ4 Greatest integer function Rational function x 2000 Ϫ20 Ϫ10 10 20 Ϫ2000 f(x) ϭ 3000 ϩ 5eϪ0.4x x ... Cataloging-in-Publication Data Beecher, Judith A Algebra and trigonometry / Judith A Beecher, Judith A Penna, Marvin L Bittinger. ? ?4th ed p cm ISBN-13: 978-0-321-69398-3 ISBN-10: 0-321-69398-1 Algebra? ??Textbooks Trigonometry? ??Textbooks... for a positive learning experience, Judy Beecher Judy Penna Marv Bittinger Algebra and Trigonometry, Fourth Edition, covers college-level algebra and trigonometry and is appropriate for a one-... DESARGALOS GRATIS Algebra and Trigonometry Judith A Beecher Indiana University Purdue University Indianapolis Judith A Penna Indiana University Purdue University Indianapolis Marvin L Bittinger Indiana