Solution Manual for Algebra and Trigonometry Enhanced with Graphing Utilities, 8th Edition

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SOLUTIONSMANUALTIMBRITTJackson State Community CollegeALGEBRA ANDTRIGONOMETRYENHANCED WITHGRAPHINGUTILITIESEIGHTHEDITIONMichael SullivanChicago State UniversityMichael Sullivan IIIJoliet Junior College

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Table of ContentsChapter RReviewR.1Real Numbers .......................................................................................................................... 1R.2Algebra Essentials ................................................................................................................... 5R.3Geometry Essentials .............................................................................................................. 11R.4Polynomials ........................................................................................................................... 16R.5Factoring Polynomials ........................................................................................................... 23R.6Synthetic Division ................................................................................................................. 28R.7Rational Expressions ............................................................................................................. 30R.8nth Roots; Rational Exponents .............................................................................................. 40Chapter 1Graphs, Equations, and Inequalities1.1Graphing Utilities; Introduction to Graphing Equations ....................................................... 501.2Solving Equations Using a Graphing Utility; Linear and Rational Equations....................... 571.3Quadratic Equations .............................................................................................................. 711.4Complex Numbers; Quadratic Equations in the Complex Number System .......................... 901.5Radical Equations; Equations Quadratic in Form; Absolute Value Equations;Factorable Equations ............................................................................................................. 961.6Problem Solving: Interest, Mixture, Uniform Motion, and Constant Rate Jobs.................. 1171.7Solving Inequalities ............................................................................................................. 125Chapter Review............................................................................................................................ 137Chapter Test ................................................................................................................................. 147Chapter Projects ........................................................................................................................... 151Chapter 2Graphs2.1The Distance and Midpoint Formulas ................................................................................. 1522.2Intercepts; Symmetry; Graphing Key Equations ................................................................. 1652.3Lines .................................................................................................................................... 1822.4Circles.................................................................................................................................. 2002.5Variation .............................................................................................................................. 215Chapter Review............................................................................................................................ 220Chapter Test ................................................................................................................................. 226Cumulative Review...................................................................................................................... 228Chapter Projects ........................................................................................................................... 230Chapter 3Functions and Their Graphs3.1Functions ............................................................................................................................. 2313.2The Graph of a Function...................................................................................................... 2493.3Properties of Functions ........................................................................................................ 2583.4Library of Functions; Piecewise-defined Functions ............................................................ 2753.5Graphing Techniques: Transformations .............................................................................. 2873.6Mathematical Models: Building Functions.......................................................................... 305Chapter Review............................................................................................................................ 313Chapter Test ................................................................................................................................. 320Cumulative Review...................................................................................................................... 323Chapter Projects ........................................................................................................................... 327

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Chapter 4Linear and Quadratic Functions4.1Properties of Linear Functions and Linear Models.............................................................. 3294.2Building Linear Models from Data...................................................................................... 3404.3Quadratic Functions and Their Properties ........................................................................... 3464.4Build Quadratic Models from Verbal Descriptions and from Data ..................................... 3704.5Inequalities Involving Quadratic Functions......................................................................... 378Chapter Review............................................................................................................................ 397Chapter Test ................................................................................................................................. 405Cumulative Review...................................................................................................................... 408Chapter Projects ........................................................................................................................... 411Chapter 5Polynomial and Rational Functions5.1Polynomial Functions .......................................................................................................... 4145.2The Graph of a Polynomial Function; Models .................................................................... 4255.3The Real Zeros of a Polynomial Function ........................................................................... 4455.4Complex Zeros; Fundamental Theorem of Algebra ............................................................ 4825.5Properties of Rational Functions ......................................................................................... 4915.6The Graph of a Rational Function ....................................................................................... 5015.7Polynomial and Rational Inequalities .................................................................................. 557Chapter Review............................................................................................................................ 579Chapter Test ................................................................................................................................. 593Cumulative Review...................................................................................................................... 597Chapter Projects ........................................................................................................................... 602Chapter 6Exponential and Logarithmic Functions6.1Composite Functions ........................................................................................................... 6036.2One-to-One Functions; Inverse Functions........................................................................... 6216.3Exponential Functions ......................................................................................................... 6446.4Logarithmic Functions......................................................................................................... 6656.5Properties of Logarithms ..................................................................................................... 6876.6Logarithmic and Exponential Equations.............................................................................. 6966.7Financial Models ................................................................................................................. 7176.8Exponential Growth and Decay Models; Newton’s Law; Logistic Growthand Decay Models ............................................................................................................... 7256.9Building Exponential, Logarithmic, and Logistic Models from Data ................................. 736Chapter Review............................................................................................................................ 741Chapter Test ................................................................................................................................. 763Cumulative Review...................................................................................................................... 757Chapter Projects ........................................................................................................................... 760

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Chapter 7Trigonometric Functions7.1Angles, Arc Length, and Circular Motion ........................................................................... 7637.2Right Triangle Trigonometry............................................................................................... 7727.3Computing the Values of Trigonometric Functions of Acute Angles ................................. 7887.4Trigonometric Functions of Any Angle............................................................................... 8017.5Unit Circle Approach; Properties of the Trigonometric Functions...................................... 8177.6Graphs of the Sine and Cosine Functions............................................................................ 8267.7Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions................................... 8487.8Phase Shift; Sinusoidal Curve Fitting.................................................................................. 858Chapter Review............................................................................................................................ 871Chapter Test ................................................................................................................................. 880Cumulative Review...................................................................................................................... 884Chapter Projects ........................................................................................................................... 888Chapter 8Analytic Trigonometry8.1The Inverse Sine, Cosine, and Tangent Functions............................................................... 8918.2The Inverse Trigonometric Functions (Continued) ............................................................. 9058.3Trigonometric Equations ..................................................................................................... 9178.4Trigonometric Identities ...................................................................................................... 9388.5Sum and Difference Formulas ............................................................................................. 9518.6Double-Angle and Half-Angle Formulas ............................................................................ 9768.7Product-to-Sum and Sum-to-Product Formulas................................................................. 1004Chapter Review.......................................................................................................................... 1017Chapter Test ............................................................................................................................... 1032Cumulative Review.................................................................................................................... 1037Chapter Projects ......................................................................................................................... 1043Chapter 9Applications of Trigonometric Functions9.1Applications Involving Right Triangles ............................................................................ 10479.2The Law of Sines............................................................................................................... 10559.3The Law of Cosines ........................................................................................................... 10709.4Area of a Triangle.............................................................................................................. 10829.5Simple Harmonic Motion; Damped Motion; Combining Waves ...................................... 1092Chapter Review.......................................................................................................................... 1102Chapter Test ............................................................................................................................... 1108Cumulative Review.................................................................................................................... 1111Chapter Projects ......................................................................................................................... 1117Chapter 10Polar Coordinates; Vectors10.1 Polar Coordinates .............................................................................................................. 112110.2 Polar Equations and Graphs............................................................................................... 113010.3 The Complex Plane; De Moivre’s Theorem...................................................................... 115910.4 Vectors............................................................................................................................... 117210.5 The Dot Product................................................................................................................. 1185Chapter Review.......................................................................................................................... 1191Chapter Test ............................................................................................................................... 1200Cumulative Review.................................................................................................................... 1203Chapter Projects ......................................................................................................................... 1205

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Chapter 11Analytic Geometry11.2 The Parabola ...................................................................................................................... 120911.3 The Ellipse......................................................................................................................... 122511.4 The Hyperbola ................................................................................................................... 124211.5 Rotation of Axes; General Form of a Conic ...................................................................... 126211.6 Polar Equations of Conics ................................................................................................. 127511.7 Plane Curves and Parametric Equations ............................................................................ 1284Chapter Review.......................................................................................................................... 1299Chapter Test ............................................................................................................................... 1308Cumulative Review.................................................................................................................... 1313Chapter Projects ......................................................................................................................... 1315Chapter 12Systems of Equations and Inequalities12.1 Systems of Linear Equations: Substitution and Elimination ............................................. 131912.2 Systems of Linear Equations: Matrices ............................................................................. 134212.3 Systems of Linear Equations: Determinants...................................................................... 136712.4 Matrix Algebra .................................................................................................................. 138112.5 Partial Fraction Decomposition ......................................................................................... 140012.6 Systems of Nonlinear Equations........................................................................................ 141912.7 Systems of Inequalities ...................................................................................................... 144712.8 Linear Programming.......................................................................................................... 1462Chapter Review.......................................................................................................................... 1476Chapter Test ............................................................................................................................... 1491Cumulative Review.................................................................................................................... 1499Chapter Projects ......................................................................................................................... 1503Chapter 13Sequences; Induction; the Binomial Theorem13.1 Sequences .......................................................................................................................... 150513.2 Arithmetic Sequences ........................................................................................................ 151913.3 Geometric Sequences; Geometric Series........................................................................... 152813.4 Mathematical Induction ..................................................................................................... 153913.5 The Binomial Theorem...................................................................................................... 1549Chapter Review.......................................................................................................................... 1556Chapter Test ............................................................................................................................... 1560Cumulative Review.................................................................................................................... 1563Chapter Projects ......................................................................................................................... 1566Chapter 14Counting and Probability14.1 Counting ............................................................................................................................ 156914.2 Permutations and Combinations ........................................................................................ 157214.3 Probability ......................................................................................................................... 1577Chapter Review.......................................................................................................................... 1584Chapter Test ............................................................................................................................... 1586Cumulative Review.................................................................................................................... 1587Chapter Projects ......................................................................................................................... 1590

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1Chapter RReviewSection R.11.rational2.45 634303313.Distributive4.c5.a6.b7.True8.False; The Zero-Product Property states that if aproduct equals 0, then at least one of the factorsmust equal 0.9.False; 6 is the Greatest Common Factor of 12and 18. The Least Common Multiple is thesmallest value that both numbers will divideevenly. The LCM for 12 and 18 is 36.10.True11.1, 3, 4,5, 92, 4, 6, 7,81, 2,3, 4, 5, 6, 7,8, 9AB12.1, 3, 4,5, 91, 3, 4, 61, 3, 4, 5, 6, 9AC13. 1, 3, 4,5, 92, 4, 6, 7,84AB14.1, 3, 4,5, 91, 3, 4, 61, 3, 4AC15.()1, 3, 4,5, 92, 4, 6, 7,81,3, 4, 61, 2,3, 4,5, 6, 7,8,91,3, 4, 61, 3, 4, 6ABC16. ()1, 3, 4,5, 92, 4, 6, 7,81,3, 4, 641, 3, 4, 61,3, 4, 6ABC17.0, 2, 6, 7, 8A18.0, 2, 5, 7, 8, 9C19. 1, 3, 4, 5, 92, 4, 6, 7, 840, 1, 2, 3, 5, 6, 7, 8, 9AB20.2, 4, 6, 7, 81, 3, 4, 61, 2, 3, 4, 6, 7, 80, 5, 9BC21.0, 2, 6, 7, 80, 1, 3, 5, 90, 1, 2, 3, 5, 6, 7, 8, 9AB22.0, 1, 3, 5, 90, 2, 5, 7, 8, 90, 5, 9BC23.a.2,5b.6, 2,5c.16,,1.333...1.3, 2,52 d. e.16,,1.333...1.3,, 2,52 24.a. 1b.0,1c.5 , 2.060606...2.06,1.25, 0,13d.5

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Chapter R:Review2e.5 , 2.060606...2.06,1.25, 0,1,5325.a. 1b.0,1c.1 1 10,1,,,2 3 4d.Nonee.1 1 10,1,,,2 3 426.a.Noneb.1c.1.3,1.2,1.1,1d.Nonee.1.3,1.2,1.1,127.a.Noneb.Nonec.Noned.12,,21,2e.12,,21,228.a.Noneb.Nonec.110.32d.2,2e.12,2,10.3229.a.18.953b.18.95230.a.25.861b.25.86131.a.28.653b.28.65332.a.99.052b.99.05233.a.0.063b.0.06234.a.0.054b.0.05335.a.9.999b.9.99836.a.1.001b.1.00037.a.0.429b.0.42838.a.0.556b.0.55539.a.34.733b.34.73340.a.16.200b.16.20041.32542.5 21043.23 4x44.322y45.312y46.24 6x47.26x48.26y49.62x50.26x51.94252752.64323553.64 36126 54.84 288055.185 21810856.10010 2100208057.112113433358.14132222

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Section R.1:Real Numbers359. 63 52326152161711 60.2834232836328183210320323   61.4956 73414423564231431762.14 3221122211211 63. 1062 283210645210252107210144 64.25 46342206118618612     65.1153212266.1154933367.4812653268.2421532 69.3 103 2 535215 3 7255327770.535 359 103 3 5 2333516271.6102 3 5 22325275 5 3 952553445972.21 1003 7 4 25325325 37 4252532873.321582345202074.418311326675.7449328187565676.8151613515192181877.51103131812363678.2864046159454579.5825643913241512012040  80.329451421424281.329812015606082.631215335147070 83.5185275 9 35918119 2 111127 3915222 1184.5215355 7 5572127 3 2235 572563 2

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Chapter R:Review485.1417417211372121212186.24122 22222235635 3 235 3 2315252102102123515151515154 34 345 35 3587.33233636232 481484842812312315888888.513513 513 513 621623 223 22515142222289.64624xx90.4 2184xx91.244xxxx92.243412xxxx93.31312 3222242422 222 32312222xxxxx94.21213 23333363633 23 231233 22xxxxx95.222442868xxxxxxx96.22515565xxxxxxx97.2292727186321163xxxxxxx98.2231531553145xxxxxxx99.228228161016xxxxxxx100.224224868xxxxxxx101.2223 (5 )36031536015604 xxkxxxxkxxxkxk102.2222222222222()(3 )41233412(3)3412(3)3412(2 )3412242xkxkxxxkxkxkxxxxkkkxxxxkkkxxxxkkxxkk103.23232355xxxxxxx104.23 421214since multiplication comes before addition in theorder of operations for real numbers.2345 420since operations inside parentheses come beforemultiplication in the order of operations for realnumbers.105.2 3 42 1224  2 32 46848106.4371257, but434 53 220626132.6251010105107.Subtraction is not commutative; forexample:231132 .

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Section R.2:Algebra Essentials5108.Subtraction is not associative; forexample:52124521.109.Division is not commutative; for example:2332.110.Division is not associative; forexample:1222623, but122212112.111.The Symmetric Property implies that if 2 =x,thenx= 2.112.From theprinciple of substitution,if5x, then  222552525530xxxxxxx113.There are no real numbers that are both rationaland irrational, since an irrational number, bydefinition, is a number that cannot be expressedas the ratio of two integers; that is, not a rationalnumberEvery real number is either a rational number oran irrational number, since the decimal form of areal number either involves an infinitelyrepeating pattern of digits or an infinite, non-repeating string of digits.114.The sum of an irrational number and a rationalnumber must be irrational. Otherwise, theirrational number would then be the difference oftwo rational numbers, and therefore would haveto be rational.115.Answers will vary.116.Since 1 day = 24 hours, we compute12997541.541624.Now we only need to consider the decimal partof the answer in terms of a 24 hour day. That is,0.54162413hours. So it must be 13 hourslater than 12 noon, which makes the time 1 AMCST.117.Answers will vary.Section R.21.variable2.origin3.strict4.base; exponent (or power)5.31.2345678106.d7.a8.b9.True10.False; the absolute value of a real number isnonnegative.00which is not a positivenumber.11.False; a number in scientific notation isexpressed as the product of a number, x,110xor101x , and a power of 10.12.True13.523414.13322315.10216.5617.12 18.532 19.3.14 20.21.41

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Chapter R:Review621.10.5222.10.33323.20.67324.10.25425.0x26.0z27.2x28.5y 29.1x30.2x31.Graph on the number line:2x 32.Graph on the number line:4x33.Graph on the number line:1x 34.Graph on the number line:7x35.(,)(0,1)1011d C Dd36.(,)(0,3)3033d C Ad37.(,)(1,3)3122d D Ed38.(,)(0,3)3033d C Ed39.(,)( 3,3)3( 3)66d A Ed 40.(,)(1,1)1122d D Bd41.222 3264xy  42.33(2)3633xy  43.525(2)(3)230228xy  44.22(2)(2)(3)462xxy   45.2(2)4242355xxy46.23112355xyxy 47.3(2)2(3)66320022355xyy48.2(2)343237333xy 49.3(2)11xy 50.3(2)55xy 51.32325xy 52.32321xy 53.33133xx54.22122yy 55.454(3)5(2)12102222xy56.323(3)2(2)9455xy57.454(3)5(2)1210121022xy 

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Section R.2:Algebra Essentials758.323 3223 32 29413xy59.21xxPart (c) must be excluded. The value0xmustbe excluded from the domain because it causesdivision by 0.60.21xxPart (c) must be excluded. The value0xmustbe excluded from the domain because it causesdivision by 0.61.2(3)(3)9xxxxxPart (a) ,3x, must be excluded because itcauses the denominator to be 0.62.29xxNone of the given values are excluded. Thedomain is all real numbers.63.221xxNone of the given values are excluded. Thedomain is all real numbers.64.332(1)(1)1xxxxxParts (b) and (d) must be excluded. The values1, and1xx must be excluded from thedomain because they cause division by 0.65.223510510(1)(1)xxxxx xxxxParts (b), (c), and (d) must be excluded. Thevalues0,1, and1xxx must be excludedfrom the domain because they cause division by0.66.22329191(1)xxxxxxx xPart (c) must be excluded. The value0xmustbe excluded from the domain because it causesdivision by 0.67.45x5xmust be exluded because it makes thedenominator equal 0.Domain5x x68.64x4x must be excluded sine it makes thedenominator equal 0.Domain4x x 69.4xx4x must be excluded sine it makes thedenominator equal 0.Domain4x x 70.26xx6xmust be excluded sine it makes thedenominator equal 0.Domain6x x71.555(32)(3232)(0)0 C999CF72.555(32)(21232)(180)100 C999CF73.555(32)(7732)(45)25 C999CF74.55(32)(432)995 ( 36)920 CCF 75.2( 9)( 9)( 9)81 76.224(4)16  77.22114164

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Chapter R:Review878.22114164  79.64642211333393 80.2323144444 81.131334446482.31313222883.2100101084.2366685.2444 86.2333 87.2242489981xxx88.122211444xxx 89.42222121422xx yxyx yy90.333113333yxyxyxyx91.2523541134xyyxyxyxx y92.22 11 231231xyxyxyx yx y 93.253533372723 1573222122(4)()16( 3)27162716271627   yx zy x zx y zx y zxyzx yzx zy94.21211344241 11621624()428481212xy zxyzx yx yxyzxyzx y z  95.22233266132223233923224yxxxxyyxyy96.3332222223326363255666562161255yxxyxyxxyy97.12 22241xxyy 98.13133322yxyx99.222221415xy 100. 2222214 14x y101.2222124xy 102. 2222111xy 103.222xx104.22xx105.222221415xy 106.2221213xyxy 107.1122yx

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Section R.2:Algebra Essentials9108.211xy109.If2,x323223542 23 25 24161210410xxxIf1,x323223542 13 15 1423540xxx110.If1,x32324324 13 11243128xxxIf2,x32324324 23 22232122244xxx111.4444(666)666381222(222)112.333333331(0.1) (20)2 101012101028113.6(8.2)304, 006.671114.5(3.7)693.440115.3(6.1)0.004116.5(2.2)0.019117.6(2.8)481.890118.6(2.8)481.890 119.4(8.11)0.000120.4(8.11)0.000 121.2454.24.54210122.132.143.21410123.20.0131.310124.30.004214.2110125.432,1553.215510126.421, 2102.12110127.40.0004234.2310128.20.05145.1410129.46.151061,500130.39.7109700131.31.214100.001214132.49.88100.000988133.81.110110, 000, 000134.24.11210411.2135.28.1100.081136.16.453100.6453137.Alw138.2Plw139.Cd140.12Abh141.234Ax142.3Px143.343Vr

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Chapter R:Review10144.24Sr145.3Vx146.26Sx147.a.If1000,x4000240002(1000)40002000$6000CxThe cost of producing 1000 watches is$6000.b.If2000,x4000240002(2000)40004000$8000CxThe cost of producing 2000 watches is$8000.148.210801202560325$98His balance at the end of the month was $98.149.We want the difference betweenxand 4 to be atleast 6 units. Since we don’t care whether thevalue forxis larger or smaller than 4, we takethe absolute value of the difference. We want theinequality to be non-strict since we are dealingwith an ‘at least’ situation. Thus, we have46x150.We want the difference betweenxand 2 to bemore than 5 units. Since we don’t care whetherthe value forxis larger or smaller than 2, wetake the absolute value of the difference. Wewant the inequality to be strict since we aredealing with a ‘more than’ situation. Thus, wehave25x151.a.110108110225x108 volts is acceptable.b.110104110665x104 volts isnotacceptable.152.a.220214220668x214 volts is acceptable.b.22020922011118x209 volts isnotacceptable.153.a.32.99930.0010.0010.01xA radius of 2.999 centimeters is acceptable.b.32.8930.110.110.01xA radius of 2.89 centimeters isnotacceptable.154.a.98.69798.61.61.61.5x97˚F is unhealthy.b.98.610098.61.41.41.5x100˚F isnotunhealthy.155.The distance from Earth to the Moon is about8410400, 000, 000meters.156.The height of Mt. Everest is about388488.84810meters.157.The wavelength of visible light is about75100.0000005meters.158.The diameter of an atom is about101100.0000000001meters.159.The diameter is about20.04034.0310inches.160.The tiniest motor is less than50.00004410millimeters tall.

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