Algebra trigonometry

Algebra and Trigonometry Eighth Edition Ron Larson The Pennsylvania State University The Behrend College With the assistance of David C Falvo The Pennsylvania State University The Behrend College Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Algebra and Trigonometry, Eighth Edition Ron Larson Publisher: Charlie VanWagner Acquiring Sponsoring Editor: Gary Whalen Development Editor: Stacy Green Assistant Editor: Cynthia Ashton Editorial Assistant: Guanglei Zhang © 2011, 2007 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, 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Inc Brooks/Cole 10 Davis Drive Belmont, CA 94002-3098 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan Locate your local office at: international.cengage.com/region Cengage Learning products are represented in Canada by Nelson Education, Ltd For your course and learning solutions, visit www.cengage.com Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in the United States of America 13 12 11 10 09 Contents A Word from the Author (Preface) vii chapter P Prerequisites P.1 Review of Real Numbers and Their Properties P.2 Exponents and Radicals 15 P.3 Polynomials and Special Products 28 P.4 Factoring Polynomials 37 P.5 Rational Expressions 45 P.6 The Rectangular Coordinate System and Graphs 55 Chapter Summary 66 Review Exercises 68 Chapter Test 71 Proofs in Mathematics 72 Problem Solving 73 chapter Equations, Inequalities, and Mathematical Modeling 75 1.1 Graphs of Equations 76 1.2 Linear Equations in One Variable 87 1.3 Modeling with Linear Equations 96 1.4 Quadratic Equations and Applications 107 1.5 Complex Numbers 122 1.6 Other Types of Equations 129 1.7 Linear Inequalities in One Variable 140 1.8 Other Types of Inequalities 150 Chapter Summary 160 Review Exercises 162 Chapter Test 165 Proofs in Mathematics 166 Problem Solving 167 chapter Functions and Their Graphs 169 2.1 Linear Equations in Two Variables 170 2.2 Functions 185 2.3 Analyzing Graphs of Functions 200 2.4 A Library of Parent Functions 212 2.5 Transformations of Functions 219 2.6 Combinations of Functions: Composite Functions 229 2.7 Inverse Functions 238 Chapter Summary 248 Review Exercises 250 Chapter Test 253 Cumulative Test for Chapters P–2 254 Proofs in Mathematics 256 Problem Solving 257 iii iv Contents chapter Polynomial Functions 259 3.1 Quadratic Functions and Models 260 3.2 Polynomial Functions of Higher Degree 270 3.3 Polynomial and Synthetic Division 284 3.4 Zeros of Polynomial Functions 293 3.5 Mathematical Modeling and Variation 308 Chapter Summary 320 Review Exercises 322 Chapter Test 326 Proofs in Mathematics 327 Problem Solving 329 chapter Rational Functions and Conics 331 4.1 Rational Functions and Asymptotes 332 4.2 Graphs of Rational Functions 340 4.3 Conics 349 4.4 Translations of Conics 362 Chapter Summary 370 Review Exercises 372 Chapter Test 375 Proofs in Mathematics 376 Problem Solving 377 chapter Exponential and Logarithmic Functions 379 5.1 Exponential Functions and Their Graphs 380 5.2 Logarithmic Functions and Their Graphs 391 5.3 Properties of Logarithms 401 5.4 Exponential and Logarithmic Equations 408 5.5 Exponential and Logarithmic Models 419 Chapter Summary 432 Review Exercises 434 Chapter Test 437 Cumulative Test for Chapters 3–5 438 Proofs in Mathematics 440 Problem Solving 441 chapter Trigonometry 6.1 6.2 6.3 6.4 6.5 6.6 443 Angles and Their Measure 444 Right Triangle Trigonometry 456 Trigonometric Functions of Any Angle 467 Graphs of Sine and Cosine Functions 479 Graphs of Other Trigonometric Functions 490 Inverse Trigonometric Functions 501 Contents 6.7 Applications and Models 511 Chapter Summary 522 Chapter Test 527 Problem Solving 529 chapter Analytic Trigonometry Review Exercises 524 Proofs in Mathematics 528 531 7.1 Using Fundamental Identities 532 7.2 Verifying Trigonometric Identities 540 7.3 Solving Trigonometric Equations 547 7.4 Sum and Difference Formulas 558 7.5 Multiple-Angle and Product-to-Sum Formulas 565 Chapter Summary 576 Review Exercises 578 Chapter Test 581 Proofs in Mathematics 582 Problem Solving 585 chapter Additional Topics in Trigonometry 587 8.1 Law of Sines 588 8.2 Law of Cosines 597 8.3 Vectors in the Plane 605 8.4 Vectors and Dot Products 618 8.5 Trigonometric Form of a Complex Number 628 Chapter Summary 638 Review Exercises 640 Chapter Test 644 Cumulative Test for Chapters 6–8 645 Proofs in Mathematics 647 Problem Solving 651 chapter Systems of Equations and Inequalities 653 9.1 Linear and Nonlinear Systems of Equations 654 9.2 Two-Variable Linear Systems 665 9.3 Multivariable Linear Systems 677 9.4 Partial Fractions 690 9.5 Systems of Inequalities 698 9.6 Linear Programming 709 Chapter Summary 718 Review Exercises 720 Chapter Test 725 Proofs in Mathematics 726 Problem Solving 727 v vi Contents chapter 10 Matrices and Determinants 729 10.1 Matrices and Systems of Equations 730 10.2 Operations with Matrices 744 10.3 The Inverse of a Square Matrix 759 10.4 The Determinant of a Square Matrix 768 10.5 Applications of Matrices and Determinants 776 Chapter Summary 788 Review Exercises 790 Chapter Test 795 Proofs in Mathematics 796 Problem Solving 797 chapter 11 Sequences, Series, and Probability 799 11.1 Sequences and Series 800 11.2 Arithmetic Sequences and Partial Sums 811 11.3 Geometric Sequences and Series 821 11.4 Mathematical Induction 831 11.5 The Binomial Theorem 841 11.6 Counting Principles 849 11.7 Probability 859 Chapter Summary 872 Review Exercises 874 Chapter Test 877 Cumulative Test for Chapters 9–11 878 Proofs in Mathematics 880 Problem Solving 883 Appendix A Errors and the Algebra of Calculus Answers to Odd-Numbered Exercises and Tests Index A123 Index of Applications (web) Appendix B Concepts in Statistics (web) B.1 B.2 B.3 Representing Data Measures of Central Tendency and Dispersion Least Squares Regression A9 A1 A Word from the Author Welcome to the Eighth Edition of Algebra and Trigonometry! We are proud to offer you a new and revised version of our textbook With this edition, we have listened to you, our users, and have incorporated many of your suggestions for improvement 8th 4th 7th 3rd 6th 2nd 5th 1st In the Eighth Edition, we continue to offer instructors and students a text that is pedagogically sound, mathematically precise, and still comprehensible There are many changes in the mathematics, art, and design; the more significant changes are noted here • New Chapter Openers Each Chapter Opener has three parts, In Mathematics, In Real Life, and In Careers In Mathematics describes an important mathematical topic taught in the chapter In Real Life tells students where they will encounter this topic in real-life situations In Careers relates application exercises to a variety of careers • New Study Tips and Warning/Cautions Insightful information is given to students in two new features The Study Tip provides students with useful information or suggestions for learning the topic The Warning/Caution points out common mathematical errors made by students • New Algebra Helps Algebra Help directs students to sections of the textbook where they can review algebra skills needed to master the current topic • New Side-by-Side Examples Throughout the text, we present solutions to many examples from multiple perspectives—algebraically, graphically, and numerically The side-by-side format of this pedagogical feature helps students to see that a problem can be solved in more than one way and to see that different methods yield the same result The side-by-side format also addresses many different learning styles vii viii A Word from the Author • New Capstone Exercises Capstones are conceptual problems that synthesize key topics and provide students with a better understanding of each section’s concepts Capstone exercises are excellent for classroom discussion or test prep, and teachers may find value in integrating these problems into their reviews of the section • New Chapter Summaries The Chapter Summary now includes an explanation and/or example of each objective taught in the chapter • Revised Exercise Sets The exercise sets have been carefully and extensively examined to ensure they are rigorous and cover all topics suggested by our users Many new skill-building and challenging exercises have been added For the past several years, we’ve maintained an independent website— CalcChat.com—that provides free solutions to all odd-numbered exercises in the text Thousands of students using our textbooks have visited the site for practice and help with their homework For the Eighth Edition, we were able to use information from CalcChat.com, including which solutions students accessed most often, to help guide the revision of the exercises I hope you enjoy the Eighth Edition of Algebra and Trigonometry As always, I welcome comments and suggestions for continued improvements Acknowledgments I would like to thank the many people who have helped me prepare the text and the supplements package Their encouragement, criticisms, and suggestions have been invaluable Thank you to all of the instructors who took the time to review the changes in this edition and to provide suggestions for improving it Without your help, this book would not be possible Reviewers Chad Pierson, University of Minnesota-Duluth; Sally Shao, Cleveland State University; Ed Stumpf, Central Carolina Community College; Fuzhen Zhang, Nova Southeastern University; Dennis Shepherd, University of Colorado, Denver; Rhonda Kilgo, Jacksonville State University; C Altay Özgener, Manatee Community College Bradenton; William Forrest, Baton Rouge Community College; Tracy Cook, University of Tennessee Knoxville; Charles Hale, California State Poly University Pomona; Samuel Evers, University of Alabama; Seongchun Kwon, University of Toledo; Dr Arun K Agarwal, Grambling State University; Hyounkyun Oh, Savannah State University; Michael J McConnell, Clarion University; Martha Chalhoub, Collin County Community College; Angela Lee Everett, Chattanooga State Tech Community College; Heather Van Dyke, Walla Walla Community College; Gregory Buthusiem, Burlington County Community College; Ward Shaffer, College of Coastal Georgia; Carmen Thomas, Chatham University; Emily J Keaton My thanks to David Falvo, The Behrend College, The Pennsylvania State University, for his contributions to this project My thanks also to Robert Hostetler, The Behrend College, The Pennsylvania State University, and Bruce Edwards, University of Florida, for their significant contributions to previous editions of this text I would also like to thank the staff at Larson Texts, Inc who assisted with proofreading the manuscript, preparing and proofreading the art package, and checking and typesetting the supplements On a personal level, I am grateful to my spouse, Deanna Gilbert Larson, for her love, patience, and support Also, a special thanks goes to R Scott O’Neil If you have suggestions for improving this text, please feel free to write to me Over the past two decades I have received many useful comments from both instructors and students, and I value these comments very highly Ron Larson ix A126 Index systems, 667, 678 operations that produce, 678 Errors involving dividing out, A2 involving exponents, A2 involving fractions, A1 involving parentheses, A1 involving radicals, A2 Euler’s Formula, 635 Evaluate an algebraic expression, Evaluating trigonometric functions of any angle, 470 Even function, 206 trigonometric, 474 Even/odd identities, 532 Event(s), 859 certain, 860 complement of, 866 probability of, 866 impossible, 860 independent, 865 probability of, 865 mutually exclusive, 863 probability of, 860 the union of two, 863 Existence theorems, 293 Expanding a binomial, 844 by cofactors, 771 logarithmic expressions, 403 Expected value, 884 Experiment, 859 outcomes of, 859 sample space of, 859 Exponent(s), 15 errors involving, A2 negative, writing with, A3 properties of, 15 rational, 23 Exponential decay model, 419 Exponential equations, solving, 408 Exponential form, 15 Exponential function, 380 f with base a, 380 graph of, 381 natural, 384 one-to-one property, 382 Exponential growth model, 419 Exponential notation, 15 Exponentiating, 411 Expression algebraic, fractional, 45 rational, 45 Extended Principle of Mathematical Induction, 833 Extracting square roots, 108 Extraneous solution, 90, 131 Extrapolation, linear, 178 F Factor(s) damping, 495 of an integer, 11 of a polynomial, 274, 297, 328 prime, 298 quadratic, 298 repeated linear, 692 quadratic, 694 scaling, 481 and terms, inserting, A4 Factor Theorem, 288, 327 Factorial, 802 Factoring, 37 completely, 37 by grouping, 41 polynomials, guidelines for, 41 solving a quadratic equation by, 107 special polynomial forms, 38 unusual, A3 Family of functions, 220 Far point, 377 Feasible solutions, 709, 710 Fibonacci sequence, 802 Finding a formula for the nth term of a sequence, 836 intercepts algebraically, 91 an inverse function, 242 an inverse matrix, 761 nth roots of a complex number, 634 test intervals for a polynomial, 150 vertical and horizontal asymptotes of a rational function, 334 Finite sequence, 800 Finite series, 805 First differences, 838 Fixed cost, 176 Fixed point, 555 Focus (foci) of an ellipse, 352 of a hyperbola, 354 of a parabola, 350 FOIL Method, 29 Formula(s), 101 change-of-base, 401 common, 101 for area, perimeter, circumference, and volume, 101 for compound interest, 101, 386 Distance, 57 for distance traveled, 101 double-angle, 565, 583 Euler’s, 635 half-angle, 568 Heron’s Area, 600, 649 Midpoint, 58, 72 for the nth term of a sequence, 836 power-reducing, 567, 583 product-to-sum, 569 Quadratic, 111 radian measure, 449 recursion, 813 reduction, 560 for simple interest, 101 sum and difference, 558, 582 sum-to-product, 570, 584 for temperature, 101 Four ways to represent a function, 186 Fractal, 168, 883 Fraction(s) addition of with like denominators, 11 with unlike denominators, 11 complex, 49 division of, 11 equivalent, 11 generate, 11 errors involving, A1 multiplication of, 11 operations of, 11 partial, 690 decomposition, 690 properties of, 11 rules of signs for, 11 subtraction of with like denominators, 11 with unlike denominators, 11 writing as a sum, A3 Fractional expression, 45 Fractional parts of degrees minute, 446 second, 446 Frequency, 514 Function(s), 185, 193 algebraic, 380 arccosine, 503 arcsine, 501, 503 arctangent, 503 arithmetic combinations of, 229 characteristics of, 185 combinations of, 229 common logarithmic, 392 composition of, 231 constant, 186, 203, 213 continuous, 270 cosecant, 456, 467 cosine, 456, 467 cotangent, 456, 467 cubic, 214 decreasing, 203 defined, 193 difference of, 229 domain of, 185, 193 even, 206 exponential, 380 Index family of, 220 four ways to represent, 186 graph of, 200 greatest integer, 215 of half-angles, 565 Heaviside, 258 identity, 213 implied domain of, 190, 193 increasing, 203 inverse, 238, 239 cosine, 503 finding, 242 sine, 501, 503 tangent, 503 trigonometric, 503 linear, 212 logarithmic, 391 of multiple angles, 565 name of, 187, 193 natural exponential, 384 natural logarithmic, 395 notation, 187, 193 objective, 709 odd, 206 one-to-one, 241 parent, 216 period of, 474 periodic, 474 piecewise-defined, 188 polynomial, 260 power, 271 product of, 229 quadratic, 260 quotient of, 229 range of, 185, 193 rational, 332 reciprocal, 214 representation, 186 secant, 456, 467 sine, 456, 467 square root, 214 squaring, 213 step, 215 sum of, 229 summary of terminology, 193 tangent, 456, 467 transcendental, 380 transformations of, 219 nonrigid, 223 rigid, 223 trigonometric, 456, 467 undefined, 193 value of, 187, 193 Vertical Line Test, 201 zeros of, 202 Fundamental Counting Principle, 850 Fundamental Theorem of Algebra, 293 of Arithmetic, 11 Fundamental trigonometric identities, 459, 532 G Gaussian elimination, 678, 679 with back-substitution, 736 Gaussian model, 419 Gauss-Jordan elimination, 737 General form of the equation of a line, 178 of a quadratic equation, 111 Generalizations about nth roots of real numbers, 20 Generate equivalent fractions, 11 Generating equivalent equations, 88 Geometric sequence, 821 common ratio of, 821 nth term of, 822 sum of a finite, 824, 881 Geometric series, 825 sum of an infinite, 825 Graph, 76 of cosecant function, 493, 496 of cosine function, 483, 496 of cotangent function, 492, 496 of an equation, 76 of an exponential function, 381 of a function, 200 of an inequality, 140, 698 in two variables, 698 intercepts of, 79 of inverse cosine function, 503 of an inverse function, 240 of inverse sine function, 503 of inverse tangent function, 503 of a line, 170 of a logarithmic function, 393 point-plotting method, 76 of a polynomial function, x-intercept of, 274 of a rational function, 340 guidelines for analyzing, 340 reflecting, 221 of secant function, 493, 496 shifting, 219 of sine function, 483, 496 symmetry of a, 80 of tangent function, 490, 496 Graphical interpretations of solutions, 668 Graphical method, for solving a system of equations, 658 Graphical tests for symmetry, 80 Greatest integer function, 215 Guidelines for analyzing graphs of rational functions, 340 for factoring polynomials, 41 for solving the basic equation of a partial fraction decomposition, 695 A127 for verifying trigonometric identities, 540 H Half-angle formulas, 568 Half-angles, functions of, 565 Half-life, 387 Harmonic motion, simple, 514, 515 Heaviside function, 258 Heron’s Area Formula, 600, 649 Hidden equality, 96 Hole, in the graph of a rational function, 342 Hooke’s Law, 317 Horizontal asymptote, 333 of a rational function, 334 Horizontal component of v, 610 Horizontal line, 178 Horizontal Line Test, 241 Horizontal shifts, 219 of conics, 362 Horizontal shrink, 223 of a trigonometric function, 482 Horizontal stretch, 223 of a trigonometric function, 482 Horizontal translation of a trigonometric function, 483 Human memory model, 397 Hyperbola, 333, 354 asymptotes of, 356 branches of, 354 center of, 354 conjugate axis of, 355 foci of, 354 standard form of the equation of, 354, 362 transverse axis of, 354 vertices of, 354 Hypotenuse of a right triangle, 456 Hypothesis, 166, 836 I i, imaginary unit, 122 Idempotent square matrix, 797 Identities cofunction, 532 even/odd, 532 Pythagorean, 459, 532 quotient, 459, 532 reciprocal, 459, 532 trigonometric fundamental, 459, 532 guidelines for verifying, 540 Identity, 87, 540 function, 213 matrix of order n ϫ n, 751 If-then form, 166 Imaginary axis of the complex plane, 628 Imaginary number, 122 pure, 122 Imaginary part of a complex number, 122 A128 Index Imaginary unit i, 122 Implied domain, 190, 193 Impossible event, 860 Improper rational expression, 285 Inclusive or, 11 Inconsistent system of linear equations, 668, 736 Increasing annuity, 826 Increasing function, 203 Independent events, 865 probability of, 865 Independent system of linear equations, 668 Independent variable, 187, 193 Index of a radical, 19 of summation, 804 Indirect proof, 726 Induction, mathematical, 831 Inductive, 771 Inequality (inequalities), absolute value, 144 solution of, 144 double, 4, 143 equivalent, 141 graph of, 140, 698 linear, 142, 699 polynomial, 150 properties of, 141 rational, 154 satisfy, 140 solution of, 140, 698 solution set of, 140 solution of a system of, 700 solution set, 700 solving, 140 symbol, Infinite geometric series, 825 sum of, 825 Infinite sequence, 800 Infinite series, 805 Infinite wedge, 702 Infinity negative, positive, Initial point, 605 Initial side of an angle, 444 Inserting factors and terms, A4 Integer(s), divisors of, 11 factors of, 11 irreducible over, 37 sums of powers of, 837 Intercept form of the equation of a line, 180 Intercepts, 79 finding algebraically, 91 Interest compound, 101 formulas for, 386 compounded n times per year, 385 continuously compounded, 385 Intermediate Value Theorem, 277 Interpolation, linear, 178 Intersection, points of, 658 Interval(s), bounded, endpoints of, using inequalities to represent, on the real number line, unbounded, Inverse, 166 additive, multiplicative, of a matrix, 759 Inverse function, 238, 239 cosine, 503 finding, 242 graph of, 240 Horizontal Line Test, 241 sine, 501, 503 tangent, 503 Inverse of a matrix, 759 finding, 761 Inverse properties of logarithms, 392 of natural logarithms, 396 of trigonometric functions, 505 Inverse trigonometric functions, 503 Inverse variation, 312 Inversely proportional, 312 Invertible matrix, 760 Irrational number, Irreducible over the integers, 37 over the rationals, 298 over the reals, 298 J Joint variation, 313 Jointly proportional, 313 K Key numbers of a polynomial inequality, 150 of a rational inequality, 154 Key points of the graph of a trigonometric function, 480 intercepts, 480 maximum points, 480 minimum points, 480 L Latus rectum of an ellipse, 360 Law of Cosines, 597, 648 alternative form, 597, 648 standard form, 597, 648 Law of Sines, 588, 647 Law of Tangents, 647 Law of Trichotomy, Leading coefficient of a polynomial, 28 Leading Coefficient Test, 272 Leading 1, 734 Least squares regression line, 309, 675 parabola, 688 Length of a circular arc, 449 of a directed line segment, 605 of a vector, 606 Lift, 652 Like radicals, 22 Like terms of a polynomial, 29 Limit of summation lower, 804 upper, 804 Line(s) in the plane general form of the equation of, 178 graph of, 170 horizontal, 178 intercept form of the equation of, 180 least squares regression, 309, 675 parallel, 175 perpendicular, 175 point-slope form of the equation of, 174, 178 secant, 205 segment, directed, 605 slope of, 170, 172 slope-intercept form of the equation of, 170, 178 summary of equations, 178 tangent, 378 two-point form of the equation of, 174, 178, 781 vertical, 171, 178 Linear combination of vectors, 610 Linear depreciation, 177 Linear equation, 78 general form, 178 graph of, 170 intercept form, 180 in one variable, 87 point-slope form, 174, 178 slope-intercept form, 170, 178 summary of, 178 in two variables, 170 two-point form, 174, 178, 781 Linear extrapolation, 178 Linear factor, repeated, 692 Linear Factorization Theorem, 293, 328 Linear function, 212 Linear inequality, 142, 699 Linear interpolation, 178 Linear programming, 709 problem optimal solution, 709 solving, 710 Index Linear speed, 449 Linear system consistent, 668 dependent, 668 inconsistent, 668, 736 independent, 668 nonsquare, 682 number of solutions, 680 row operations, 678 row-echelon form, 677 square, 682 Literal equation, 101 Local maximum, 204 Local minimum, 204 Locus, 349 Logarithm(s) change-of-base formula, 401 natural, properties of, 396, 402, 440 inverse, 396 one-to-one, 396 power, 402, 440 product, 402, 440 quotient, 402, 440 properties of, 392, 402, 440 inverse, 392 one-to-one, 392 power, 402, 440 product, 402, 440 quotient, 402, 440 Logarithmic equations, solving, 408 Logarithmic expressions condensing, 403 expanding, 403 Logarithmic function, 391 with base a, 391 common, 392 graph of, 393 natural, 395 Logarithmic model, 419 Logistic curve, 424 growth model, 317, 419 Long division of polynomials, 284 Lower bound, 301 Lower limit of summation, 804 Lower triangular matrix, 775 M Magnitude, of a directed line segment, 605 of a vector, 606 Main diagonal entries of a square matrix, 730 Major axis of an ellipse, 352 Mandelbrot Set, 168 Marginal cost, 176 Mathematical induction, 831 Extended Principle of, 833 Principle of, 832 Mathematical model, 96 Mathematical modeling, 96 Matrix (matrices), 730 addition, 745 properties of, 747 additive identity, 748 adjoining, 761 augmented, 731 coded row, 782 coefficient, 731, 752 cofactor of, 770 column, 730 constant, 752 determinant of, 763, 768, 771 diagonal, 758, 775 elementary row operations, 732 entry of a, 730 equal, 744 idempotent, 797 identity, 751 inverse of, 759 finding, 761 invertible, 760 lower triangular, 775 main diagonal entries of a, 730 minor of, 770 multiplication, 749 properties of, 751 nonsingular, 760 order of a, 730 reduced row-echelon form, 734 representation of, 744 row, 730 row-echelon form, 734 row-equivalent, 732 scalar identity, 747 scalar multiplication, 745 properties of, 747 singular, 760 square, 730 stochastic, 757 subtraction, 746 transpose of, 798 triangular, 775 uncoded row, 782 upper triangular, 775 zero, 748 Maximum local, 204 relative, 204 value of a quadratic function, 265 Mean, arithmetic, 810 Measure of an angle, 445 degree, 445 radian, 447 Method of elimination, 665, 666 of substitution, 654 A129 Midpoint Formula, 58, 72 Midpoint of a line segment, 58 Minimum local, 204 relative, 204 value of a quadratic function, 265 Minor axis of an ellipse, 352 Minor of a matrix, 770 Minors and cofactors of a square matrix, 770 Minute, fractional part of a degree, 446 Miscellaneous common formulas, 101 Model mathematical, 96 verbal, 96 Modulus of a complex number, 629 Monomial, 28 Multiple angles, functions of, 565 Multiplication of complex numbers, 630 of fractions, 11 of matrices, 749 properties of, 751 scalar of matrices, 745 of vectors, 607 Multiplicative identity of a real number, Multiplicative inverse, of a matrix, 759 of a real number, Multiplicity, 274 Multiplier effect, 829 Mutually exclusive events, 863 N n factorial, 802 Name of a function, 187, 193 Natural base, 384 Natural exponential function, 384 Natural logarithm properties of, 396, 402, 440 inverse, 396 one-to-one, 396 power, 402, 440 product, 402, 440 quotient, 402, 440 Natural logarithmic function, 395 Natural numbers, Near point, 377 Negation, 166 properties of, 10 Negative angle, 444 exponents, writing with, A3 infinity, number, principal square root of, 126 of a vector, 607 Newton’s Law of Cooling, 317, 430 Newton’s Law of Universal Gravitation, 317 A130 Index Nonnegative number, Nonrigid transformations, 223 Nonsingular matrix, 760 Nonsquare system of linear equations, 682 Normally distributed, 423 Notation double subscript, 730 exponential, 15 function, 187, 193 scientific, 17 sigma, 804 summation, 804 nth partial sum, 805, 815 of an arithmetic sequence, 815 nth root(s) of a, 19 of a complex number, 633, 634 generalizations about, 20 principal, 19 of unity, 635 nth term of an arithmetic sequence, 812 of a geometric sequence, 822 of a sequence, finding a formula for, 836 Number(s) complex, 122 composite, 11 imaginary, 122 pure, 122 irrational, key, 150, 154 natural, negative, principal square root of, 126 nonnegative, of outcomes, 860 prime, 11 rational, real, whole, Number of permutations of n elements, 851 taken r at a time, 852 Number of solutions of a linear system, 680 Numerator, rationalizing, 23 O Objective function, 709 Oblique asymptote, 343 Oblique triangle, 588 area of, 592 Obtuse angle, 445 Odd/even identities, 532 Odd function, 206 trigonometric, 474 One cycle of a sine curve, 479 One-to-one correspondence, One-to-one function, 241 One-to-one property of exponential functions, 382 of logarithms, 392 of natural logarithms, 396 Operations of fractions, 11 that produce equivalent systems, 678 Opposite side of a right triangle, 456 Optimal solution of a linear programming problem, 709 Optimization, 709 Order of a matrix, 730 on the real number line, Ordered pair, 55 Ordered triple, 677 Origin, 3, 55 of the real number line, of the rectangular coordinate system, 55 symmetric with respect to, 80 Orthogonal vectors, 620 Outcomes, 859 number of, 860 P Parabola, 260, 350 axis of, 261, 350 directrix of, 350 focus of, 350 least squares regression, 688 standard form of the equation of, 350, 362, 376 vertex of, 261, 350 Parallel lines, 175 Parallelogram law for vector addition, 607 Parent functions, 216 Parentheses, errors involving, A1 Partial fraction, 690 decomposition, 690 Partial sum, nth, 805, 815 Pascal’s Triangle, 843 Perfect cube, 20 square, 20 square trinomial, 38, 39 Perihelion, 369 Perimeter, common formulas for, 101 Period of a function, 474 of sine and cosine functions, 482 Periodic function, 474 Permutation(s), 851 distinguishable, 853 of n elements, 851 taken r at a time, 852 Perpendicular lines, 175 vectors, 620 Phase shift, 483 Piecewise-defined function, 188 Plotting, on the real number line, Point(s) break-even, 659 collinear, 65, 780 test for, 780 of diminishing returns, 282 equilibrium, 672, 703 fixed, 555 initial, 605 of intersection, 658 solution, 76 terminal, 605 Point-plotting method, 76 Point-slope form of the equation of a line, 174, 178 Polar form of a complex number, 629 Polygon, diagonal of, 858 Polynomial(s), 28 coefficient of, 28 completely factored, 37 constant term, 28 degree of, 28 equation, 129 second-degree, 107 solution of, 274 factoring special forms, 38 factors of, 274, 297, 328 finding test intervals for, 150 guidelines for factoring, 41 inequality, 150 irreducible, 37 leading coefficient of, 28 like terms, 29 long division of, 284 operations with, 29 prime, 37 prime quadratic factor, 298 standard form of, 28 synthetic division, 287 Polynomial function, 260 Leading Coefficient Test, 272 real zeros of, 274 standard form, 275 of x with degree n, 260 x-intercept of the graph of, 274 zeros of, 273 Position equation, 114, 683 Positive angle, 444 infinity, Power, 15 of a complex number, 632 Power function, 271 Power property of logarithms, 402, 440 of natural logarithms, 402, 440 Power-reducing formulas, 567, 583 Powers of integers, sums of, 837 Index Prime factor of a polynomial, 298 factorization, 11 number, 11 polynomial, 37 quadratic factor, 298 Principal nth root of a, 19 of a number, 19 Principal square root of a negative number, 126 Principle of Mathematical Induction, 832 Extended, 833 Probability of a complement, 866 of an event, 860 of independent events, 865 of the union of two events, 863 Producer surplus, 703 Product of functions, 229 of trigonometric functions, 565 of two complex numbers, 630 Product property of logarithms, 402, 440 of natural logarithms, 402, 440 Product-to-sum formulas, 569 Projection of a vector, 622 Proof, 72 by contradiction, 726 indirect, 726 without words, 796 Proper rational expression, 285 Properties of absolute value, of the dot product, 618, 650 of equality, 10 of exponents, 15 of fractions, 11 of inequalities, 141 inverse, of trigonometric functions, 505 of logarithms, 392, 402, 440 inverse, 392 one-to-one, 392 power, 402, 440 product, 402, 440 quotient, 402, 440 of matrix addition and scalar multiplication, 747 of matrix multiplication, 751 of natural logarithms, 396, 402, 440 inverse, 396 one-to-one, 396 power, 402, 440 product, 402, 440 quotient, 402, 440 of negation, 10 one-to-one, exponential functions, 382 of radicals, 20 of sums, 805, 880 of vector addition and scalar multiplication, 609 of zero, 11 Proportional directly, 310 to the nth power, 311 inversely, 312 jointly, 313 Proportionality, constant of, 310 Pure imaginary number, 122 Pythagorean identities, 459, 532 Pythagorean Theorem, 57, 116, 528 Q Quadrants, 55 Quadratic equation, 78, 107 complex solutions of, 126 discriminant, 112 general form of, 111 solutions of, 112 solving by completing the square, 109 by extracting square roots, 108 by factoring, 107 using the Quadratic Formula, 111 Quadratic factor prime, 298 repeated, 694 Quadratic Formula, 111 Quadratic function, 260 maximum value, 265 minimum value, 265 standard form of, 263 Quadratic type equations, 130, 549 Quotient difference, 192, 49, 846 of functions, 229 of two complex numbers, 630 Quotient identities, 459, 532 Quotient property of logarithms, 402, 440 of natural logarithms, 402, 440 R Radian, 447 conversion to degrees, 448 Radian measure formula, 449 Radical(s) errors involving, A2 index of, 19 like, 22 properties of, 20 simplest form, 21 symbol, 19 Radicand, 19 Radius of a circle, 82 A131 Random selection with replacement, 849 without replacement, 849 Range of the cosine function, 474 of a function, 185, 193 of the sine function, 474 Rate, 176 Rate of change, 176 average, 205 Ratio, 176 Rational exponent, 23 Rational expression(s), 45 improper, 285 proper, 285 Rational function, 332 asymptotes of, 334 domain of, 332 graph of, guidelines for analyzing, 340 hole in the graph, 342 Rational inequality, 154 test intervals, 154 Rational number, Rational Zero Test, 294 Rationalizing a denominator, 21, 22, 542 a numerator, 23 Real axis of the complex plane, 628 Real number(s), absolute value of, classifying, division of, subset of, subtraction of, Real number line, bounded intervals on, distance between two points, interval on, order on, origin of, plotting on, unbounded intervals on, Real part of a complex number, 122 Real zeros of a polynomial function, 274 Reciprocal function, 214 Reciprocal identities, 459, 532 Rectangular coordinate system, 55 Recursion formula, 813 Recursive sequence, 802 Reduced row-echelon form of a matrix, 734 Reducible over the reals, 298 Reduction formulas, 560 Reference angle, 469 Reflection, 221 of a trigonometric function, 482 Regression, least squares line, 309, 675 parabola, 688 Relation, 185 A132 Index Relative maximum, 204 Relative minimum, 204 Remainder, uses in synthetic division, 289 Remainder Theorem, 288, 327 Repeated linear factor, 692 Repeated quadratic factor, 694 Repeated zero, 274 Representation of functions, 186 of matrices, 744 Resultant of vector addition, 607 Right triangle adjacent side of, 456 definitions of trigonometric functions, 456 hypotenuse of, 456 opposite side of, 456 solving, 461 Rigid transformations, 223 Root(s) of a complex number, 633, 634 cube, 19 nth, 19 principal nth, 19 square, 19 Row matrix, 730 coded, 782 uncoded, 782 Row operations, 678 elementary, 732 Row-echelon form, 677 of a matrix, 734 reduced, 734 Row-equivalent matrices, 732 Rules of signs for fractions, 11 S Sample space, 859 Satisfy the inequality, 140 Scalar, 607, 745 multiple, 745 Scalar Identity Property for matrices, 747 Scalar multiplication of matrices, 745 properties of, 747 of a vector, 607 properties of, 609 Scaling factor, 481 Scatter plot, 56 Scientific notation, 17 Scribner Log Rule, 663 Secant function, 456 of any angle, 467 graph of, 493, 496 Secant line, 205 Second differences, 838 Second, fractional part of a degree, 446 Second-degree polynomial equation, 107 Sector of a circle, 451 area of, 451 Sequence, 800 arithmetic, 811 Fibonacci, 802 finite, 800 first differences of, 838 geometric, 821 infinite, 800 nth partial sum of, 805 recursive, 802 second differences of, 838 terms of, 800 Series, 805 finite, 805 geometric, 825 infinite, 805 geometric, 825 Shifting graphs, 219 Shrink horizontal, 223 vertical, 223 Sierpinski Triangle, 883 Sigma notation, 804 Sigmoidal curve, 424 Simple harmonic motion, 514, 515 frequency, 514 Simple interest formula, 101 Simplest form, of an expression involving radicals, 21 Sine curve, 479 amplitude of, 481 one cycle of, 479 Sine function, 456 of any angle, 467 common angles, 470 domain of, 474 graph of, 483, 496 inverse, 501, 503 period of, 482 range of, 474 special angles, 458 Sines, cosines, and tangents of special angles, 458 Singular matrix, 760 Sketching the graph of an equation by point plotting, 76 of an inequality in two variables, 698 Slant asymptote, 343 Slope of a line, 170, 172 Slope-intercept form of the equation of a line, 170, 178 Solution(s), 76 of an absolute value inequality, 144 of an equation, 76, 87 extraneous, 90, 131 feasible, 709, 710 of an inequality, 140, 698 of a linear programming problem, optimal, 709 of a linear system, number of, 680 of a polynomial equation, 274 of a quadratic equation, 112 complex, 126 of a system of equations, 654 graphical interpretations, 668 of a system of inequalities, 700 solution set, 700 Solution point, 76 Solution set of an inequality, 140 of a system of inequalities, 700 Solving an absolute value inequality, 144 the basic equation of a partial fraction decomposition, 695 an equation, 87 exponential and logarithmic equations, 408 an inequality, 140 a linear programming problem, 710 a polynomial inequality, 151 a quadratic equation by completing the square, 109 by extracting square roots, 108 by factoring, 107 using the Quadratic Formula, 111 a rational inequality, 154 right triangles, 461 a system of equations, 654 Cramer’s Rule, 776, 777 Gaussian elimination, 678, 679 with back-substitution, 736 Gauss-Jordan elimination, 737 graphical method, 658 method of elimination, 665, 666 method of substitution, 654 a trigonometric equation, 547 Special angles cosines of, 458 sines of, 458 tangents of, 458 Special products, 30 Speed angular, 449 linear, 449 Square of a binomial, 30 of trigonometric functions, 565 Square matrix, 730 determinant of, 768, 771 diagonal, 775 idempotent, 797 lower triangular, 775 main diagonal entries of, 730 minors and cofactors of, 770 Index triangular, 775 upper triangular, 775 Square root(s), 19 extracting, 108 function, 214 of a negative number, 126 principal, of a negative number, 126 Square system of linear equations, 682 Squaring function, 213 Standard form of a complex number, 122 of the equation of a circle, 82, 362 of a conic, 362 of an ellipse, 352, 362 of a hyperbola, 354, 362 of a parabola, 350, 362, 376 of Law of Cosines, 597, 648 of a polynomial, 28 of a polynomial function, 275 of a quadratic function, 263 Standard position of an angle, 444 of a vector, 606 Standard unit vector, 610 Step function, 215 Stochastic matrix, 757 Straight-line depreciation, 177 Strategies for solving exponential and logarithmic equations, 408 Stretch horizontal, 223 vertical, 223 Subsets, Substitution, method of, 654 Substitution Principle, Subtraction of complex numbers, 123 of fractions with like denominators, 11 with unlike denominators, 11 of matrices, 746 of real numbers, Sum(s) of complex numbers, 123 of a finite arithmetic sequence, 814, 881 of a finite geometric sequence, 824, 881 of functions, 229 of an infinite geometric series, 825 nth partial, 805, 815 of powers of integers, 837 properties of, 805, 880 of square differences, 309 of two cubes, 38 of vectors, 607 Sum and difference formulas, 558, 582 Sum and difference of same terms, 30 Summary of equations of lines, 178 of function terminology, 193 Summation index of, 804 lower limit of, 804 notation, 804 upper limit of, 804 Sum-to-product formulas, 570, 584 Supplementary angles, 446 Surplus consumer, 703 producer, 703 Symbol “approximately equal to,” inequality, radical, 19 union, 144 Symmetry, 80 algebraic tests for, 80 axis of, of a parabola, 261 graphical tests for, 80 with respect to the origin, 80 with respect to the x-axis, 80 with respect to the y-axis, 80 Synthetic division, 287 uses of the remainder in, 289 System of equations, 654 equivalent, 667, 678 solution of, 654 solving, 654 with a unique solution, 764 System of inequalities, solution of, 700 solution set, 700 System of linear equations consistent, 668 dependent, 668 inconsistent, 668, 736 independent, 668 nonsquare, 682 number of solutions, 680 row operations, 678 row-echelon form, 677 square, 682 T Tangent function, 456 of any angle, 467 common angles, 470 graph of, 490, 496 inverse, 503 special angles, 458 Tangent line, 378 Temperature formula, 101 Term of an algebraic expression, constant, 8, 28 degree of, 28 A133 of a sequence, 800 variable, Terminal point, 605 Terminal side of an angle, 444 Terms, inserting factors and, A4 Test(s) for collinear points, 780 Horizontal Line, 241 Leading Coefficient, 272 Rational Zero, 294 for symmetry algebraic, 80 graphical, 80 Vertical Line, 201 Test intervals polynomial inequality, 150 rational inequality, 154 Theorem of Algebra, Fundamental, 293 of Arithmetic, Fundamental, 11 Binomial, 841, 882 DeMoivre’s, 632 Descartes’s Rule of Signs, 300 existence, 293 Factor, 288, 327 Intermediate Value, 277 Linear Factorization, 293, 328 Pythagorean, 57, 116, 528 Remainder, 288, 327 Thrust, 652 Transcendental function, 380 Transformations of functions, 219 nonrigid, 223 rigid, 223 Translating key words and phrases, 97 Translations of conics, 362 Transpose of a matrix, 798 Transverse axis of a hyperbola, 354 Triangle area of using a determinant, 779 Heron’s Area Formula, 600, 649 oblique, 588 area of, 592 Triangular matrix, 775 Trigonometric equations, solving, 547 Trigonometric form of a complex number, 629 argument of, 629 modulus of, 629 Trigonometric functions, 456, 467 of any angle, 467 evaluating, 470 cosecant, 456, 467 cosine, 456, 467 cotangent, 456, 467 even, 474 horizontal shrink of, 482 horizontal stretch of, 482 A134 Index horizontal translation of, 483 inverse, 503 inverse properties of, 505 key points, 480 intercepts, 480 maximum points, 480 minimum points, 480 odd, 474 product of, 565 reflection of, 482 right triangle definitions of, 456 secant, 456, 467 sine, 456, 467 square of, 565 tangent, 456, 467 vertical shrink of, 481 vertical stretch of, 481 vertical translation of, 484 Trigonometric identities cofunction, 532 even/odd, 532 fundamental, 459, 532 guidelines for verifying, 540 Pythagorean, 459, 532 quotient, 459, 532 reciprocal, 459, 532 Trigonometric values of common angles, 470 Trigonometry, 444 Trinomial, 28 perfect square, 38, 39 Two-point form of the equation of a line, 174, 178, 781 U Unbounded intervals, Uncoded row matrices, 782 Undefined, 193 Union symbol, 144 Union of two events, probability of, 863 Unit analysis, 98 Unit circle, 473 Unit vector, 606 in the direction of v, 609 standard, 610 Unity, nth roots of, 635 Unusual factoring, A3 Upper bound, 301 Upper limit of summation, 804 Upper and Lower Bound Rules, 301 Upper triangular matrix, 775 Uses of the remainder in synthetic division, 289 V Value of a function, 187, 193 Variable, dependent, 187, 193 independent, 187, 193 term, Variation combined, 312 constant of, 310 direct, 310 as an nth power, 311 inverse, 312 joint, 313 in sign, 300 Vary directly, 310 as nth power, 311 Vary inversely, 312 Vary jointly, 313 Vector(s) addition of, 607 properties of, 609 resultant of, 607 angle between two, 619, 650 component form of, 606 components of, 606, 621, 622 difference of, 607 directed line segment representation, 605 direction angle of, 611 dot product of, 618 properties of, 618, 650 equal, 606 horizontal component of, 610 length of, 606 linear combination of, 610 magnitude of, 606 negative of, 607 orthogonal, 620 parallelogram law, 607 perpendicular, 620 in the plane, 605 projection of, 622 resultant of, 607 scalar multiplication of, 607 properties of, 609 standard position of, 606 sum of, 607 unit, 606 in the direction of v, 609 standard, 610 v in the plane, 605 vertical component of, 610 zero, 606 Verbal model, 96 Vertex (vertices) of an angle, 444 of an ellipse, 352 of a hyperbola, 354 of a parabola, 261, 350 Vertical asymptote, 333 of a rational function, 334 Vertical component of v, 610 Vertical line, 171, 178 Vertical Line Test, 201 Vertical shifts, 219 of conics, 362 Vertical shrink, 223 of a trigonometric function, 481 Vertical stretch, 223 of a trigonometric function, 481 Vertical translation of a trigonometric function, 484 Volume, common formulas for, 101 W Wedge, infinite, 702 Whole numbers, With replacement, 849 Without replacement, 849 Work, 624 Writing a fraction as a sum, A3 with negative exponents, A3 X x-axis, 55 symmetric with respect to, 80 x-coordinate, 55 x-intercepts, 79 finding algebraically, 91 of the graph of a polynomial function, 274 Y y-axis, 55 symmetric with respect to, 80 y-coordinate, 55 y-intercepts, 79 finding algebraically, 91 Z Zero(s) of a function, 202 matrix, 748 multiplicity of, 274 of a polynomial function, 273, 274 bounds for, 301 real, 274 properties of, 11 repeated, 274 vector, 606 Zero polynomial, 28 Zero-Factor Property, 11 GRAPHS OF PARENT FUNCTIONS Linear Function Absolute Value Function x, x Ն f ͑x͒ ϭ ԽxԽ ϭ ΆϪx, f ͑x͒ ϭ mx ϩ b y Square Root Function f ͑x͒ ϭ Ίx x < y y f(x) = ⏐x⏐ x −2 (− mb , 0( (− mb , 0( f(x) = mx + b, m>0 (0, b) 2 −1 f(x) = mx + b, m0 x −1 −1 Domain: ͑Ϫ ϱ, ϱ͒ Range: ͑Ϫ ϱ, ϱ͒ x-intercept: ͑Ϫb͞m, 0͒ y-intercept: ͑0, b͒ Increasing when m > Decreasing when m < y x x (0, 0) −1 f(x) = f(x) = ax , a < (0, 0) −3 −2 −1 −2 −2 −3 −3 Domain: ͑Ϫ ϱ, ϱ͒ Range ͑a > 0͒: ͓0, ϱ͒ Range ͑a < 0͒ : ͑Ϫ ϱ, 0͔ Intercept: ͑0, 0͒ Decreasing on ͑Ϫ ϱ, 0͒ for a > Increasing on ͑0, ϱ͒ for a > Increasing on ͑Ϫ ϱ, 0͒ for a < Decreasing on ͑0, ϱ͒ for a < Even function y-axis symmetry Relative minimum ͑a > 0͒, relative maximum ͑a < 0͒, or vertex: ͑0, 0͒ x f(x) = x Domain: ͑Ϫ ϱ, ϱ͒ Range: ͑Ϫ ϱ, ϱ͒ Intercept: ͑0, 0͒ Increasing on ͑Ϫ ϱ, ϱ͒ Odd function Origin symmetry Rational (Reciprocal) Function f ͑x͒ ϭ x Exponential Function Logarithmic Function f ͑x͒ ϭ ax, a > f ͑x͒ ϭ loga x, a > y y y f(x) = x f(x) = a −x (0, 1) f(x) = a x x −1 f(x) = loga x (1, 0) x x −1 Domain: ͑Ϫ ϱ, 0͒ ʜ ͑0, ϱ) Range: ͑Ϫ ϱ, 0͒ ʜ ͑0, ϱ) No intercepts Decreasing on ͑Ϫ ϱ, 0͒ and ͑0, ϱ͒ Odd function Origin symmetry Vertical asymptote: y-axis Horizontal asymptote: x-axis Domain: ͑Ϫ ϱ, ϱ͒ Range: ͑0, ϱ͒ Intercept: ͑0, 1͒ Increasing on ͑Ϫ ϱ, ϱ͒ for f ͑x͒ ϭ ax Decreasing on ͑Ϫ ϱ, ϱ͒ for f ͑x͒ ϭ aϪx Horizontal asymptote: x-axis Continuous Domain: ͑0, ϱ͒ Range: ͑Ϫ ϱ, ϱ͒ Intercept: ͑1, 0͒ Increasing on ͑0, ϱ͒ Vertical asymptote: y-axis Continuous Reflection of graph of f ͑x͒ ϭ ax in the line y ϭ x Sine Function f ͑x͒ ϭ sin x Cosine Function f ͑x͒ ϭ cos x Tangent Function f ͑x͒ ϭ tan x y y y 3 f(x) = sin x 2 f(x) = cos x 1 x −π f(x) = tan x π π 2π x −π − π π −2 −2 −3 −3 Domain: ͑Ϫ ϱ, ϱ͒ Range: ͓Ϫ1, 1͔ Period: 2␲ x-intercepts: ͑n␲, 0͒ y-intercept: ͑0, 0͒ Odd function Origin symmetry π 2π Domain: ͑Ϫ ϱ, ϱ͒ Range: ͓Ϫ1, 1͔ Period: 2␲ ␲ x-intercepts: ϩ n␲ , y-intercept: ͑0, 1͒ Even function y-axis symmetry ΂ x π − π 3π ␲ ϩ n␲ Range: ͑Ϫ ϱ, ϱ͒ Period: ␲ x-intercepts: ͑n␲, 0͒ y-intercept: ͑0, 0͒ Vertical asymptotes: ␲ x ϭ ϩ n␲ Odd function Origin symmetry Domain: all x ΃ π Cosecant Function f ͑x͒ ϭ csc x Secant Function f ͑x͒ ϭ sec x f(x) = csc x = y sin x y Cotangent Function f ͑x͒ ϭ cot x f(x) = sec x = cos x f(x) = cot x = y 3 2 1 tan x x x −π π π 2π −π π − π π 3π 2π x −π π − π π 2π −2 −3 Domain: all x n␲ Range: ͑Ϫ ϱ, Ϫ1͔ ʜ ͓1, ϱ͒ Period: 2␲ No intercepts Vertical asymptotes: x ϭ n␲ Odd function Origin symmetry Domain: all x ␲ ϩ n␲ Range: ͑Ϫ ϱ, Ϫ1͔ ʜ ͓1, ϱ͒ Period: 2␲ y-intercept: ͑0, 1͒ Vertical asymptotes: ␲ x ϭ ϩ n␲ Even function y-axis symmetry Domain: all x n␲ Range: ͑Ϫ ϱ, ϱ͒ Period: ␲ ␲ ϩ n␲ , x-intercepts: Vertical asymptotes: x ϭ n␲ Odd function Origin symmetry Inverse Sine Function f ͑x͒ ϭ arcsin x Inverse Cosine Function f ͑x͒ ϭ arccos x Inverse Tangent Function f ͑x͒ ϭ arctan x y ΂ y π ΃ y π π f(x) = arccos x x −1 −2 x −1 f(x) = arcsin x −π Domain: ͓Ϫ1, 1͔ ␲ ␲ Range: Ϫ , 2 Intercept: ͑0, 0͒ Odd function Origin symmetry ΄ ΅ f(x) = arctan x −π x −1 Domain: ͓Ϫ1, 1͔ Range: ͓0, ␲͔ ␲ y-intercept: 0, ΂ ΃ Domain: ͑Ϫ ϱ, ϱ͒ ␲ ␲ Range: Ϫ , 2 Intercept: ͑0, 0͒ Horizontal asymptotes: ␲ yϭ± Odd function Origin symmetry ΂ ΃ y Definition of the Six Trigonometric Functions Right triangle definitions, where < ␪ < ␲͞2 e Opposite nus te ypo H θ Adjacent opp hyp adj cos ␪ ϭ hyp opp tan ␪ ϭ adj sin ␪ ϭ (− 12 , 23 ) π (− 22 , 22 ) 3π 23π 120° 5π 135° − , ( 2) 150° hyp opp hyp sec ␪ ϭ adj adj cot ␪ ϭ opp csc ␪ ϭ Circular function definitions, where ␪ is any angle y r y csc ␪ ϭ sin ␪ ϭ 2 r y r= x +y (x , y ) x r sec ␪ ϭ cos ␪ ϭ r r x θ y y x x cot ␪ ϭ tan ␪ ϭ x x y ( ( 12 , 23 ) 90° π , 22 ) π ( 60° 45° π ( , ) 2 30° (0, 1) 0° x 360° 2π (1, 0) 330°11π 315° , − 12 300° 74π (−1, 0) π 180° 7π 210° 225° − 2, −2 5π 240° ) (− , − 4π ) 2 − 12 , ( − 270° ) 3π 5π (0, −1) ( ) ( 22 , − 22 ) ( 12 , − 23 ) Double-Angle Formulas Reciprocal Identities csc u csc u ϭ sin u sin u ϭ sec u sec u ϭ cos u cos u ϭ cot u cot u ϭ tan u tan u ϭ sin u cos u cot u ϭ cos u sin u Pythagorean Identities sin2 u ϩ cos2 u ϭ 1 ϩ tan2 u ϭ sec2 u ΂␲2 Ϫ u΃ ϭ cos u ␲ cos΂ Ϫ u΃ ϭ sin u ␲ tan΂ Ϫ u΃ ϭ cot u ΂␲2 Ϫ u΃ ϭ tan u ␲ sec΂ Ϫ u΃ ϭ csc u ␲ csc΂ Ϫ u΃ ϭ sec u cot sin u ϩ sin v ϭ sin ΂u ϩ2 v΃ cos΂u Ϫ2 v΃ sin u Ϫ sin v ϭ cos ΂u ϩ2 v΃ sin΂u Ϫ2 v΃ cos u ϩ cos v ϭ cos ΂u ϩ2 v΃ cos΂u Ϫ2 v΃ cos u Ϫ cos v ϭ Ϫ2 sin ΂u ϩ2 v΃ sin΂u Ϫ2 v΃ Product-to-Sum Formulas Even/Odd Identities sin͑Ϫu͒ ϭ Ϫsin u cos͑Ϫu͒ ϭ cos u tan͑Ϫu͒ ϭ Ϫtan u Ϫ cos 2u ϩ cos 2u cos u ϭ Ϫ cos 2u tan u ϭ ϩ cos 2u Sum-to-Product Formulas ϩ cot2 u ϭ csc2 u Cofunction Identities sin Power-Reducing Formulas sin2 u ϭ Quotient Identities tan u ϭ sin 2u ϭ sin u cos u cos 2u ϭ cos2 u Ϫ sin2 u ϭ cos2 u Ϫ ϭ Ϫ sin2 u tan u tan 2u ϭ Ϫ tan2 u cot͑Ϫu͒ ϭ Ϫcot u sec͑Ϫu͒ ϭ sec u csc͑Ϫu͒ ϭ Ϫcsc u Sum and Difference Formulas sin͑u ± v͒ ϭ sin u cos v ± cos u sin v cos͑u ± v͒ ϭ cos u cos v ϯ sin u sin v tan u ± tan v tan͑u ± v͒ ϭ ϯ tan u tan v sin u sin v ϭ ͓cos͑u Ϫ v͒ Ϫ cos͑u ϩ v͔͒ cos u cos v ϭ ͓cos͑u Ϫ v͒ ϩ cos͑u ϩ v͔͒ sin u cos v ϭ ͓sin͑u ϩ v͒ ϩ sin͑u Ϫ v͔͒ cos u sin v ϭ ͓sin͑u ϩ v͒ Ϫ sin͑u Ϫ v͔͒ FORMULAS FROM GEOMETRY Triangle: c Sector of Circular Ring: a h h ϭ a sin ␪ θ b Area ϭ bh 2 c ϭ a ϩ b Ϫ 2ab cos ␪ (Law of Cosines) Area ϭ ␪pw p ϭ average radius, w ϭ width of ring, ␪ in radians Right Triangle: Ellipse: c Pythagorean Theorem c2 ϭ a2 ϩ b2 a Circumference Ϸ 2␲ Equilateral Triangle: hϭ Area ϭ Ί3s s Volume ϭ h b a Ίa ϩ b2 Ah h A ϭ area of base A s Parallelogram: Right Circular Cone: Area ϭ bh ␲ r 2h Lateral Surface Area ϭ ␲rΊr ϩ h h Trapezoid: h Area ϭ ͑a ϩ b͒ h Volume ϭ b a Frustum of Right Circular Cone: h ␲ ͑r ϩ rR ϩ R 2͒h Volume ϭ Lateral Surface Area ϭ ␲s͑R ϩ r͒ b a r r s h Circle: Right Circular Cylinder: ␲r Area ϭ Circumference ϭ 2␲ r r ␲r2h Volume ϭ Lateral Surface Area ϭ 2␲ rh r Sector of Circle: ␪r Area ϭ s ϭ r␪ ␪ in radians R h b h Sphere: Volume ϭ ␲ r 3 s θ r Surface Area ϭ 4␲ r r Wedge: Circular Ring: Area ϭ ␲ ͑ Ϫ ͒ ϭ 2␲ pw p ϭ average radius, w ϭ width of ring R2 w Cone: s θ Area ϭ ␲ ab b Ί3s p r2 r p R w A ϭ B sec ␪ A ϭ area of upper face, B ϭ area of base A θ B ALGEBRA Factors and Zeros of Polynomials: Given the polynomial p͑x͒ ϭ an x n ϩ anϪ1 x nϪ1 ϩ ϩ a x ϩ a If p͑b͒ ϭ 0, then b is a zero of the polynomial and a solution of the equation p͑x͒ ϭ Furthermore, ͑x Ϫ b͒ is a factor of the polynomial Fundamental Theorem of Algebra: An nth degree polynomial has n (not necessarily distinct) zeros Quadratic Formula: If p͑x͒ ϭ ax ϩ bx ϩ c, a ͑ ͒ x ϭ Ϫb ± Ίb2 Ϫ 4ac ͞2a and b Ϫ 4ac Ն 0, then the real zeros of p are Special Factors: Examples Ϫ ϭ ͑x Ϫ a͒͑x ϩ a͒ Ϫ a ϭ ͑x Ϫ a͒͑x ϩ ax ϩ a 2͒ x ϩ a ϭ ͑x ϩ a͒͑x Ϫ ax ϩ a 2͒ x Ϫ ϭ ͑x Ϫ 3͒͑x ϩ 3͒ x Ϫ ϭ ͑x Ϫ 2͒͑x ϩ 2x ϩ 4͒ x2 Ϫ Ί 4x ϩ Ί 16 x ϩ ϭ ͑x ϩ Ί ͒͑ ͒ x2 a2 x3 x Ϫ a ϭ ͑x Ϫ a͒͑x ϩ a͒͑x ϩ a 2͒ x ϩ a ϭ ͑x ϩ Ί2 ax ϩ a 2͒͑x Ϫ Ί2 ax ϩ a 2͒ x n Ϫ a n ϭ ͑x Ϫ a͒͑x nϪ1 ϩ axnϪ2 ϩ ϩ a nϪ1͒, for n odd x n ϩ a n ϭ ͑x ϩ a͒͑x nϪ1 Ϫ ax nϪ2 ϩ ϩ a nϪ1͒, for n odd x 2n Ϫ a 2n ϭ ͑x n Ϫ a n͒͑x n ϩ a n͒ x Ϫ ϭ ͑x Ϫ Ί2 ͒͑x ϩ Ί2 ͒͑x ϩ 2͒ x ϩ ϭ ͑x ϩ 2x ϩ 2͒͑x Ϫ 2x ϩ 2͒ x Ϫ ϭ ͑x Ϫ 1͒͑x ϩ x ϩ x ϩ x ϩ 1͒ x ϩ ϭ ͑x ϩ 1͒͑x Ϫ x ϩ x Ϫ x ϩ x Ϫ x ϩ 1͒ x Ϫ ϭ ͑x Ϫ 1͒͑x ϩ 1͒ Binomial Theorem: Examples ͑x ϩ 3͒2 ϭ x ϩ 6x ϩ ͑x Ϫ 5͒2 ϭ x Ϫ 10x ϩ 25 ͑x ϩ 2͒3 ϭ x ϩ 6x ϩ 12x ϩ ͑x Ϫ 1͒3 ϭ x Ϫ 3x ϩ 3x Ϫ ͑x ϩ Ί2 ͒4 ϭ x ϩ 4Ί2 x ϩ 12x ϩ 8Ί2 x ϩ ͑x Ϫ 4͒4 ϭ x Ϫ 16x ϩ 96x Ϫ 256x ϩ 256 ͑x ϩ a͒ ϭ ϩ 2ax ϩ ͑x Ϫ a͒2 ϭ x Ϫ 2ax ϩ a ͑x ϩ a͒3 ϭ x ϩ 3ax ϩ 3a 2x ϩ a ͑x Ϫ a͒3 ϭ x Ϫ 3ax ϩ 3a 2x Ϫ a ͑x ϩ a͒4 ϭ x ϩ 4ax ϩ 6a 2x ϩ 4a ϩ a ͑x Ϫ a͒4 ϭ x Ϫ 4ax ϩ 6a 2x Ϫ 4a 3x ϩ a n͑n Ϫ 1͒ nϪ2 a x ͑x ϩ a͒n ϭ xn ϩ naxnϪ1 ϩ ϩ ϩ nanϪ1x ϩ a n 2! n͑n Ϫ 1͒ nϪ2 a x ͑x Ϫ a͒n ϭ x n Ϫ nax nϪ1 ϩ Ϫ ± na nϪ1x ϯ a n 2! x2 a2 ͑x ϩ 1͒5 ϭ x ϩ 5x ϩ 10x ϩ 10x ϩ 5x ϩ ͑x Ϫ 1͒6 ϭ x Ϫ 6x5 ϩ 15x Ϫ 20x ϩ 15x Ϫ 6x ϩ Rational Zero Test: If p͑x͒ ϭ an x n ϩ anϪ1x nϪ1 ϩ ϩ a1 x ϩ a0 has integer coefficients, then every rational zero of p͑x͒ ϭ is of the form x ϭ r͞s, where r is a factor of a0 and s is a factor of an Exponents and Radicals: a0 ϭ 1, a aϪx ϭ ax a xa y ϭ a xϩy ax ϭ a xϪy ay ΂ab΃ ͑a x͒ y ϭ a xy Ίa ϭ a1͞2 n n Ί Ί ab ϭ Ί anb ͑ab͒ x ϭ a xb x n Ί a ϭ a1͞n Ί΂ab΃ ϭ x ϭ ax bx ͑ n n Ί am ϭ am͞n ϭ Ί a n ͒m n a Ί n Ί b Conversion Table: centimeter Ϸ meter Ϸ Ϸ kilometer Ϸ liter Ϸ newton Ϸ 0.394 inch 39.370 inches 3.281 feet 0.621 mile 0.264 gallon 0.225 pound joule Ϸ 0.738 foot-pound gram Ϸ 0.035 ounce kilogram Ϸ 2.205 pounds inch Ϸ 2.540 centimeters foot Ϸ 30.480 centimeters Ϸ 0.305 meter mile Ϸ 1.609 kilometers gallon Ϸ 3.785 liters pound Ϸ 4.448 newtons foot-lb Ϸ 1.356 joules ounce Ϸ 28.350 grams pound Ϸ 0.454 kilogram ... common mathematical errors made by students • New Algebra Helps Algebra Help directs students to sections of the textbook where they can review algebra skills needed to master the current topic... Prerequisites Algebraic Expressions One characteristic of algebra is the use of letters to represent numbers The letters are variables, and combinations of letters and numbers are algebraic expressions... are algebraic expressions Here are a few examples of algebraic expressions 5x, 2x Ϫ 3, x2 , ϩ2 7x ϩ y Definition of an Algebraic Expression An algebraic expression is a collection of letters (variables)