Solution Manual for Precalculus Enhanced with Graphing Utilities, 8th Edition

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SOLUTIONSMANUALTIMBRITTJackson State Community CollegePRECALCULUSENHANCED WITHGRAPHINGUTILITIESEIGHTHEDITIONMichael SullivanChicago State UniversityMichael Sullivan IIIJoliet Junior College

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Table of ContentsChapter 1Graphs1.1Graphing Utilities; Introduction to Graphing Equations ......................................................... 11.2The Distance and Midpoint Formulas ..................................................................................... 81.3Intercepts; Symmetry; Graphing Key Equations ................................................................... 201.4Solving Equations Using a Graphing Utility ......................................................................... 361.5Lines ...................................................................................................................................... 411.6Circles.................................................................................................................................... 58Chapter Review.............................................................................................................................. 72Chapter Test ................................................................................................................................... 79Chapter Projects ............................................................................................................................. 82Chapter 2Functions and Their Graphs2.1Functions ............................................................................................................................... 832.2The Graph of a Function...................................................................................................... 1012.3Properties of Functions ........................................................................................................ 1102.4Library of Functions; Piecewise-defined Functions ............................................................ 1272.5Graphing Techniques: Transformations .............................................................................. 1392.6Mathematical Models: Building Functions.......................................................................... 157Chapter Review............................................................................................................................ 165Chapter Test ................................................................................................................................. 172Cumulative Review...................................................................................................................... 175Chapter Projects ........................................................................................................................... 179Chapter 3Linear and Quadratic Functions3.1Properties of Linear Functions and Linear Models.............................................................. 1813.2Building Linear Functions from Data.................................................................................. 1923.3Quadratic Functions and Their Properties ........................................................................... 1983.4Build Quadratic Models from Verbal Descriptions and from Data ..................................... 2223.5Inequalities Involving Quadratic Functions......................................................................... 230Chapter Review............................................................................................................................ 249Chapter Test ................................................................................................................................. 257Cumulative Review...................................................................................................................... 260Chapter Projects ........................................................................................................................... 263Chapter 4Polynomial and Rational Functions4.1Polynomial Functions .......................................................................................................... 2664.2The Graph of a Polynomial Function: Models .................................................................... 2774.3The Real Zeros of a Polynomial Function ........................................................................... 2974.4Complex Zeros; Fundamental Theorem of Algebra ............................................................ 3344.5Properties of Rational Functions ......................................................................................... 3434.6The Graph of a Rational Function ....................................................................................... 3534.7Polynomial and Rational Inequalities .................................................................................. 409Chapter Review............................................................................................................................ 431Chapter Test ................................................................................................................................. 445Cumulative Review...................................................................................................................... 449Chapter Projects ........................................................................................................................... 454

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Chapter 5Exponential and Logarithmic Functions5.1Composite Functions ........................................................................................................... 4555.2One-to-One Functions; Inverse Functions........................................................................... 4735.3Exponential Functions ......................................................................................................... 4965.4Logarithmic Functions......................................................................................................... 5175.5Properties of Logarithms ..................................................................................................... 5395.6Logarithmic and Exponential Equations.............................................................................. 5485.7Financial Models ................................................................................................................. 5695.8Exponential Growth and Decay Models; Newton’s Law; Logistic Growthand Decay Models ............................................................................................................... 5775.9Building Exponential, Logarithmic, and Logistic Models from Data ................................. 588Chapter Review............................................................................................................................ 593Chapter Test ................................................................................................................................. 605Cumulative Review...................................................................................................................... 609Chapter Projects ........................................................................................................................... 612Chapter 6Trigonometric Functions6.1Angles, Arc length, and Circular Motion ............................................................................ 6156.2Trigonometric Functions; Unit Circle Approach................................................................. 6246.3Properties of the Trigonometric Functions .......................................................................... 6426.4Graphs of the Sine and Cosine Functions............................................................................ 6566.5Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions................................... 6786.6Phase Shift; Sinusoidal Curve Fitting.................................................................................. 688Chapter Review............................................................................................................................ 701Chapter Test ................................................................................................................................. 709Cumulative Review...................................................................................................................... 712Chapter Projects ........................................................................................................................... 716Chapter 7Analytic Trigonometry7.1The Inverse Sine, Cosine, and Tangent Functions............................................................... 7207.2The Inverse Trigonometric Functions (Continued) ............................................................. 7347.3Trigonometric Equations .................................................................................................... 7467.4Trigonometric Identities ...................................................................................................... 7677.5Sum and Difference Formulas ............................................................................................. 7807.6Double-angle and Half-angle Formulas............................................................................... 8057.7Product-to-Sum and Sum-to-Product Formulas................................................................... 833Chapter Review............................................................................................................................ 846Chapter Test ................................................................................................................................. 861Cumulative Review...................................................................................................................... 866Chapter Projects ........................................................................................................................... 872

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Chapter 8Applications of Trigonometric Functions8.1Right Triangle Trigonometry; Applications ........................................................................ 8768.2The Law of Sines................................................................................................................. 8908.3The Law of Cosines ............................................................................................................. 9058.4Area of a Triangle................................................................................................................ 9178.5Simple Harmonic Motion; Damped Motion; Combining Waves ........................................ 927Chapter Review............................................................................................................................ 937Chapter Test ................................................................................................................................. 943Cumulative Review...................................................................................................................... 947Chapter Projects ........................................................................................................................... 953Chapter 9Polar Coordinates; Vectors9.1Polar Coordinates ................................................................................................................ 9579.2Polar Equations and Graphs................................................................................................. 9669.3The Complex Plane; De Moivre’s Theorem........................................................................ 9969.4Vectors............................................................................................................................... 10089.5The Dot Product................................................................................................................. 10219.6Vectors in Space ................................................................................................................ 10279.7The Cross Product ............................................................................................................. 1034Chapter Review.......................................................................................................................... 1045Chapter Test ............................................................................................................................... 1054Cumulative Review.................................................................................................................... 1058Chapter Projects ......................................................................................................................... 1060Chapter 10Analytic Geometry10.2 The Parabola ...................................................................................................................... 106410.3 The Ellipse......................................................................................................................... 108010.4 The Hyperbola ................................................................................................................... 109710.5 Rotation of Axes; General Form of a Conic ...................................................................... 111710.6 Polar Equations of Conics ................................................................................................. 113010.7 Plane Curves and Parametric Equations ............................................................................ 1139Chapter Review.......................................................................................................................... 1154Chapter Test ............................................................................................................................... 1163Cumulative Review.................................................................................................................... 1168Chapter Projects ......................................................................................................................... 1170Chapter 11Systems of Equations and Inequalities11.1 Systems of Linear Equations: Substitution and Elimination ............................................. 117411.2 Systems of Linear Equations: Matrices ............................................................................. 119711.3 Systems of Linear Equations: Determinants...................................................................... 122211.4 Matrix Algebra .................................................................................................................. 123611.5 Partial Fraction Decomposition ......................................................................................... 125511.6 Systems of Nonlinear Equations........................................................................................ 127411.7 Systems of Inequalities ...................................................................................................... 130211.8 Linear Programming.......................................................................................................... 1317Chapter Review.......................................................................................................................... 1331Chapter Test ............................................................................................................................... 1346Cumulative Review.................................................................................................................... 1354Chapter Projects ......................................................................................................................... 1358

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Chapter 12Sequences; Induction; the Binomial Theorem12.1 Sequences .......................................................................................................................... 136012.2 Arithmetic Sequences ........................................................................................................ 137412.3 Geometric Sequences; Geometric Series........................................................................... 138312.4 Mathematical Induction ..................................................................................................... 139412.5 The Binomial Theorem...................................................................................................... 1404Chapter Review.......................................................................................................................... 1411Chapter Test ............................................................................................................................... 1415Cumulative Review.................................................................................................................... 1418Chapter Projects ......................................................................................................................... 1421Chapter 13Counting and Probability13.1 Counting ............................................................................................................................ 142413.2 Permutations and Combinations ........................................................................................ 142713.3 Probability ......................................................................................................................... 1432Chapter Review.......................................................................................................................... 1439Chapter Test ............................................................................................................................... 1441Cumulative Review.................................................................................................................... 1442Chapter Projects ......................................................................................................................... 1445Chapter 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function14.1 Finding Limits Using Tables and Graphs .......................................................................... 144814.2 Algebra Techniques for Finding Limits............................................................................. 145414.3 One-sided Limits; Continuity ............................................................................................ 145814.4 The Tangent Problem; The Derivative .............................................................................. 146614.5 The Area Problem; The Integral ........................................................................................ 1476Chapter Review.......................................................................................................................... 1490Chapter Test ............................................................................................................................... 1497Chapter Projects ......................................................................................................................... 1500Appendix A: ReviewA.1Algebra Essentials ............................................................................................................. 1506A.2Geometry Essentials .......................................................................................................... 1511A.3Polynomials ....................................................................................................................... 1517A.4Synthetic Division ............................................................................................................. 1525A.5Rational Expressions ......................................................................................................... 1527A.6Solving Equations.............................................................................................................. 1532A.7Complex Numbers; Quadratic Equations in the Complex Number System ...................... 1547A.8Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications... 1553A.9Interval Notation; Solving Inequalities.............................................................................. 1561A.10nth Roots; Rational Exponents .......................................................................................... 1572Appendix B: The Limit of a Sequence; Infinite Series.................................................... 1583

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1Chapter 1GraphsSection 1.11.02.3.x-coordinate;y-coordinate; quadrants4.False; points that lie in Quadrant IV will have apositivex-coordinate and a negativey-coordinate.The point1, 4lies in Quadrant II.5.d6.c7.(a)Quadrant II(b)x-axis(c)Quadrant III(d)Quadrant I(e)y-axis(f)Quadrant IV8.(a)Quadrant I(b)Quadrant III(c)Quadrant II(d)Quadrant I(e)y-axis(f)x-axis9.The points will be on a vertical line that is twounits to the right of they-axis.10.The points will be on a horizontal line that isthree units above the x-axis.11.1, 4; Quadrant II12.(3, 4); Quadrant I13.(3, 1); Quadrant I

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Chapter 1:Graphs214.6,4; Quadrant III15.min11max5scl1min3max6scl1XXXYYY  16.min3max7scl1min4max9scl1XXXYYY  17.min30max50scl10min90max50scl10XXXYYY  18.min90max30scl10min50max70scl10XXXYYY  19.min10max110scl10min10max160scl10XXXYYY  20.min20max110scl10min10max60scl10XXXYYY  21.min6max6scl2min4max4scl2XXXYYY  22.min3max3scl1min2max2scl1XXXYYY  23.min6max6scl2min1max3scl1XXXYYY  24.min9max9scl3min12max4scl4XXXYYY  25.min3max9scl1min2max10scl2XXXYYY26.min22max10scl2min4max8scl1XXXYYY  

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Section 1.1:Graphing Utilities; Introduction to Graphing Equations327.4yxx40000041111040( 1)1011 (0, 0) is on the graph of the equation.28.32yxx30020003112 111 3112 111 (0, 0) and (1, –1) are on the graph of theequation.29.229yx223099922039018220( 3)9018(0, 3) is on the graph of the equation.30.31yx321182310111301100  (0, 1) and (–1, 0) are on the graph of theequation.31.224xy220244422(2)24842222444(0, 2) and2,2are on the graph of theequation.32.2244xy2204 14442224 0444221224454(0, 1) and (2, 0) are on the graph of the equation.33.(–1, 0), (1, 0)34.(0, 1)35., 0 ,, 0 , (0, 1)22   36.(–2, 0), (2, 0), (0, –3)37.1, 0,0, 2,0,238.(2, 0), (0, 2), (–2, 0), (0, –2)39.(–4,0), (–1,0), (4, 0), (0, –3)40.(–2, 0), (2, 0), (0, 3)41.2yx42.6yx43.28yx

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Chapter 1:Graphs444.39yx45.21yx46.29yx47.24yx 48.21yx 49.236xy50.5210xy

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Section 1.1:Graphing Utilities; Introduction to Graphing Equations551.29436xy52.244xy53.213yxThe x-intercept is6.5xand the y-intercept is13y .54.314yx The x-intercept is4.67xand the y-intercept is14y.55.2215yxThe x-intercepts are2.74x and2.74x.The y-intercept is15y .56.2319yx 

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Chapter 1:Graphs6The x-intercepts are2.52x and2.52x.The y-intercept is19y.57.3433243or2xxyyThe x-intercept is14.33xand the y-interceptis21.5y .58.4824582or5xxyyThe x-intercept is20.5xand the y-intercept is16.4y.59.225375337or3xxyyThe x-intercepts are2.72x and2.72x.The y-intercept is12.33y.60.222352335or3xxyyThe x-intercepts are4.18x and4.18x.The y-intercept is11.67y .61.If2, 5is shifted 3 units right then the xcoordinate would be23. If it is shifted 2 unitsdown then the y-coordinate would be5( 2) .

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Section 1.1:Graphing Utilities; Introduction to Graphing Equations7Thus the new point would be23, 5( 2)(5,3) .62.If1, 6is shifted 2 units left then the xcoordinate would be1( 2)  . If it is shifted 4units up then the y-coordinate would be64.Thus the new point would be1( 2), 64( 3,10)   .63.a.25 feetb.23.2 ft; 34.1 ftc.(0,6), The shot is released at a height of 6feet; (48.7, 0), the shot hits the ground aftertraveling a horizontal distance of 48.7 feet.64.a.20 metersb.12 seconds; 36 metersc.(0, 2), The discus is released at a height of 2meters; (18, 0), the discus hits the groundafter 18 seconds.65.a.$19.95; $19.95b.$182.45c.(0, 19.95), The membership plan costs$19.95 per month.66.a.1.5 milesb.1 milec.(0, 2), Caleb’s friend lives 2 miles from hishouse; (28, 0), it takes Caleb 28 minutes toride home.67.( 1, 0), (1, 0), (0,1), (0,1)68.( 2, 0), (3, 0), (0,2), (0, 0), (0, 2)69.Answers will vary70.a.b.Since2xxfor allx, the graphs of2andyxyxare the same.c.For2yx, the domain of the variablexis0x; foryx, the domain of thevariablexis all real numbers. Thus,2only for0.xxxd.For2yx, the range of the variableyis0y; foryx, the range of the variableyis all real numbers. Also,2only if0.xxx71.Answers will vary

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Chapter 1:Graphs872.Answers will vary. A complete graph presentsenough of the graph to the viewer so they can“see” the rest of the graph as an obviouscontinuation of what is shown.73.Answers will vary.Section 1.21.5388 2.22342553.22211601213600372161Since the sum of the squares of two of the sidesof the triangle equals the square of the third side,the triangle is a right triangle.4.0, 05.12bh6.true7.midpoint8.False; the distance between two points is nevernegative.9.True;1212,22xxyyM 10.a11.221222(,)(20)(10)21415d P P12.221222(,)( 20)(10)( 2)1415d P P13.221222(,)( 21)(21)( 3)19110d P P14.221222(,)2( 1)(21)319110d P P 15. 221222(,)(53)4428464682 17d P P 16. 221222(,)214034916255d P P 17.221222(,)4( 7)(03)11(3)1219130  d P P18.221222(,)422( 3)2542529d P P 19.221222(,)(65)1( 2)131910 d P P20.221222(,)6(4)2( 3)1051002512555d P P  21.221222(,)2.3( 0.2)1.1(0.3)2.50.86.250.646.892.62 d P P22.221222(,)0.31.21.12.3( 1.5)( 1.2)2.251.443.691.92 d P P23.22122222(,)(0)(0)()()d P Pababab 24.221222222(,)(0)(0)()()22d P Paaaaaaaa 

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Section 1.2:The Distance and Midpoint Formulas925.( 2,5),(1,3),( 1, 0)ABC  222222222222(,)1( 2)(35)3( 2)9413(,)11(03)( 2)( 3)4913(,)1( 2)(05)1( 5)12526d A Bd B Cd A C       Verifying that∆ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)1313261313262626d A Bd B Cd A CThe area of a triangle is12Abh. In thisproblem, 1(,)(,)2111313132213 square units2Ad A Bd B C26.( 2, 5),(12, 3),(10,11)ABC 222222222222(,)12( 2)(35)14( 2)1964200102(,)1012( 113)( 2)( 14)4196200102(,)10( 2)( 115)12(16)14425640020d A Bd B Cd A C       Verifying that∆ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)10210220200200400400400d A Bd B Cd A CThe area of a triangle is12Abh. In thisproblem, 1(,)(,)21 102 10221 100 2100 square units2Ad A Bd B C

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Chapter 1:Graphs1027.(5,3),(6, 0),(5,5)ABC 222222222222(,)6(5)(03)11(3)1219130(,)56(50)(1)512526(,)5(5)(53)1021004104226d A Bd B Cd A C   Verifying that∆ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)1042613010426130130130d A Cd B Cd A BThe area of a triangle is12Abh. In thisproblem, 1(,)(,)211042621 2262621 2 26226 square unitsAd A Cd B C28.( 6, 3),(3,5),( 1, 5)ABC  222222222222(,)3( 6)( 53)9( 8)8164145(,)13(5( 5))( 4)1016100116229(,)1(6)(53)5225429d A Bd B Cd A C       Verifying that∆ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)29229145294 2914529116145145145d A Cd B Cd A BThe area of a triangle is12Abh. In thisproblem, 1(,)(,)2129 22921 2 29229 square unitsAd A Cd B C

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