Skip to content
Textbook Guides

Solution Manual for Precalculus, 8th Edition

Take the stress out of textbook problems with Solution Manual for Precalculus, 8th Edition, a complete guide to solving every question.

Preview 16 of 1468 pages

See the first 16 pages exactly as they appear in the document.

  1. Page 1
    Read page 1 as text

    ’ S S OLUTIONS M ANUAL T IM B RITT Jackson State Community College P RECALCULUS N INTH E DITION Michael Sullivan Chicago State University

  2. Page 2
  3. Page 3
    Read page 3 as text

    Table of Contents Preface Chapter 1 Graphs 1.1 The Distance and Midpoint Formulas......................................................................................... 1 1.2 Graphs of Equations in Two Variables; Intercepts; Symmetry ................................................ 12 1.3 Lines ......................................................................................................................................... 24 1.4 Circles ....................................................................................................................................... 41 Chapter Review ............................................................................................................................... 52 Chapter Test ..................................................................................................................................... 62 Chapter Projects ............................................................................................................................... 64 Chapter 2 Functions and Their Graphs 2.1 Functions .................................................................................................................................. 66 2.2 The Graph of a Function ........................................................................................................... 80 2.3 Properties of Functions ............................................................................................................. 87 2.4 Library of Functions; Piecewise-defined Functions ............................................................... 102 2.5 Graphing Techniques: Transformations ................................................................................. 113 2.6 Mathematical Models: Building Functions ............................................................................. 129 Chapter Review ............................................................................................................................. 135 Chapter Test ................................................................................................................................... 148 Cumulative Review ....................................................................................................................... 152 Chapter Projects ............................................................................................................................. 155 Chapter 3 Linear and Quadratic Functions 3.1 Linear Functions and Their Properties.................................................................................... 157 3.2 Linear Models: Building Linear Functions from Data ........................................................... 167 3.3 Quadratic Functions and Their Properties .............................................................................. 172 3.4 Build Quadratic Models from Verbal Descriptions and from Data ........................................ 190 3.5 Inequalities Involving Quadratic Functions ............................................................................ 198 Chapter Review ............................................................................................................................. 216 Chapter Test ................................................................................................................................... 229 Cumulative Review........................................................................................................................ 230 Chapter Projects ............................................................................................................................. 233 Chapter 4 Polynomial and Rational Functions 4.1 Polynomial Functions and Models.......................................................................................... 236 4.2 Properties of Rational Functions............................................................................................. 258 4.3 The Graph of a Rational Function .......................................................................................... 265 4.4 Polynomial and Rational Inequalities ..................................................................................... 312 4.5 The Real Zeros of a Polynomial Function .............................................................................. 328 4.6 Complex Zeros; Fundamental Theorem of Algebra ............................................................... 355 Chapter Review ............................................................................................................................. 362 Chapter Test ................................................................................................................................... 391 Cumulative Review........................................................................................................................ 395 Chapter Projects ............................................................................................................................. 400

  4. Page 4
    Read page 4 as text

    Chapter 5 Exponential and Logarithmic Functions 5.1 Composite Functions .............................................................................................................. 402 5.2 One-to-One Functions; Inverse Functions .............................................................................. 418 5.3 Exponential Functions ............................................................................................................ 437 5.4 Logarithmic Functions ............................................................................................................ 456 5.5 Properties of Logarithms ........................................................................................................ 476 5.6 Logarithmic and Exponential Equations................................................................................. 484 5.7 Financial Models .................................................................................................................... 501 5.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growth and Decay Models .................................................................................................................... 508 5.9 Building Exponential, Logarithmic, and Logistic Models from Data .................................... 515 Chapter Review ............................................................................................................................. 520 Chapter Test ................................................................................................................................... 540 Cumulative Review........................................................................................................................ 544 Chapter Projects ............................................................................................................................. 548 Chapter 6 Trigonometric Functions 6.1 Angles and Their Measure ...................................................................................................... 550 6.2 Trigonometric Functions: Unit Circle Approach .................................................................... 557 6.3 Properties of the Trigonometric Functions ............................................................................. 572 6.4 Graphs of the Sine and Cosine Functions ............................................................................... 584 6.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions...................................... 603 6.6 Phase Shift; Sinusoidal Curve Fitting ..................................................................................... 611 Chapter Review ............................................................................................................................. 622 Chapter Test ................................................................................................................................... 637 Cumulative Review........................................................................................................................ 640 Chapter Projects ............................................................................................................................. 644 Chapter 7 Analytic Trigonometry 7.1 The Inverse Sine, Cosine, and Tangent Functions .................................................................. 648 7.2 The Inverse Trigonometric Functions (Continued) ................................................................ 658 7.3 Trigonometric Equations ........................................................................................................ 669 7.4 Trigonometric Identities ......................................................................................................... 688 7.5 Sum and Difference Formulas ................................................................................................ 699 7.6 Double-angle and Half-angle Formulas .................................................................................. 721 7.7 Product-to-Sum and Sum-to-Product Formulas...................................................................... 744 Chapter Review ............................................................................................................................. 754 Chapter Test ................................................................................................................................... 778 Cumulative Review........................................................................................................................ 782 Chapter Projects ............................................................................................................................. 787 Chapter 8 Applications of Trigonometric Functions 8.1 Right Triangle Trigonometry; Applications ........................................................................... 791 8.2 The Law of Sines .................................................................................................................... 803 8.3 The Law of Cosines ................................................................................................................ 816 8.4 Area of a Triangle ................................................................................................................... 825 8.5 Simple Harmonic Motion; Damped Motion; Combining Waves ........................................... 832 Chapter Review ............................................................................................................................. 840 Chapter Test ................................................................................................................................... 852 Cumulative Review........................................................................................................................ 855 Chapter Projects ............................................................................................................................. 861

  5. Page 5
    Read page 5 as text

    Chapter 9 Polar Coordinates; Vectors 9.1 Polar Coordinates.................................................................................................................... 865 9.2 Polar Equations and Graphs.................................................................................................... 872 9.3 The Complex Plane; DeMoivre’s Theorem ............................................................................ 900 9.4 Vectors .................................................................................................................................... 910 9.5 The Dot Product...................................................................................................................... 920 9.6 Vectors in Space ..................................................................................................................... 924 9.7 The Cross Product................................................................................................................... 930 Chapter Review ............................................................................................................................. 940 Chapter Test ................................................................................................................................... 958 Cumulative Review........................................................................................................................ 962 Chapter Projects ............................................................................................................................. 964 Chapter 10 Analytic Geometry 10.2 The Parabola ......................................................................................................................... 968 10.3 The Ellipse ............................................................................................................................ 982 10.4 The Hyperbola ...................................................................................................................... 997 10.5 Rotation of Axes; General Form of a Conic ....................................................................... 1015 10.6 Polar Equations of Conics................................................................................................... 1027 10.7 Plane Curves and Parametric Equations ............................................................................. 1034 Chapter Review ........................................................................................................................... 1045 Chapter Test ................................................................................................................................. 1063 Cumulative Review...................................................................................................................... 1068 Chapter Projects ........................................................................................................................... 1070 Chapter 11 Systems of Equations and Inequalities 11.1 Systems of Linear Equations: Substitution and Elimination............................................... 1074 11.2 Systems of Linear Equations: Matrices .............................................................................. 1093 11.3 Systems of Linear Equations: Determinants ....................................................................... 1115 11.4 Matrix Algebra.................................................................................................................... 1127 11.5 Partial Fraction Decomposition .......................................................................................... 1144 11.6 Systems of Nonlinear Equations ......................................................................................... 1157 11.7 Systems of Inequalities ....................................................................................................... 1184 11.8 Linear Programming ........................................................................................................... 1198 Chapter Review ........................................................................................................................... 1209 Chapter Test ................................................................................................................................. 1235 Cumulative Review...................................................................................................................... 1244 Chapter Projects ........................................................................................................................... 1248 Chapter 12 Sequences; Induction; the Binomial Theorem 12.1 Sequences ........................................................................................................................... 1250 12.2 Arithmetic Sequences ......................................................................................................... 1259 12.3 Geometric Sequences; Geometric Series ............................................................................ 1265 12.4 Mathematical Induction ...................................................................................................... 1275 12.5 The Binomial Theorem ....................................................................................................... 1282 Chapter Review ........................................................................................................................... 1287 Chapter Test ................................................................................................................................. 1294 Cumulative Review...................................................................................................................... 1297 Chapter Projects ........................................................................................................................... 1300

  6. Page 6
    Read page 6 as text

    Chapter 13 Counting and Probability 13.1 Counting ............................................................................................................................. 1303 13.2 Permutations and Combinations ......................................................................................... 1304 13.3 Probability ........................................................................................................................... 1308 Chapter Review ........................................................................................................................... 1313 Chapter Test ................................................................................................................................. 1315 Cumulative Review...................................................................................................................... 1317 Chapter Projects ........................................................................................................................... 1320 Chapter 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function 14.1 Finding Limits Using Tables and Graphs ........................................................................... 1322 14.2 Algebra Techniques for Finding Limits .............................................................................. 1327 14.3 One-sided Limits; Continuous Functions ........................................................................... 1331 14.4 The Tangent Problem; The Derivative................................................................................ 1339 14.5 The Area Problem; The Integral ......................................................................................... 1348 Chapter Review ........................................................................................................................... 1362 Chapter Test ................................................................................................................................. 1374 Chapter Projects ........................................................................................................................... 1378 Appendix A Review A.1 Algebra Essentials ................................................................................................................ 1383 A.2 Geometry Essentials............................................................................................................. 1388 A.3 Polynomials ......................................................................................................................... 1393 A.4 Synthetic Division ................................................................................................................ 1402 A.5 Rational Expressions............................................................................................................ 1404 A.6 Solving Equations ................................................................................................................ 1409 A.7 Complex Numbers; Quadratic Equations in the Complex Number System ........................ 1423 A.8 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications ..... 1428 A.9 Interval Notation; Solving Inequalities ................................................................................ 1436 A.10 n th Roots; Rational Exponents............................................................................................ 1447 Appendix B Graphing Utilities B.1 The Viewing Rectangle ........................................................................................................ 1456 B.2 Using a Graphing Utility to Graph Equations ...................................................................... 1457 B.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry ............................ 1461 B.5 Square Screens ..................................................................................................................... 1463

  7. Page 7
    Read page 7 as text

    1 Chapter 1 Graphs Section 1.1 1. 0 2. ( ) 5 3 8 8 − − = = 3. 2 2 3 4 25 5 + = = 4. 2 2 2 11 60 121 3600 3721 61 + = + = = Since the sum of the squares of two of the sides of the triangle equals the square of the third side, the triangle is a right triangle. 5. 1 2 bh 6. true 7. x- coordinate; y -coordinate 8. quadrants 9. midpoint 10. False; the distance between two points is never negative. 11. False; points that lie in Quadrant IV will have a positive x -coordinate and a negative y -coordinate. The point ( ) 1, 4 − lies in Quadrant II. 12. True; 1 2 1 2 , 2 2 x x y y M + +   =     13. (a) Quadrant II (b) x -axis (c) Quadrant III (d) Quadrant I (e) y -axis (f) Quadrant IV 14. (a) Quadrant I (b) Quadrant III (c) Quadrant II (d) Quadrant I (e) y -axis (f) x -axis 15. The points will be on a vertical line that is two units to the right of the y -axis.

  8. Page 8
    Read page 8 as text

    Chapter 1: Graphs 2 16. The points will be on a horizontal line that is three units above the x -axis. 17. 2 2 1 2 2 2 ( , ) (2 0) (1 0) 2 1 4 1 5 d P P = − + − = + = + = 18. 2 2 1 2 2 2 ( , ) ( 2 0) (1 0) ( 2) 1 4 1 5 d P P = − − + − = − + = + = 19. 2 2 1 2 2 2 ( , ) ( 2 1) (2 1) ( 3) 1 9 1 10 d P P = − − + − = − + = + = 20. ( ) 2 2 1 2 2 2 ( , ) 2 ( 1) (2 1) 3 1 9 1 10 d P P = − − + − = + = + = 21. ( ) ( ) ( ) 2 2 1 2 2 2 ( , ) (5 3) 4 4 2 8 4 64 68 2 17 d P P = − + − − = + = + = = 22. ( ) ( ) ( ) ( ) 2 2 1 2 2 2 ( , ) 2 1 4 0 3 4 9 16 25 5 d P P = − − + − = + = + = = 23. ( ) 2 2 1 2 2 2 ( , ) 6 ( 3) (0 2) 9 ( 2) 81 4 85 d P P = − − + − = + − = + = 24. ( ) ( ) 2 2 1 2 2 2 ( , ) 4 2 2 ( 3) 2 5 4 25 29 d P P = − + − − = + = + = 25. ( ) 2 2 1 2 2 2 ( , ) (6 4) 4 ( 3) 2 7 4 49 53 d P P = − + − − = + = + = 26. ( ) ( ) 2 2 1 2 2 2 ( , ) 6 ( 4) 2 ( 3) 10 5 100 25 125 5 5 d P P = − − + − − = + = + = = 27. 2 2 1 2 2 2 2 2 ( , ) (0 ) (0 ) ( ) ( ) d P P a b a b a b = − + − = − + − = + 28. 2 2 1 2 2 2 2 2 2 ( , ) (0 ) (0 ) ( ) ( ) 2 2 d P P a a a a a a a a = − + − = − + − = + = = 29. ( 2,5), (1,3), ( 1, 0) A B C = − = = − ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 ( , ) 1 ( 2) (3 5) 3 ( 2) 9 4 13 ( , ) 1 1 (0 3) ( 2) ( 3) 4 9 13 ( , ) 1 ( 2) (0 5) 1 ( 5) 1 25 26 d A B d B C d A C = − − + − = + − = + = = − − + − = − + − = + = = − − − + − = + − = + = Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem: [ ] [ ] [ ] ( ) ( ) ( ) 2 2 2 2 2 2 ( , ) ( , ) ( , ) 13 13 26 13 13 26 26 26 d A B d B C d A C + = + = + = = The area of a triangle is 1 2 A bh = ⋅ . In this problem,

  9. Page 9
    Read page 9 as text

    Section 1.1: The Distance and Midpoint Formulas 3 [ ] [ ] 1 ( , ) ( , ) 2 1 1 13 13 13 2 2 13 square units 2 A d A B d B C = ⋅ ⋅ = ⋅ = ⋅ ⋅ = 30. ( 2, 5), (12, 3), (10, 11) A B C = − = = − ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 ( , ) 12 ( 2) (3 5) 14 ( 2) 196 4 200 10 2 ( , ) 10 12 ( 11 3) ( 2) ( 14) 4 196 200 10 2 ( , ) 10 ( 2) ( 11 5) 12 ( 16) 144 256 400 20 d A B d B C d A C = − − + − = + − = + = = = − + − − = − + − = + = = = − − + − − = + − = + = = Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem: [ ] [ ] [ ] ( ) ( ) ( ) 2 2 2 2 2 2 ( , ) ( , ) ( , ) 10 2 10 2 20 200 200 400 400 400 d A B d B C d A C + = + = + = = The area of a triangle is 1 2 A bh = . In this problem, [ ] [ ] 1 ( , ) ( , ) 2 1 10 2 10 2 2 1 100 2 2 100 square units A d A B d B C = ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ = 31. ( 5,3), (6, 0), (5,5) A B C = − = = ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 ( , ) 6 ( 5) (0 3) 11 ( 3) 121 9 130 ( , ) 5 6 (5 0) ( 1) 5 1 25 26 ( , ) 5 ( 5) (5 3) 10 2 100 4 104 2 26 d A B d B C d A C = − − + − = + − = + = = − + − = − + = + = = − − + − = + = + = = Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem: [ ] [ ] [ ] ( ) ( ) ( ) 2 2 2 2 2 2 ( , ) ( , ) ( , ) 104 26 130 104 26 130 130 130 d A C d B C d A B + = + = + = = The area of a triangle is 1 2 A bh = . In this

  10. Page 10
    Read page 10 as text

    Chapter 1: Graphs 4 problem, [ ] [ ] 1 ( , ) ( , ) 2 1 104 26 2 1 2 26 26 2 1 2 26 2 26 square units A d A C d B C = ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ = 32. ( 6, 3), (3, 5), ( 1, 5) A B C = − = − = − ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 ( , ) 3 ( 6) ( 5 3) 9 ( 8) 81 64 145 ( , ) 1 3 (5 ( 5)) ( 4) 10 16 100 116 2 29 ( , ) 1 ( 6) (5 3) 5 2 25 4 29 d A B d B C d A C = − − + − − = + − = + = = − − + − − = − + = + = = = − − − + − = + = + = Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem: [ ] [ ] [ ] ( ) ( ) ( ) 2 2 2 2 2 2 ( , ) ( , ) ( , ) 29 2 29 145 29 4 29 145 29 116 145 145 145 d A C d B C d A B + = + = + ⋅ = + = = The area of a triangle is 1 2 A bh = . In this problem, [ ] [ ] 1 ( , ) ( , ) 2 1 29 2 29 2 1 2 29 2 29 square units A d A C d B C = ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ = 33. (4, 3), (0, 3), (4, 2) A B C = − = − = ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 ( , ) (0 4) 3 ( 3) ( 4) 0 16 0 16 4 ( , ) 4 0 2 ( 3) 4 5 16 25 41 ( , ) (4 4) 2 ( 3) 0 5 0 25 25 5 d A B d B C d A C = − + − − − = − + = + = = = − + − − = + = + = = − + − − = + = + = = Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem: [ ] [ ] [ ] ( ) 2 2 2 2 2 2 ( , ) ( , ) ( , ) 4 5 41 16 25 41 41 41 d A B d A C d B C + = + = + = = The area of a triangle is 1 2 A bh = . In this

  11. Page 11
    Read page 11 as text

    Section 1.1: The Distance and Midpoint Formulas 5 problem, [ ] [ ] 1 ( , ) ( , ) 2 1 4 5 2 10 square units A d A B d A C = ⋅ ⋅ = ⋅ ⋅ = 34. (4, 3), (4, 1), (2, 1) A B C = − = = ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 ( , ) (4 4) 1 ( 3) 0 4 0 16 16 4 ( , ) 2 4 1 1 ( 2) 0 4 0 4 2 ( , ) (2 4) 1 ( 3) ( 2) 4 4 16 20 2 5 d A B d B C d A C = − + − − = + = + = = = − + − = − + = + = = = − + − − = − + = + = = Verifying that ∆ ABC is a right triangle by the Pythagorean Theorem: [ ] [ ] [ ] ( ) 2 2 2 2 2 2 ( , ) ( , ) ( , ) 4 2 2 5 16 4 20 20 20 d A B d B C d A C + = + = + = = The area of a triangle is 1 2 A bh = . In this problem, [ ] [ ] 1 ( , ) ( , ) 2 1 4 2 2 4 square units A d A B d B C = ⋅ ⋅ = ⋅ ⋅ = 35. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 4 4 3 5 , 2 2 8 0 , 2 2 (4, 0) x x y y x y + +   =     − + +   =       =     = 36. The coordinates of the midpoint are: ( ) 1 2 1 2 ( , ) , 2 2 2 2 0 4 , 2 2 0 4 , 2 2 0, 2 x x y y x y + +   =     − + +   =       =     = 37. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 3 6 2 0 , 2 2 3 2 , 2 2 3 ,1 2 x x y y x y + +   =     − + +   =       =       =     38. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 2 4 3 2 , 2 2 6 1 , 2 2 1 3, 2 x x y y x y + +   =     + − +   =     −   =       = −    

  12. Page 12
    Read page 12 as text

    Chapter 1: Graphs 6 39. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 4 6 3 1 , 2 2 2 10 , 2 2 (5, 1) x x y y x y + +   =     + − +   =     −   =     = − 40. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 4 2 3 2 , 2 2 2 1 , 2 2 1 1, 2 x x y y x y + +   =     − + − +   =     − −   =       = − −     41. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 0 0 , 2 2 , 2 2 x x y y x y a b a b + +   =     + +   =       =     42. The coordinates of the midpoint are: 1 2 1 2 ( , ) , 2 2 0 0 , 2 2 , 2 2 x x y y x y a a a a + +   =     + +   =       =     43. The x coordinate would be 2 3 5 + = and the y coordinate would be 5 2 3 − = . Thus the new point would be ( ) 5,3 . 44. The new x coordinate would be 1 2 3 − − = − and the new y coordinate would be 6 4 10 + = . Thus the new point would be ( ) 3,10 − 45. a. If we use a right triangle to solve the problem, we know the hypotenuse is 13 units in length. One of the legs of the triangle will be 2+3=5. Thus the other leg will be: 2 2 2 2 2 5 13 25 169 144 12 b b b b + = + = = = Thus the coordinates will have an y value of 1 12 13 − − = − and 1 12 11 − + = . So the points are ( ) 3,11 and ( ) 3, 13 − . b. Consider points of the form ( ) 3, y that are a distance of 13 units from the point ( ) 2, 1 − − . ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 1 2 1 2 2 2 2 2 2 3 ( 2) 1 5 1 25 1 2 2 26 d x x y y y y y y y y = − + − = − − + − − = + − − = + + + = + + ( ) ( )( ) 2 2 2 2 2 2 13 2 26 13 2 26 169 2 26 0 2 143 0 11 13 y y y y y y y y y y = + + = + + = + + = + − = − + 11 0 11 y y − = = or 13 0 13 y y + = = − Thus, the points ( ) 3,11 and ( ) 3, 13 − are a distance of 13 units from the point ( ) 2, 1 − − . 46. a. If we use a right triangle to solve the problem, we know the hypotenuse is 17 units in length. One of the legs of the triangle will be 2+6=8. Thus the other leg will be: 2 2 2 2 2 8 17 64 289 225 15 b b b b + = + = = = Thus the coordinates will have an x value of 1 15 14 − = − and 1 15 16 + = . So the points are ( ) 14, 6 − − and ( ) 16, 6 − .

  13. Page 13
    Read page 13 as text

    Section 1.1: The Distance and Midpoint Formulas 7 b. Consider points of the form ( ) , 6 x − that are a distance of 17 units from the point ( ) 1, 2 . ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 1 2 1 2 2 2 2 2 2 1 2 6 2 1 8 2 1 64 2 65 d x x y y x x x x x x x = − + − = − + − − = − + + = − + + = − + ( ) ( )( ) 2 2 2 2 2 2 17 2 65 17 2 65 289 2 65 0 2 224 0 14 16 x x x x x x x x x x = − + = − + = − + = − − = + − 14 0 14 x x + = = − or 16 0 16 x x − = = Thus, the points ( ) 14, 6 − − and ( ) 16, 6 − are a distance of 13 units from the point ( ) 1, 2 . 47. Points on the x -axis have a y -coordinate of 0. Thus, we consider points of the form ( ) , 0 x that are a distance of 6 units from the point ( ) 4, 3 − . ( ) ( ) ( ) ( ) ( ) 2 2 2 1 2 1 2 2 2 2 2 2 4 3 0 16 8 3 16 8 9 8 25 d x x y y x x x x x x x = − + − = − + − − = − + + − = − + + = − + ( ) 2 2 2 2 2 2 2 6 8 25 6 8 25 36 8 25 0 8 11 ( 8) ( 8) 4(1)( 11) 2(1) 8 64 44 8 108 2 2 8 6 3 4 3 3 2 x x x x x x x x x = − + = − + = − + = − − − − ± − − − = ± + ± = = ± = = ± 4 3 3 x = + or 4 3 3 x = − Thus, the points ( ) 4 3 3,0 + and ( ) 4 3 3, 0 − are on the x -axis and a distance of 6 units from the point ( ) 4, 3 − . 48. Points on the y -axis have an x -coordinate of 0. Thus, we consider points of the form ( ) 0, y that are a distance of 6 units from the point ( ) 4, 3 − . ( ) ( ) ( ) ( ) 2 2 2 1 2 1 2 2 2 2 2 2 4 0 3 4 9 6 16 9 6 6 25 d x x y y y y y y y y y = − + − = − + − − = + + + = + + + = + + ( ) 2 2 2 2 2 2 2 6 6 25 6 6 25 36 6 25 0 6 11 ( 6) (6) 4(1)( 11) 2(1) 6 36 44 6 80 2 2 6 4 5 3 2 5 2 x x x x x x x x x = + + = + + = + + = + − − ± − − = − ± + − ± = = − ± = = − ± 6 2 5 x = − + or 6 2 5 x = − − Thus, the points ( ) 0, 6 2 5 − + and ( ) 0, 6 2 5 − −

  14. Page 14
    Read page 14 as text

    Chapter 1: Graphs 8 are on the y- axis and a distance of 6 units from the point ( ) 4, 3 − . 49. ( ) 1 2 1 2 , , 2 2 x x y y M x y + +   = =     . ( ) 1 1 1 , ( 3, 6) P x y = = − and ( , ) ( 1, 4) x y = − , so 1 2 2 2 2 2 3 1 2 2 3 1 x x x x x x + = − + − = − = − + = and 1 2 2 2 2 2 6 4 2 8 6 2 y y y y y y + = + = = + = Thus, 2 (1, 2) P = . 50. ( ) 1 2 1 2 , , 2 2 x x y y M x y + +   = =     . ( ) 2 2 2 , (7, 2) P x y = = − and ( , ) (5, 4) x y = − , so 1 2 1 1 1 2 7 5 2 10 7 3 x x x x x x + = + = = + = and 1 2 1 1 1 2 ( 2) 4 2 8 ( 2) 6 y y y y y y + = + − − = − = + − − = Thus, 1 (3, 6) P = − . 51. The midpoint of AB is: ( ) 0 6 0 0 , 2 2 3, 0 D + +   =     = The midpoint of AC is: ( ) 0 4 0 4 , 2 2 2, 2 E + +   =     = The midpoint of BC is: ( ) 6 4 0 4 , 2 2 5, 2 F + +   =     = ( ) 2 2 2 2 ( , ) 0 4 (3 4) ( 4) ( 1) 16 1 17 d C D = − + − = − + − = + = ( ) 2 2 2 2 ( , ) 2 6 (2 0) ( 4) 2 16 4 20 2 5 d B E = − + − = − + = + = = 2 2 2 2 ( , ) (2 0) (5 0) 2 5 4 25 29 d A F = − + − = + = + = 52. Let 1 2 (0, 0), (0, 4), ( , ) P P P x y = = = ( ) ( ) ( ) 2 2 1 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 , (0 0) (4 0) 16 4 , ( 0) ( 0) 4 16 , ( 0) ( 4) ( 4) 4 ( 4) 16 d P P d P P x y x y x y d P P x y x y x y = − + − = = = − + − = + = → + = = − + − = + − = → + − = Therefore, ( ) 2 2 2 2 4 8 16 8 16 2 y y y y y y y = − = − + = = which gives 2 2 2 2 16 12 2 3 x x x + = = = ± Two triangles are possible. The third vertex is ( ) ( ) 2 3, 2 or 2 3, 2 − . 53. Let ( ) 1 0, 0 P = , ( ) 2 0, P s = , ( ) 3 , 0 P s = , and ( ) 4 , P s s = . y x (0, ) s (0, 0) ( , 0) s ( , ) s s The points 1 P and 4 P are endpoints of one diagonal and the points 2 P and 3 P are the

  15. Page 15
    Read page 15 as text

    Section 1.1: The Distance and Midpoint Formulas 9 endpoints of the other diagonal. 1,4 0 0 , , 2 2 2 2 s s s s M + +     = =         2,3 0 0 , , 2 2 2 2 s s s s M + +     = =         The midpoints of the diagonals are the same. Therefore, the diagonals of a square intersect at their midpoints. 54. Let ( ) 1 0, 0 P = , ( ) 2 , 0 P a = , and 3 3 , 2 2 a a P   =       . To show that these vertices form an equilateral triangle, we need to show that the distance between any pair of points is the same constant value. ( ) ( ) ( ) ( ) ( ) 2 2 1 2 2 1 2 1 2 2 2 , 0 0 0 d P P x x y y a a a = − + − = − + − = = ( ) ( ) ( ) 2 2 2 3 2 1 2 1 2 2 2 2 2 2 , 3 0 2 2 3 4 4 4 4 d P P x x y y a a a a a a a a = − + −     = − + −           = + = = = ( ) ( ) ( ) 2 2 1 3 2 1 2 1 2 2 2 2 2 2 , 3 0 0 2 2 3 4 4 4 4 d P P x x y y a a a a a a a = − + −     = − + −           = + = = = Since all three distances have the same constant value, the triangle is an equilateral triangle. Now find the midpoints: 1 2 2 3 1 3 4 5 6 0 0 0 , , 0 2 2 2 3 3 3 0 , 2 2 , 4 4 2 2 3 0 0 3 2 2 , , 2 2 4 4 P P P P P P a a P M a a a a a P M a a a a P M + +     = = =               + +   = = =               + +       = = =             ( ) 2 2 4 5 2 2 2 2 3 3 , 0 4 2 4 3 4 4 3 16 16 2 a a a d P P a a a a a     = − + −               = +           = + = ( ) 2 2 4 6 2 2 2 2 3 , 0 4 2 4 3 4 4 3 16 16 2 a a a d P P a a a a a     = − + −               = − +           = + = ( ) 2 2 5 6 2 2 2 3 3 3 , 4 4 4 4 0 2 4 2 a a a a d P P a a a     = − + −             = +     = = Since the sides are the same length, the triangle is equilateral. 55. 2 2 1 2 2 2 ( , ) ( 4 2) (1 1) ( 6) 0 36 6 d P P = − − + − = − + = = ( ) 2 2 2 3 2 2 ( , ) 4 ( 4) ( 3 1) 0 ( 4) 16 4 d P P = − − − + − − = + − = = 2 2 1 3 2 2 ( , ) ( 4 2) ( 3 1) ( 6) ( 4) 36 16 52 2 13 d P P = − − + − − = − + − = + = = Since [ ] [ ] [ ] 2 2 2 1 2 2 3 1 3 ( , ) ( , ) ( , ) d P P d P P d P P + = , the triangle is a right triangle.

  16. Page 16
    Read page 16 as text

    Chapter 1: Graphs 10 56. ( ) 2 2 1 2 2 2 ( , ) 6 ( 1) (2 4) 7 ( 2) 49 4 53 d P P = − − + − = + − = + = ( ) 2 2 2 3 2 2 ( , ) 4 6 ( 5 2) ( 2) ( 7) 4 49 53 d P P = − + − − = − + − = + = ( ) 2 2 1 3 2 2 ( , ) 4 ( 1) ( 5 4) 5 ( 9) 25 81 106 d P P = − − + − − = + − = + = Since [ ] [ ] [ ] 2 2 2 1 2 2 3 1 3 ( , ) ( , ) ( , ) d P P d P P d P P + = , the triangle is a right triangle. Since ( ) ( ) 1 2 2 3 , , d P P d P P = , the triangle is isosceles. Therefore, the triangle is an isosceles right triangle. 57. ( ) ( ) 2 2 1 2 2 2 ( , ) 0 ( 2) 7 ( 1) 2 8 4 64 68 2 17 d P P = − − + − − = + = + = = ( ) 2 2 2 3 2 2 ( , ) 3 0 (2 7) 3 ( 5) 9 25 34 d P P = − + − = + − = + = ( ) ( ) 2 2 1 3 2 2 ( , ) 3 ( 2) 2 ( 1) 5 3 25 9 34 d P P = − − + − − = + = + = Since 2 3 1 3 ( , ) ( , ) d P P d P P = , the triangle is isosceles. Since [ ] [ ] [ ] 2 2 2 1 3 2 3 1 2 ( , ) ( , ) ( , ) d P P d P P d P P + = , the triangle is also a right triangle. Therefore, the triangle is an isosceles right triangle. 58. ( ) ( ) 2 2 1 2 2 2 ( , ) 4 7 0 2 ( 11) ( 2) 121 4 125 5 5 d P P = − − + − = − + − = + = = ( ) 2 2 2 3 2 2 ( , ) 4 ( 4) (6 0) 8 6 64 36 100 10 d P P = − − + − = + = + = = ( ) ( ) 2 2 1 3 2 2 ( , ) 4 7 6 2 ( 3) 4 9 16 25 5 d P P = − + − = − + = + = = Since [ ] [ ] [ ] 2 2 2 1 3 2 3 1 2 ( , ) ( , ) ( , ) d P P d P P d P P + = , the triangle is a right triangle. 59. Using the Pythagorean Theorem: 2 2 2 2 2 90 90 8100 8100 16200 16200 90 2 127.28 feet d d d d + = + = = = = ≈ 90 90 90 90 d 60. Using the Pythagorean Theorem: 2 2 2 2 2 60 60 3600 3600 7200 7200 60 2 84.85 feet d d d d + = + = → = = = ≈ 60 60 60 60 d 61. a. First: (90, 0), Second: (90, 90), Third: (0, 90)

1452 more pages in the full document

Unlock it once and it stays in your library, ready to chat with or turn into flashcards.

Unlock the full document

More Mathematics study materials

See all

Study this document with CramX

Once it is in your library, every tool can work from it.