Aufmann college algebra trigonometry 7th txtbk

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Your success in college algebra and trigonometry is important to us To guide you to that success, we have created a textbook with features that promote learning and support various learning styles These features are highlighted below We encourage you to examine these features and use them to successfully complete this course Prepare for This Section These exercises test your understanding of prerequisite skills and concepts that were covered earlier in the text Mastery of these concepts is required for success in the following section Motivating Applications Large selections of contemporary applications from many different disciplines demonstrate the utility of mathematics Engaging Examples Examples are designed to capture your attention and help you master important concepts Annotated Examples Step-by-step solutions are provided for each example Try Exercises A reference to an exercise follows each worked example This exercise provides you the opportunity to test your understanding by working an exercise similar to the worked example www.elsolucionario.net Solutions to Try Exercises The complete solutions to the Try Exercises can be found in the Solutions to the Try Exercises appendix, starting page S1 Visualize the Solution When appropriate, both algebraic and graphical solutions are provided to help visualize the mathematics of the example and to create a link between the two Mid-Chapter Quizzes These quizzes will help you assess your understanding of the concepts studied earlier in the chapter They provide a mini-review of the chapter material Chapter Test Prep This is a summary of the major concepts discussed in the chapter and will help you prepare for the chapter test For each concept, there is a reference to a worked example illustrating how the concept is used and at least one exercise in the chapter review relating to that concept www.elsolucionario.net This page intentionally left blank www.elsolucionario.net COLLEGE ALGEBRA AND TRIGONOMETRY www.elsolucionario.net This page intentionally left blank www.elsolucionario.net COLLEGE ALGEBRA AND TRIGONOMETRY Chad Ehlers/Getty Images SEVENTH EDITION Richard N Aufmann Vernon C Barker Richard D Nation Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States www.elsolucionario.net College Algebra and Trigonometry, Seventh Edition Richard N Aufmann, Vernon C Barker, Richard D Nation Acquisitions Editor: Gary Whalen Senior Developmental Editor: Carolyn Crockett Assistant Editor: Stefanie Beeck © 2011, 2008 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means, graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher Editorial Assistant: Guanglei Zhang Associate Media Editor: Lynh Pham Marketing Manager: Myriah Fitzgibbon Marketing Assistant: Angela Kim Marketing Communications Manager: Katy Malatesta Content Project Manager: Jennifer Risden For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Further permissions questions can be e-mailed to permissionrequest@cengage.com Creative Director: Rob Hugel Library of Congress Control Number: 2009938510 Art Director: Vernon Boes ISBN-13: 978-1-4390-4860-3 Print Buyer: Karen Hunt ISBN-10: 1-4390-4860-6 Rights Acquisitions Account Manager, Text: Roberta Broyer Rights Acquisitions Account Manager, Image: Don Schlotman Production Service: Graphic World Inc Brooks/Cole 20 Davis Drive Belmont, CA 94002-3098 USA Text Designer: Diane Beasley Photo Researcher: PrepressPMG Copy Editor: Graphic World Inc Illustrators: Network Graphics; Macmillan Publishing Solutions Cover Designer: Lisa Henry Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan Locate your local office at www.cengage.com/global Cengage Learning products are represented in Canada by Nelson Education, Ltd Cover Image: Chad Ehlers/Getty Images Compositor: MPS Limited, A Macmillan Company To learn more about Brooks/Cole, visit www.cengage.com/brookscole Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in the United States of America 13 12 11 10 09 www.elsolucionario.net CONTENTS CHAPTER P Preliminary Concepts P.1 P.2 P.3 P.4 P.5 P.6 The Real Number System Integer and Rational Number Exponents 17 Polynomials 32 Mid-Chapter P Quiz 39 Factoring 40 Rational Expressions 49 Complex Numbers 59 Exploring Concepts with Technology 66 Chapter P Test Prep 67 Chapter P Review Exercises 70 Chapter P Test 73 CHAPTER Equations and Inequalities 75 1.1 Linear and Absolute Value Equations 76 1.2 Formulas and Applications 83 1.3 Quadratic Equations 96 Mid-Chapter Quiz 109 1.4 Other Types of Equations 110 1.5 Inequalities 123 1.6 Variation and Applications 136 Exploring Concepts with Technology 144 Chapter Test Prep 145 Chapter Review Exercises 148 Chapter Test 151 Cumulative Review Exercises 152 CHAPTER Functions and Graphs 153 2.1 Two-Dimensional Coordinate System and Graphs 154 2.2 Introduction to Functions 166 2.3 Linear Functions 186 Mid-Chapter Quiz 200 2.4 Quadratic Functions 200 2.5 Properties of Graphs 213 2.6 Algebra of Functions 227 2.7 Modeling Data Using Regression 237 Exploring Concepts with Technology 248 Chapter Test Prep 249 Chapter Review Exercises 253 Chapter Test 257 Cumulative Review Exercises 258 v www.elsolucionario.net I2 INDEX Commutative property, 12–13 Complementary angles, 429 Completing the square, 99–100, 163 Complex conjugates, 63 Complex fractions, 55–56 Complex numbers, 59–64 argument of, 617 fifth roots of, 624–625 modulus of, 617 polar form of, 617 power of, 623 product property of, 618–620 quotient property of, 618–620 in standard form, 618 trigonometric form of, 616–622 Complex plane, 616 Complex solutions, of quadratic equation, 103 Complex zeros, of polynomial, 299–305 Composite numbers, Composition of functions, 230–234 with inverse function, 336–338 logarithmic and exponential, 359 Compound inequalities, 125–126 Compound interest, 394–397, 866–867 Compressing graphs, 221–223 Computational complexity, 37 Computer algebra systems, 323–324 Concavity, 405–406 Conditional equations, 78 Conic(s) focus–directrix definitions of, 692 general equation of a nondegenerate, 670 identification theorem, 674 polar equations of, 691–695 rotation theorem for, 670–673 Conic sections degenerate, 634 description of, 633 ellipses See Ellipse hyperbolas See Hyperbola parabolas See Parabola Conjugate of complex number, 63 of radical expression, 28 Conjugate axis, 659 Conjugate Pair Theorem, 302–304 Consistent systems of equations, 718, 729, 732 Constant functions, 175 Constant matrices, 780 Constant of proportionality, 137 Constant polynomials, 33 Constant sequences, 852 Constant terms, 11, 33 Constraints, 762–764 Continuous curves, 175, 271 Contradictions, 78 Conversion factor, 432 Coordinate(s) description of, 7, 154–155 polar See Polar coordinate system rectangular, 154–155, 686–688 Coordinate axis real number line, symmetry with respect to, 213–214, 217 in the plane, 154 in three dimensions, 728 Correlation coefficient, 240–242, 407–409 cosϪ1 x, 550 Cosecant, 443 Cosecant function graph of, 486–487, 495 inverse, 552 period of, 467 Cosine definition of, 443, 446 double-angle identity for, 532 identity verification using, 517–519 See also Law of Cosines Cosine function absolute value of, 478 as cofunctions, 525 graph of, 476–479 harmonic motion See Harmonic motion period of, 467 Cost, 88, 132–133, 194, 199 Cotangent, 443 Cotangent function graph of, 484–486 inverse, 552 period of, 468 Coterminal angles, 430–431 Counting Principle, 883–884, 887 Counting problems, 887 Cramer’s Rule, 833–836 Critical value method, 127–130 Cube roots, 45, 624 Cubes, sum or difference of, 45 Cubic equations, 110, 294–295 Cubic regression model, 280–282 Curve, 697–698 Curve fitting, 736–737 See also Interpolating polynomials Cycloid, 701 Damped harmonic motion, 503–504 Damping factor, 496 De Moivre’s Theorem, 622–625, 705 Decibels (dB), 379 Decimal(s) definition of, repeating, 2, 866 in scientific notation, 20 Decimal degree method, 431 Decreasing functions, 175, 335 Degenerate conic section, 634 Degenerate form, 658 Degree of angles, 428–432 definition of, 428 fractional part of, 431–432 of a monomial, 32 of a polynomial, 32 radians conversions, 434–435 Demand–supply problems, 724–725 Denominator, 12 rationalizing, 27–28 www.elsolucionario.net Dependent systems of equations in three variables, 729, 732–734, 786 in two variables, 718, 720, 724 Dependent variable, 167 Depreciation, straight-line, 178 Depressed polynomials See Reduced polynomials Descartes, René, 60, 154 Descartes’ Rule of Signs, 290–292 Determinants, 824–830 of ϫ matrix, 824–825 conditions for zero determinant, 829 of matrix in triangular form, 828 product property of, 830 solving linear systems with, 833–836 Diagonal of a matrix, 780 Difference of cubes, 45 of real numbers, 11 of two functions, 496 See also Subtraction Difference identities, 523–524, 526–528 Difference quotient, 229–230 Direct variation, 137–138 Direction angle of vector, 603 Directrix of parabola, 636, 638–639 Discontinuities, 175 Discriminant, 102–103 Disjoint sets, Displacement, 501 Distance between points in plane, 155–156 between points on real number line, 7–8 of falling object, 138, 856 in uniform motion, 89 Distributive property, 12–13 Dividend, 261 Division of complex numbers, 63–64 of exponential expressions, 18 of functions, 227 of inequality by real number, 123–124 of polynomials, 260–264 of radical expressions, 25, 27–28 of rational expressions, 51–52 of real numbers, 12 Divisor, 261 DMS method, 431–432 Domain, 167–168, 170 composition of functions and, 231, 233 of inverse function, 334, 340 of logarithmic function, 363–364 operations on functions and, 227–228 of rational expression, 50 of rational function, 307 of trigonometric functions of real numbers, 465 Dominant term, of polynomial, 272 Dot product, 609–610, 612 Double root, 97 Double solution, 97, 103 Double-angle identities, 532–534 INDEX e (base of natural exponential function), 351 Earth orbit of, 652–653 radius of, 441 Earthquakes, 373–376, 379 Eccentricity of ellipse, 651–652 of hyperbola, 663–664 Edges, 805 Eiffel Tower, 451, 453 Einstein, Albert, 1, 23 Elementary row operations, 781–783 determinant and, 828–830 Elements of a matrix, 780 of a set, Eliminating the parameter of a pair of parametric equations, 698–699 Elimination methods for linear systems, 721–724, 730–734, 784–787 for nonlinear systems, 743 Ellipse applications of, 652–653 with center at (0, 0), 646–648 with center at (h, k), 648–650 definition of, 645, 692 eccentricity of, 651–652 equation of, 650 foci of, 645, 647–649 graph of, 646, 650, 677 parametric equations for, 704 in polar form, 694 reflective property of, 653 vertices of, 646–649 Empty set, Endpoint, 428 Epicycloid, 704 Equality of matrices, 792 of ordered pairs, 155 of polynomials, 750 properties of, 14 of rational expressions, 50 Equality of Exponents Theorem, 381 Equations absolute value, 79–80 classifying, 79 conditional, 78 contradiction as, 78 cubic, 110, 294–295 definition of, 14, 76–77 of ellipse, 650 equivalent, 76 exponential, 380–383 formulas, 84–85 general equation of a nondegenerate conic, 670 general second-degree equation in two variables, 670 of a graph, 479 identity as, 78 inverse trigonometric, 555 of a line, 190–191 linear in one variable, 77–78, 80–81, 84–85 logarithmic, 384–386 of parabolas, 636, 639 parametric See Parametric equations polar See Polar equations polynomial, 110, 294–295, 304 quadratic See Quadratic equations quadratic in form, 116–117 radical, 112–114 rational, 110–112 with rational exponents, 114–115 rectangular, 688–689 transformation, 637 trigonometric See Trigonometric equations in two variables, 157–163 variations, 136–141 See also Linear systems of equations; Nonlinear systems, of equations; Solutions Equilibrium, 501 Equilibrium price, 724–725 Equivalent equations, 76 Equivalent expression, 525 Equivalent inequalities, 123 Equivalent rational expressions, 50 Equivalent systems of equations, 721–722, 730 Equivalent vectors, 602 Eratosthenes, 441 Euler, Leonhard, 2, 167, 351, 849 Even and Odd Powers of (x Ϫ c) Theorem, 278–280 Even functions, 216–217, 466–467 Events, 891–895 Existence theorems, 300 Expanding the logarithmic expression, 370–371 Expectation, 899–900 Experiment, 890 Exponent(s) equality of, 381 integer, 17–20 natural number, negative, 17 properties of, 18–19 rational, 21–23, 114–115 restriction agreement for, 18 in scientific notation, 20 simplest form of expressions with, 19–20 zero and, 17–18 Exponential decay, 391–393, 416–418 Exponential equations, 380–383 Exponential functions, 346–354 definition of, 346–347 evaluating, 347 graphs of, 347–351 logarithmic functions and, 358–361 models based on, 333, 353–354 natural, 354 properties of, 349 Exponential growth, 391–393 Exponential notation, Exponential regression, 408–409 www.elsolucionario.net I3 Exponential time algorithm, 37 Extended Principle of Mathematical Induction, 876–877 Extraneous solutions, 113 Factor Theorem, 266–267 Factorable over the integers, 40, 42 Factorials, 849–850 Factoring polynomials, 40–48 difference of squares, 44 general strategy for, 47–48 greatest common factor in, 40–41 by grouping, 46–47 solving equations with, 96–97, 110 sum or difference of cubes, 45 trinomials, 41–45 zeros and, 266–267, 276, 278, 280, 287–288, 300–301 Factoring trigonometric equations, 561 Factorization Theorem, 43 Factors of a real number, Fair game, 899 Falling objects air resistance, 384–385, 399 distance fallen, 138, 856 height of, 105–106 Family of curves, 218 Feasible solutions, 763 Feedback, 66 Fermat, Pierre de, 154, 890 Fibonacci, Leonardo, 849 Fibonacci sequence, 849, 853 Final demand, 819 Finite sets, Floor function, 175–178 Focus/foci of ellipses, 645, 647–649 of hyperbolas, 658, 661–663 of parabola, 636, 638–639 of paraboloid, 640 FOIL method, 34–35, 41 Force, 614 Formulas, 84–85, 178–179 Fraction(s) complex, 55–56 equations containing, 78, 110–112 rationalizing the denominator, 27–28 See also Rational expressions Fractional form, of synthetic division, 263 Frustum of a cone, 84 Functions algebraic operations on, 227–228 applications of, 178–179 composition of, 230–234 constant, 175 decreasing, 175 definition of, 166–167 difference quotient of, 229–230 domain of, 167–168, 170 evaluating, 168–169, 228 even, 216–217 families of, 218 of the form f(x) = a sin x + b cos x, 544–545 graphs of, 170–172 I4 INDEX Functions (Continued ) greatest integer, 175–178 identifying, 168 increasing, 175 inverse, 334–342 maximum or minimum of, 274–275 notation for, 168–170 odd, 216–217 one-to-one, 175 piecewise-defined, 169, 248–249 range of, 167–168 trigonometric See Trigonometric functions vertical line test for, 174–175 Fundamental Counting Principle, 883–884, 887 Fundamental Linear Programming Theorem, 764 Fundamental Theorem of Algebra, 299 Future value of annuity, 867 Fuzzy sets, Galileo, 248, 856 Galois, Evariste, 101 Gauss, Carl Friedrich, 299–300 Gaussian elimination, 784–787 GCF (greatest common factor), 40–41 General equation of a nondegenerate conic, 670 General second-degree equation in two variables, 670 Geometric formulas, 84–85 Geometric sequences, 847, 861–862 Geometric series, 862–869 Germain, Sophie, Germain prime number, Golden mean, 75 Googol, Gordon model of stock valuation, 867 Graph(s) amplitude of, 474 of circles, 161–163 compressing, 221–223 of conic in polar form, 693–694 of cosecant function, 486–487, 495 of cosine function, 476–479 of cotangent function, 484–486 of curve, 698–699 cycle of, 474 of cycloid, 701 definition of, 805 difference of two functions, 496 of ellipses, 646, 650, 677 equation of, 479 of equations in three variables, 728–730 of equations in two variables, 157–163 of even functions, 217 of exponential functions, 347–351 function of the form f (x) = a sin x + b cos x, 545 of functions, 170–172 of hyperbola, 663, 677 of inequalities, 755–760 of inverse functions, 335–336, 338, 556557 of lemniscate, 686 of limaỗon, 682683 of linear functions, 186–195 of linear systems, 718 of logarithmic functions, 361–363 of nonlinear systems, 740–741 of odd functions, 217 of parabolas, 201–203, 637, 677 of piecewise-defined functions, 248–249 of polar equation, 679, 681 of polynomial functions, 271–282 product of two functions, 496 of r ϭ a, 680 of rational functions, 307–320 reflections of, 220–221 of rose curve, 684 scatter plots, 154 of secant function, 488–489 of second-degree equations in two variables, 674–676 semilog, 416–418 of sine function, 473–476 stretching, 221–223 sum of two functions, 495–496 symmetries of, 201, 213–217 of tangent function, 481–484 translations of, 217–220, 492–493 vectors added using, 602 walks through, 806 Graphing calculators absolute value functions, graphing, 159 adjusting settings of, 178 combinations, 886 complex numbers, 63, 303 connected vs dot mode, 176–177, 248 CUTOUT program, 276 decimal degree measure conversion to DMS measure, 432 degree measure conversion to radian measure, 435 determinants, evaluating, 827 equations in two variables, graphing, 158–159, 161 exponential equations, solving, 381–382 exponential expressions, evaluating, 19, 352 exponential functions, graphing, 352, 354 families of curves, graphing, 218 functions, graphing, 172, 174, 178 greatest integer function, 176 inverse functions, graphing, 336–338 inverse of a matrix, 816 iterations, 66 linear systems, solving, 723, 772 LIST feature, 218 logarithmic functions, graphing, 364–365, 373 logarithms, evaluating, 364–365 logistic models, 411 matrix operations, 798 maximum and minimum, 274–276 modeling guidelines, 408 nonlinear systems, solving, 742 nth roots of z, 705 parabolas, 159, 669 permutations, 885 piecewise functions, graphing, 248–249 polar equation, 683 polynomial equations, solving, 294–295, 304 www.elsolucionario.net polynomial functions, graphing, 274–275, 290, 293 projectile path, 702 quadratic equations, solving, 105 reference angles, 458 regression analysis, 239–240, 243, 407–408 regression models, 281–282 rose curve, 684–685 row echelon form, 783, 790 scientific notation, 20 second-degree equations in two variables, 674–676 sinusoidal data, 566–567 sinusoidal families, 505 SOLVE feature, 144–145 special angles, 447 SQUARE viewing window, 338 synthetic division, 264 TABLE feature, 158 translations of figures, 812–813 ZERO feature, 161 zeros of polynomial functions, 290, 293 Greatest common factor (GCF), 40–41 Greatest integer function, 175–178 Ground speed, 607 Growth models exponential, 391–393 logistic, 397–399, 410–411 Gunning-Fog Index, 93 Hale Telescope, 643 Half-angle identities, 535–537 Half-life, 392 Half-line, 428 Half-open intervals, Halley’s comet, 656, 664 Harmonic motion damped, 503–504 definition of, 501 simple, 501–504 Heading, 588 Heron’s formula, 596–597 Herschel, Caroline, 664 Holes in graphs, 271 Homogeneous systems of equations, 735–736 Hooke’s Law, 142 Hooper, Grace Murray, 849 Horizontal asymptotes, 307–312, 314–315 Horizontal axis of symmetry, 638 Horizontal compressing and stretching, 222–223 Horizontal line test, 175 Horizontal lines, 188, 191 Horizontal translations, 218–220, 491 Hubble Space Telescope, 441 Hydronium-ion concentration, 377 Hyperbola applications of, 664–665 asymptotes of, 660, 662–663 with center at (0, 0), 659–661 with center at (h, k), 661–663 definition of, 658, 692 degenerate form of, 658 INDEX eccentricity of, 663–664 foci of, 658, 661–663 graphing of, 663, 677 navigational uses of, 664–665 in polar form, 693–694 reflective property of, 665 transverse axis of, 659, 662 vertices of, 659, 661 Hypocycloid, 704 Hypotenuse, 103–104 i (imaginary unit), 60, 64 Ideal Gas Law, 143 Identities, 78 a Ϯ b, 523–524 cofunction, 525 definition of, 515 description of, 468–470 difference, 523–524, 526–528 double-angle, 532–534 fundamental, 516 half-angle, 535–537 inverses, 555–556 odd–even, 516 power-reducing, 534–535 product-to-sum, 541–542 Pythagorean, 468–469, 516–517 ratio, 468–469, 516 reciprocal, 469, 516 sines used to verify, 517–519 sum-to-product, 542–543 verification of, 516–520, 528, 534, 537 Identity matrix, 800 Identity properties of matrices, 794, 800 of real numbers, 12–13 Ill-conditioned systems of equations, 772 Imaginary axis, 616 Imaginary numbers, 60 Imaginary part, 60 Imaginary unit, 60, 64 Inconsistent systems of equations linear, 718–720, 729, 734, 786–787 nonlinear, 743 Increasing functions, 175, 335 Independent events, 895 Independent systems of equations, 718, 720, 731–732 Independent variable, 167 Index of radical, 23 of summation, 851 Index property of radicals, 25 Induction, mathematical, 871–877 Induction axiom, 873 Induction hypothesis, 873 Inequalities, 123–133 with absolute values, 126–127, 756–757 applications of, 131–133 compound, 125–126 critical values of, 127 equivalent, 123 graphs of, 755–760 linear, 124–125, 756–759, 762–768 nonlinear, 756, 759–760 polynomial, 127–129 proof of, by mathematical induction, 876–877 properties of, 123–124 rational, 130–131 in two variables, 755–760 Infinite geometric series, 864–866 Infinite sequences, 848–849 See also Sequences Infinite sets, Infinity symbol, 6, 308–309 Initial point, 602 Initial side, 428 Inner product, 609 Input-output analysis, 819–821 Input-output matrix, 819 Integers, 2–4 Intercepted arc, 441 Intercepts, 160 See also x-intercepts; y-intercepts Interest compound, 394–397, 866–867 simple, 88–89, 394 Intermediate Value Theorem, 276–277 Interpolating polynomials, 788 Intersection of events, 893 of lines, 718, 720, 722–723 of sets, 4–5 of solution sets of inequalities, 125 Interval notation, 6–8 Inverse functions, 334–342 logarithmic and exponential, 359 Inverse of a matrix, 813–818 condition for existence of, 830 solving linear systems with, 816–818 Inverse properties of real numbers, 12 Inverse relations, 335 Inverse trigonometric equation, 555 Inverse trigonometric functions composition of, 553–556 cosecant function, 552 description of, 549–552 evaluation of, 551–552 graphs of, 556–557 polynomials used to approximate, 572 secant function, 552 Inverse variation, 138–140 Involute of a circle, 704 Irrational numbers, 2–3, 351 Isosceles triangle, 452 Iteration, 66 Joint variation, 140–141 Kepler’s Laws, 144 Latus rectum of a parabola, 644 of an ellipse, 657 Law of Cosines applications of, 593–594 dot product formula from, 610 www.elsolucionario.net I5 law of sines vs., 594 triangles solved using, 592 uses of, 593 Law of Sines ambiguous case of, 583–587 applications of, 587–588 description of, 582–583 law of cosines vs., 594 triangles solved using, 582–587 Leading coefficient of polynomial, 33, 272 Leading term of polynomial, 272 Least common denominator (LCD), 53–54 Least-squares regression line, 238–239 Lemniscate, 682, 685–686 Leontief, Wassily, 819 Libby, Willard Frank, 393 Lick Telescope, 643 Like radicals, 26 Like terms, 13, 32 Limaỗon, 682683 Line(s) equations of, 190191 parallel, 191192, 718 perpendicular, 191–192 polar equations of a, 679 slopes of, 186–190 symmetry with respect to, 201, 213–214, 217 Line of best fit See Linear regression Linear correlation coefficient, 241 Linear equations in one variable, 77–78, 80–81, 84–85 in three variables, 728–730 in two variables, 189–190 Linear Factor Theorem, 300–301 Linear functions, 186, 190–195 Linear inequalities, 124–125, 756 systems of, 757–759, 762–768 Linear motion, 498 Linear programming, 762–768 Linear regression, 237–243 Linear speed, 437–438, 440 Linear systems of equations applications of, 724–726, 736–738 condition for unique solution, 836 Cramer’s Rule used in solving, 833–836 dependent, 718, 720, 724, 729, 732–734, 786 elimination for solving of, 721–724, 730–734, 784–787 graphing calculator for solving of, 723, 772 homogeneous, 735–736 ill-conditioned, 772 inconsistent, 718–720, 729, 734, 786–787 inverses of matrices for solving of, 816–818 matrix representations of, 780–781, 801–802 in n variables, 835 nonsquare, 734–735, 787 solutions of, 718, 720, 735, 787 substitution for solving of, 719–721, 729–730 in three variables, 728–730 triangular form of, 730–734 in two variables, 718–726 Lissajous figures, 704 Lithotripter, 655 I6 INDEX Local minimum, 274 Logarithm(s) change-of-base formula for, 372–373 changing to exponential form, 359–360 common, 364–365 definition of, 359 natural, 364–365 properties of, 360–361, 369–371 Logarithmic equations, 384–386 Logarithmic functions, 358–366 applications of, 365–366, 373–377, 379 common, 364–365 definition of, 359 domains of, 363–364 exponential functions and, 358–360 graphs of, 361–364, 373 natural, 364–365 properties of, 362–363 Logarithmic scales, 373–377, 379 Logarithm-of-each-side property, 370 Logistic models, 397–399, 410–411 Long division, 262 LORAN, 667 Lovell Telescope, 643 Lucas sequence, 853 Magnitude of the vector, 601 Main diagonal of matrix, 780 Major axis, 646, 649 Marginal cost or revenue, 199 Marginal propensity to consume, 868 Mars, 657 Mathematica, 323–324 Mathematical induction, 871–877 Matrices addition of, 791–794 additive inverse of, 793–794 adjacency, 805–806 applications of, 779, 788, 806–807, 819–821 cofactors of, 825–827 definition of, 779–780 determinants of, 824–830, 833–836 elementary row operations on, 781–783 elements of, 780 equality of, 792 identity, 800 input-output, 819 inverse of, 813–818, 830 linear systems represented with, 780–781, 801–802 linear systems solved with, 784–787, 816–818 main diagonal of, 780 minors of, 825–827 multiplication of, 794–800 notation for, 792 order (dimension) of, 780 real number used in multiplication of, 794–796 row echelon form of, 781 stochastic, 837–838 subtraction of, 793–794 transformation, 779, 802–805 triangular form of, 828 zero matrix, 794 Maximum and minimum in linear programming, 763–768 of polynomial function, 273–276 of quadratic function, 204–206 Means, arithmetic, 858–859 Midpoint formula, 155–157 Millennium Wheel, 464 Minor axis, 646, 649 Minors of a matrix, 825–827 Minute, 432 Mixture problems, 90–91 Modeling data, 237–244, 407–410 Modulus, 617 Mollweide’s formulas, 600 Monomials, 32 Motion harmonic See Harmonic motion uniform, 89 See also Falling objects; Speed Motion of a point, 700 Multiple zeros of a polynomial function, 287–288 Multiplication of binomials, 35 of complex numbers, 62–63 of exponential expressions, 19 of functions, 227 of inequality by real number, 123–124 of matrices, 796–800 of matrix by real number, 794–796 of polynomials, 34 of radical expressions, 25, 28 of rational expressions, 51–52 of real numbers, 11, 13 Multiplicative identity for matrices, 800 for real numbers, 12 Multiplicative inverse of matrix, 814–816, 830 of real number, 12 Multiplier effect, 868–869 Mutually exclusive events, 893–894 n! (n factorial), 850 Napier, John, 358 Natural exponential function, 354 Natural logarithms, 364–365 Natural numbers, 3–4 Nautical mile, 441 Negative angles, 428 Negative exponents, 17 Negative infinity symbol, Negative integers, 3–4 Newton’s Method, 853 Nomograms, 379 Nonfactorable over the integers, 42–43 Nonlinear inequalities, 756 Nonlinear systems of equations, 740–745 of inequalities, 759–760 Nonsingular matrices, 816 Nonsquare systems of equations, 734–735, 787 nth roots, 623–625, 705 Null set, www.elsolucionario.net Numbers complex See Complex numbers sets of, 2–4 Numerator, 12 Numerical coefficients, 11, 32 leading, 33, 272 Objective function, 762 Oblique triangle, 582, 594 Observation angle, 571 Obtuse angles, 429 Odd functions, 216–217, 466–467, 526 Odd–even identities, 516 One-to-one functions, 175, 335, 338 One-to-one property, 370 Open intervals, Optimization problems, 762–768 Orbits of comets, 664 of planets, 652–653, 656–657 Order of Operations Agreement, 9–11, 54 Ordered pairs as coordinates, 154 equality of, 155 of function, 167, 170 of relation, 166 as solutions of equations, 157 as solutions of inequalities, 755 as solutions of systems, 718, 720–721, 735 Ordered triples, 728, 733–734 Ordinate, 154 Orientation, 700 Origin, 7, 154 symmetry with respect to, 215–217 Orthogonal vectors, 611 Parabolas applications of, 639–641 axis of symmetry of, 634–635, 637–638 definition of, 201, 634, 692 directrix of, 636, 638–639 equation in standard form of, 639 equation of, 636 focus of, 636, 638–639 graphing of, 677 reflective property of, 640 with vertex at (0, 0), 634–637 with vertex at (h, k), 637–639 vertex of, 201–203, 634 See also Quadratic functions Parabolic asymptotes, 323 Paraboloid, 640 Parallel lines, 191–192, 718 Parallel vectors, 611–612 Parallelogram, 84, 602 Parameter of family of functions, 218 time as, 700 Parametric equations brachistochrone problem, 700–701 curve and, 697–698 definition of, 697 eliminating the parameter of a pair of, 698–699 INDEX for ellipse, 704 projectile motion and, 702 time as parameter, 700 Partial fractions, 748–753 Partial sums, 850–851, 856–857, 863–864 Pascal’s Triangle, 881–882 Percent mixture problems, 90–91 Perfect cubes, 45 Perfect squares, 43 Perfect-square trinomials, 44–45, 99 Perihelion, 652 Periodic function, 467 Permutations, 884–886 Perpendicular lines, 191–192 Perpendicular vectors, 611–612 Petronas Towers, 453 pH of a solution, 376–377 pi (p), Piecewise-defined functions, 169, 248–249 Plane coordinates in, 154–155 as graph of equation, 728–729 Point(s) plotting, in coordinate plane, 154, 158–159, 170–171 of real number line, symmetry with respect to, 215, 217 Point–slope form, 190 Polar axis, 678 Polar coordinate system definition of, 678 graph of equations in, 679–686 polar axis, 678 rectangular coordinates and, 686–688 Polar equations of circle, 681–682 of conics, 691–695 definition of, 679 graph of, 681 of lemniscates, 685–686 rectangular equations and, 688–689 Polar form of complex number, 617 Pole, 678 Polynomial(s), 32–37 addition of, 33 applications of, 36–37, 275–276, 293–295 approximating inverse trigonometric functions, 572 basic terminology about, 32–33 with complex coefficients, 298–299 definition of, 32 division of, 260–264 dominant term of, 272 equality of, 750 evaluating, 36 Even and Odd Powers of (x Ϫ c) Theorem, 278–280 factoring See Factoring polynomials far-left and far-right behavior, 272–273, 279 finding, with given zeros, 304–305 graphing procedure, 279–280 graphs of, 271–282 Intermediate Value Theorem for, 276–277 interpolating, 788 maxima and minima of, 273–276 multiplication of, 34 nonfactorable over the integers, 42–43 reduced, 267–268, 292–293, 300 sign property of, 127 standard form of, 33 subtraction of, 34 See also Linear functions; Quadratic functions; Zeros of a polynomial Polynomial equations, 110, 294–295, 304 Polynomial inequalities, 127–129 Polynomial time algorithm, 37 Population growth, 391–392, 397–399 Position equation, 105 Positive angles, 428 Positive integers, 3–4 Power(s) direct variation as, 137 of exponential expressions, 18–19 of i, 64 inverse variation as, 139 of radical expressions, 24 restrictions on zero, 18 See also Exponents Power functions, 416 Power principle, 112–113 Power property, of logarithm, 370 Power-reducing identities, 534–535 Price, equilibrium, 724–725 Prime numbers, Principal, 394 Principal square root, 23 Principle of Mathematical Induction, 873 extended, 876–877 Probability, 890–896 expectation and, 899–900 guidelines for, 896 Product definition of, 11 of two functions, 496 See also Multiplication Product property of complex numbers, 618–620 of logarithm, 370 Product-to-sum identities, 541–542 Profit, 88, 132–133, 194 Projectile, 565–566, 702, 713–714 Proportionality constant, 137 Protractor, 429 p-waves, 375 Pyramid, 452 Pythagorean identities, 468–469, 516–517 Pythagorean Theorem, 103–104, 444, 616 Quadrantal angle description of, 430 trigonometric functions of, 456 Quadrants, 154 Quadratic equations, 96–106 applications of, 103–106 classifying the solutions of, 103 completing the square used in solving of, 99–100 definition of, 96 www.elsolucionario.net I7 discriminant of, 102–103 factoring used in solving of, 96–97 quadratic formula used in solving of, 101–102 square roots used in solving of, 97–98 standard form of, 96 Quadratic formula, 101–102 trigonometric equation solved using, 562–563 Quadratic functions applications of, 206–209 definition of, 201 maximum or minimum of, 204–206 range of, 204 standard form of, 202–203 See also Parabolas Quadratic in form, 45–46, 116–117 Quadratic regression, 242–244 Quartic regression model, 280–282 Quotient definition of, 12, 261 difference, 229–230 See also Division Quotient property of complex numbers, 620 of logarithm, 370 Radian angle measurements using, 432–435 definition of, 433 degree conversions, 434–435 Radical equations, 112–114 Radical expressions, 23–28 Radicand, 23 Radius, of circle, 161, 453 Random walk, 890 Range of function, 167–168 of inverse function, 334 of quadratic function, 204 of trigonometric functions of real numbers, 465 Ratio identities, 468–469, 516 Rational equations, 110–112 Rational exponents, 21–23 equations with, 114–115 Rational expressions, 49–57 application of, 56–57 arithmetic operations on, 51–54 complex fractions and, 55–56 critical values of, 130 definition of, 50 domains of, 50 least common denominators of, 53–54 order of operations agreement with, 54 partial fraction decomposition of, 748–753 properties of, 50 simplifying, 51–53 Rational functions applications of, 318–320 asymptotes of graphs of, 307–313, 316–318, 323 with common factor, 318 definition of, 307 domains of, 307 graphing procedure for, 313–316 sign property of, 313 I8 INDEX Rational inequalities, 130–131 Rational numbers, 2–3 Rational Zero Theorem, 288–289, 292–293 Rationalizing the denominator, 27–28 Ray, 428 Real number line, 7–8 Real numbers, 2–4, 11–13 trigonometric functions of, 461–472 Real part, of complex number, 60 Reciprocal, 12 Reciprocal functions, 446 Reciprocal identities, 469, 516 Rectangle, 84 Rectangular coordinates, 154–155, 686–688 Rectangular equations, 688–689 Rectangular form, 616 Rectangular solid, 84 Recursion See Iteration Recursively defined sequence, 849 Reduced polynomials, 267–268, 292–293, 300 Reduction formulas, 528–529 Reference angle, 457–459 Reflection matrices, 802–803 Reflections of graphs, 220–221 of exponential functions, 351 Reflective property of ellipse, 653 of hyperbola, 665 of parabola, 640 Reflexive property of equality, 14 Regression analysis, 237–244, 566 Regression models, 280–282 Relations, 166, 168 inverse, 335 Relative maximum and minimum, 274 Relativity theory, 1, 23, 31 Remainder, in polynomial division, 261, 264–267 Remainder Theorem, 264–266 Repeating decimals, 2, 866 Resolving a vector, 607 Restriction Agreement, 18 Resultant vector, 602 Revenue, 88, 132–133, 194 marginal, 199 Richter, Charles F., 373 Richter scale, 373–376 Right angles, 429 Right circular cone, 84 Right circular cylinder, 84 Right triangles, 103–104 applications involving, 447–449 trigonometric functions, 442–444 Roots, 623–624 cube, 45, 624 double, 97 of an equation, 76, 287 fifth, 624–625 nth, 623–625, 705 in radical expressions, 23–28 rational exponents and, 21 See also Solution(s); Square roots; Zeros of a polynomial Rose curves, 682, 684 Rotation matrices, 779, 804–805 Rotation of axes definition of, 670 formulas for, 671 Rounding numbers, 177, 448 Row echelon form, 781 Row matrices, 796 Row operations, 781–783, 828–830 Rule of Signs, Descartes’, 290–292 Sample spaces, 890, 892 Satellite, 471 Satellite dish, 640–642 Satisfying an equation, 76 Saturn, 656 Scalar, 601 Scalar multiplication description of, 602, 605 of matrix, 794–796 Scalar product, 609 Scalar projection, 611 Scalar quantities, 601 Scatter diagram, 281 Scatter plots, 154, 405–406, 567 See also Regression analysis Scientific notation, 20–21 Secant, 443, 446 Secant function graph of, 488–489 inverse, 552 period of, 467 Second, 432 Second-degree equations in two variables general, 670 graphing of, 674–676 Sector, 441 Seismograms, 375–376, 379 Semilog graphs, 416–418 Semimajor axis, 646 Semiminor axis, 646 Semiperimeter, 592 Sequences, 851 alternating, 849 arithmetic, 854–859 constant, 852 Fibonacci, 849, 853 geometric, 847, 861–862 infinite, 848–849 Lucas, 853 recursively defined, 849 Series, 851 arithmetic, 855–859 geometric, 862–869 Set(s), 2–5 disjoint, elements of, empty (null), fuzzy, infinite, intersection of, 4–5 interval notation for, 6–8 of numbers, 2–4 union of, 4–5 www.elsolucionario.net Set-builder notation, Sign diagrams, 128 Sign property of rational functions, 313 Significant digits, 448 Signs of rational expressions, 50 Similar triangles, 87–88 Simple harmonic motion, 501–504 Simple interest, 88–89, 394 Simple zero, 287 Simplifying complex fractions, 55–56 exponential expressions, 19–20, 23 radical expressions, 23–28 rational expressions, 51–53 trigonometric expressions, 469 variable expressions, 11–14 sinϪ1 x, 549 Sine definition of, 443 identity verification using, 517 See also Law of Sines Sine function as cofunctions, 525 graph of, 473–476 harmonic motion See Harmonic motion inverse, 550 period of, 467 Sine regression, 566–567 Singular matrices, 816 Sinusoidal data, 566–567 Slant asymptotes, 316–318 Slope-intercept form, 188–190 Slopes, 186–190 SMOG readability formula, 92–93 Smooth continuous curves, 271 Solution(s) definition of, 76 double, 97 of equation in two variables, 157 extraneous, 113 feasible, in linear programming, 763 polar equation, 679 of quadratic equation, 97, 103 of system of equations, 718, 720–721, 735, 787 x-intercepts and, 190 See also Zeros of a polynomial Solution set of inequality in one variable, 123, 125–126 of inequality in two variables, 755 of system of inequalities, 757 Sonic boom, 668 Sound wave, 480 Special product formulas, 35 Speed average for a round trip, 56–57 average over a time interval, 230 in uniform motion, 89 Sphere, 84 Spring constant, 502 Square(s) difference of, 44 formula, 84 Square matrices, 780 INDEX Square roots of negative numbers, 60 of perfect squares, 43–44 of real numbers, 23–24 solving quadratic equations with, 97–98 Sørenson, Søren, 376 Standard form of the equation of a circle, 162–163 Standard position, 429 Statute mile, 441 Step, 805 Step functions, 176 Stochastic matrices, 837–838 Stock valuation, 867–868 Straight angles, 429 Stretching graphs, 221–223 of exponential functions, 351 Subsets, Substitution methods for equations quadratic in form, 116–117 for linear systems, 719–721, 729–730 for nonlinear systems, 741, 744 Substitution property of equality, 14 Subtraction of complex numbers, 61 of functions, 227 of matrices, 793–794 of polynomials, 34 of rational expressions, 51, 53–54 of real number from inequality, 123 of real numbers, 11 Sum See Addition Sum identities, 523–524, 526–528 Sum of first n positive integers, 858 Sum of two functions, 495–496 Summation notation, 851–852 Sum-to-product identities, 542–543 Supplementary angles, 429 Supply–demand problems, 724–725 s-waves, 375 Sylvester, James, 780 Symmetric property of equality, 14 Symmetries of graphs of function and its inverse, 336 of polynomial functions, 280 of rational functions, 314–315 with respect to a line, 201, 213–214, 217 with respect to a point, 215, 217 See also Axis of symmetry Synthetic division, 262–264 bounds for real zeros and, 289–290 with complex numbers, 303 Systems of linear equations See Linear systems of equations of linear inequalities, 757–759 of nonlinear equations, 740–745 of nonlinear inequalities, 759–760 Tangent, 443 Tangent function double-angle identity for, 532 graph of, 481–484 period of, 468 Tangent lines, concavity and, 405 Telescope, 712 Terminal point, 602 Terminal side, 428 Terms of polynomial, 32 of sequence, 848 of sum, 11 of variable expression, 11 Test value, 128 Third-degree equations See Cubic equations Tides, 499 Time as parameter, 700 in uniform motion, 89 Touch-tone phones, 515 Tower, 452 Transformation equations, 637 Transformation matrix, 779, 802–805 Transitive property of equality, 14 of inequalities, 123 Translation matrices, 802 Translations graphing uses of, 492–493 horizontal, 491 of trigonometric functions, 491–495 vertical, 491 Translations of graphs, 217–220 of exponential functions, 350–351 of logarithmic functions, 364–365 Transverse axis, 659, 662 Triangle(s), 84, 87–88, 103–104 area of, 453, 595–596 Isosceles, 452 Law of Cosines used to solve, 592 Law of Sines used to solve, 582–587 oblique, 582, 594 Pascal’s, 881–882 right See Right triangles similar, 87–88 Triangular form of matrix, 828 of system of equations, 730–734 Trigonometric equations factoring used to solve, 561 inverse, 555 quadratic formula used to solve, 562–563 solving, 560–566 squaring each side of, 561–562 Trigonometric expression difference of, 543 evaluation of, 524, 554 simplifying, 527, 533 Trigonometric form of complex numbers, 616–622 Trigonometric functions absolute value of, 458 acute angle, 443 of angles See Angle(s) angular speed, 437–438 applications of, 427 arcs, 435–436 complex number written in form of, 617 www.elsolucionario.net I9 composition of, 553–556 definition of, 442 evaluation of, 527, 533, 535–537 even, 466–467 inverse See Inverse trigonometric functions linear speed, 437–438 list of, 442–444 odd, 466–467 of quadrantal angles, 456 of real numbers, 427, 461–472 signs of, 456–457 of special angles, 444–447 translations of, 491–495 Trigonometric identities See Identities Trinomials definition of, 33 factoring, 41–43, 45 perfect-square, 44–45, 99 quadratic in form, 45–46 Trivial solution of linear system, 735 Turning points, 272–273 Uniform motion, 89 See also Speed Union of events, 893 of sets, 4–5 of solution sets of inequalities, 125 Unit circle, 462, 467, 469 Unit fraction, 432 Unit vectors, 604–606 Upper- and Lower-Bound Theorem, 289–290 Variable, 167–168 Variable expressions, 11–14 Variable part, 11 Variable terms, 11 Variation, 136–141 Vector additive inverse of a, 604 angle between, 610 applications of, 607–608 components of, 603–604, 608 definition of, 601 direction angle of, 603 dot product of, 609–610, 612 equivalent, 602 fundamental operations, 604 horizontal component of, 606 magnitude of, 601, 609–610 orthogonal, 611 parallel, 611–612 perpendicular, 611–612 resolving the, 607 resultant, 602 scalar multiplication of, 602 triangle method for adding, 602 unit, 604–606 vertical component of, 606 zero, 604 Vector quantities, 601 Velocity See Speed Venus, 656 Verhulst population models, 66, 397 I10 INDEX Vertex at (0, 0), 634–637 of angle, 428 at (h, k), 637–639 in linear programming, 764 of parabola, 201–203, 634 of paraboloid, 640 Vertical asymptotes, 307–311, 314–316 Vertical axis of symmetry, 637 Vertical line test, 174–175 Vertical lines, 188 Vertical stretching and compressing, 221–222 Vertical translations, 217–220, 491 Vertices description of, 805 of ellipse, 646–649 of hyperbola, 659, 661 Voltage, 499 Walk definition of, 805 in graphs, 806 length of, 805 random, 890 Washington Monument, 452 Whispering gallery, 653, 656 Work, 612 Work problems, 91–92 Wrapping function, 462–463 x-axis, 154, 647, 649, 659, 662 reflection across, 220 symmetry with respect to, 213–214 x-coordinate, 154 x-intercepts, 160 of rational functions, 314–315 real solutions and, 190 zeros of a polynomial and, 278 xy-plane, 154 xyz-coordinate system, 728 y-axis, 154, 647, 649, 659, 662 reflection across, 220 symmetry with respect to, 213–214, 217 y-coordinate, 154 y-intercepts, 160 of lines, 188–189 of polynomial functions, 279 of rational functions, 314–315 z-axis, 728 Zeller’s Congruence, 185 www.elsolucionario.net Zero, in exponential expressions, 17–18 Zero determinant, 829 Zero factorial, 850 Zero matrix, 794 Zero product principle, 96–97 Zero vector, 604 Zeros of a polynomial applications of, 293–295 complex, 299–305 complex coefficients and, 298–299 definition of, 127, 260 Descartes’ Rule of Signs and, 290–292 factors and, 266–267, 276, 278, 280, 287, 300–301 finding polynomial, given zeros, 304–305 finding with Mathematica, 323–324 guidelines for finding, 292–293 Intermediate Value Theorem and, 276–277 multiple, 287–288 number of, 288 rational, 288–289 sign of polynomial between, 127 simple, 287 upper and lower bounds for, 289–290, 298 x-intercepts and, 278 Zeros of a rational function, 313 This page intentionally left blank www.elsolucionario.net A Library of Functions 2.3 Identity function y 2.3 Linear function y f (x) = x 2.3 Constant function y f (x) = mx + b y f(x) = c (0, c) 2.2 Absolute value function m (0, b) f(x) = |x| −4 −2 x x 2 x −2 −4 −2 −2 −4 2.4 Squaring function −4 3.2 Cubing function y y 4 f (x) = x 2 −2 x 2 −4 −2 Square root function f(x) = x −2 x 4.2 Exponential function −4 −2 4.3 Logarithmic function y f(x) = log b x, b > (b, 1) (1, b) (1, 0) x (1, 0) x (1, b) f(x) = log b x, < b < x x 4.6 Logistic function 4.6 Logistic function P(t) P(t) c 3.5 Reciprocal function y c c P(t) = , a>1 + ae −bt P0 c P(t) = ,0 0, b > f (x) = y=a x x P0 t x −4 (b, 1) (0, 1) −2 x y f (x) = b x, < b < (0, 1) f (x) = x f(x) = x 4.3 Logarithmic function y f(x) = b x, b > −2 4.2 Exponential function y y −2 −4 −2 y x Cube root function t −4 www.elsolucionario.net x=b A Library of Functions 5.5 Sine function (cont’d) 5.5 Cosine function 5.6 Tangent function y y y f (x) = sin x −2π 2π x π −π − 2π π −π −1 f(x) = cos x − 2π 2π x 2π x π −π −1 −1 f(x) = tan x 5.6 Cosecant function −2π 5.6 Secant function 5.6 Cotangent function y y y 1 2π x π −π − 2π π −π −1 2π x − 2π −1 f (x) = csc x 2π x π −π −1 f (x) = sec x f(x) = cot x Conic Sections 8.1 Parabolas y y y Focus Directrix y=k−p (h, k + p) y Directrix x=h−p Vertex (h, k) Vertex (h, k) Vertex (h, k) x x x (x – h) = 4p(y – k); p < Vertex (h + a, k) Vertex (h − a, k) Focus (h − c, k) y Vertex (h, k + a) y Asymptote b (x − h) a Focus y−k= Focus (h, k + c) Center (h, k) Vertex (h − a, k) (h, k + c) Vertex (h + a, k) Center (h, k) Focus Focus (h + c, k) a b Center (h, k) Asymptote a y − k = − (x − h) b x Vertex (h, k − a) (h + c, k) x Focus (h, k − c) y−k= Vertex (h, k − a) = 1, a > b x−h b2 + y−k a2 x−h = 1, a > b Focus Asymptote b y − k = − (x − h) a x y−k Asymptote a (x − h) b Vertex (h, k + a) Focus (h − c, k) Center (h, k) x (y – k) = 4p(x – h); p < 8.3 Hyperbolas y y + (y – k) = 4p(x – h); p > 8.2 Ellipses x 2 (x – h) = 4p(y – k); p > x−h Focus (h + p, k) Focus (h + p, k) Focus (h, k + p) Directrix y=k−p Directrix x=h−p Vertex (h, k) a2 − y−k b2 www.elsolucionario.net (h, k − c) y−k =1 a − x −h b 2 =1 P.2 Properties of Exponents aman = am + n am = am - n an (amb n)p = ampb np a 2.5 Graphing Concepts (am)n = amn am p amp nb = b bnp b-p = bp Even Functions A function is an even function if f( -x) = f(x) for all x in the domain of f The graph of an even function is symmetric with respect to the y-axis P.2 Properties of Radicals n n n (1b)m = 1bm = bm/n n n 1a n a = n Ab 1b m n n 1a 1b = 1ab Odd Functions A function f is an odd function if f ( - x) = - f (x) for all x in the domain of f The graph of an odd function is symmetric with respect to the origin Vertical and Horizontal Translations If f is a function and c is a positive constant, then the graph of mn 31b = b P.4 Factoring a + 2ab + b2 = (a + b)2 a - 2ab + b2 = (a - b)2 a - b2 = (a + b)(a - b) ● y = f(x) + c is the graph of y = f (x) shifted up vertically c units ● y = f(x) - c is the graph of y = f (x) shifted down vertically c units ● y = f (x + c) is the graph of y = f (x) shifted left horizontally c units ● y = f (x - c) is the graph of y = f (x) shifted right horizontally c units a + b = (a + b)(a - ab + b ) 3 2 Reflections If f is a function then the graph of a3 - b3 = (a - b)(a2 + ab + b2) 1.5 Properties of Absolute Value Inequalities ƒ x ƒ c (c Ú 0) if and only if - c x c ƒ x ƒ c (c Ú 0) if and only if either x c or x - c 2.2 Properties of Functions A function is a set of ordered pairs in which no two ordered pairs that have the same first coordinate have different second coordinates If a and b are elements of an interval I that is a subset of the domain of a function f, then ● f is an increasing function on I if f (a) f (b) whenever a b ● f is a decreasing function on I if f (a) f (b) whenever a b ● f is a constant function on I if f (a) = f (b) for all a and b A one-to-one function satisfies the additional condition that given any y, there is one and only one x that can be paired with that given y ● y = - f (x) is the graph of y = f (x) reflected across the x-axis ● y = f( - x) is the graph of y = f (x) reflected across the y-axis Vertical Shrinking and Stretching ● ● ● If c and the graph of y = f (x) contains the point (x, y), then the graph of y = c # f(x) contains the point (x, cy) If c 1, the graph of y = c # f(x) is obtained by stretching the graph of y = f (x) away from the x-axis by a factor of c If c 1, the graph of y = c # f(x) is obtained by shrinking the graph of y = f (x) toward the x-axis by a factor of c Horizontal Shrinking and Stretching ● If a and the graph of y = f (x) contains the point (x, y), then the graph of y = f (ax) contains the point a x, yb a ● If a 1, the graph of y = f (ax) is a horizontal shrinking of the graph of y = f (x) ● If a 1, the graph of y = f (ax) is a horizontal stretching of the graph of y = f (x) www.elsolucionario.net Important Formulas 4.6 Compound Interest Formulas The distance between P1(x1, y1) and P2(x2 , y2) is P = principal invested, t = time in years, r = annual interest rate, A = balance: d(P1, P2) = 2(x2 - x1)2 + ( y2 - y1)2 The slope m of a line through P1(x1, y1) and P2(x2, y2) is m = y2 - y1 , x2 - x1 x1 Z x2 A = Pert (compounded continuously) The slope-intercept form of a line with slope m and y-intercept b is y = mx + b The point-slope formula for a line with slope m passing through P1(x1, y1) is y - y1 = m(x - x1) Quadratic Formula If a Z 0, the solutions of ax + bx + c = are x = -b Ϯ 2b2 - 4ac 2a Leg a 5.3 Definitions of Trigonometric Functions b r a cos u = r b tan u = a sin u = r b r sec u = a a cot u = b y csc u = P (a, b) b θ a x where r = 2a2 + b2 Important Theorems Pythagorean Theorem c = a2 + b2 r nt A = P a1 + n b (compounded n time per year) Hypotenuse c 5.4 Definitions of Circular Functions sin t = y Leg b Remainder Theorem If a polynomial P(x) is divided by x - c, then the remainder is P(c) Factor Theorem A polynomial P(x) has a factor (x - c) if and only if P(c) = cos t = x y tan t = x y y sec t = x x cot t = y csc t = t P (x, y) A(1, 0) r=1 x Fundamental Theorem of Algebra If P is a polynomial of degree n Ú with complex coefficients, then P has at least one complex zero 6.1 Fundamental Trigonometric Identities Binomial Theorem sin2 u + cos2 u = tan u = n n (a + b) = a + a ban - 1b + a b an - 2b 2 n + Á + a ban - kbk + Á + bn k n n 4.4 Properties of Logarithms y = logb x if and only if b = x y logb b = logb = logb (b) p = p blogb p = p log x = log10 x ln x = log e x logb (MN ) = logb M + logb N logb (M>N) = logb M - logb N logb M p = p logb M sin u cos u cot u = cos u sin u + tan2 u = sec2 u + cot u = csc2 u sin (- u) = - sin u tan (- u) = - tan u csc u = sin u cos ( - u) = cos u sec u = cos u cot u = 6.2 Sum and Difference Identities sin (a + b) = sin a cos b + cos a sin b cos (a + b) = cos a cos b - sin a sin b tan (a + b) = www.elsolucionario.net tan a + tan b - tan a tan b tan u sin (a - b) = sin a cos b - cos a sin b 6.4 Sum-to-Product Identities cos (a - b) = cos a cos b + sin a sin b x - y x + y cos 2 x - y x + y sin sin x - sin y = cos 2 x - y x + y cos cos x + cos y = cos 2 x - y x + y sin cos x - cos y = - sin 2 tan (a - b) = sin x + sin y = sin tan a - tan b + tan a tan b 6.2 Cofunction Identities sin a p p - ub = cos u cos a - ub = sin u 2 tan a p - ub = cot u 7.1–7.2 Formulas for Triangles For any triangle ABC, the following formula can be used Law of Sines b c a = = sin A sin B sin C 6.3 Double-Angle Identities sin 2a = sin a cos a cos 2a = cos a - sin a = - sin a = cos2 a - tan 2a = 2 C Law of Cosines A c = a + b - 2ab cos C tan a - tan2 a a b c Area of a Triangle K = 6.3 Power-Reducing Identities a2 sin B sin C ab sin C K = 2 sin A K = 3s(s - a)(s - b)(s - c), where s = a + b + c - cos 2a sin2 a = + cos 2a cos2 a = - cos 2a tan a = + cos 2a 11.2 Arithmetic Sequences and Series 6.3 Half-Angle Identities sin a - cos a = Ϯ A cos + cos a a = Ϯ A tan sin a - cos a a = = + cos a sin a Common difference + - = d nth-term an = a1 + (n - 1)d Sum of n terms Sn = = n (a + an) n [2a1 + (n - 1)d] 11.3 Geometric Sequences and Series Common ratio + = r nth-term an = a1r n - Sum of n terms Sn = Sum of an infinite series S = 6.4 Product-to-Sum Identities sin a cos b = sin (a + b) + sin (a - b) cos a sin b = sin (a + b) - sin (a - b) cos a cos b = cos (a + b) + cos (a - b) sin a sin b = cos (a - b) - cos (a + b) www.elsolucionario.net a1(1 - r n) - r a1 , ƒrƒ 1 - r B ... both College Algebra and College Algebra and Trigonometry, and Christi Verity wrote the solutions for the Complete Solutions Manual and the Student Solutions Manual for College Algebra and College. .. www.elsolucionario.net 847 PREFACE We are proud to offer the seventh edition of College Algebra and Trigonometry Your success in college algebra and trigonometry is important to us To guide you to that success,... blank www.elsolucionario.net COLLEGE ALGEBRA AND TRIGONOMETRY www.elsolucionario.net This page intentionally left blank www.elsolucionario.net COLLEGE ALGEBRA AND TRIGONOMETRY Chad Ehlers/Getty