,Solution manual for Precalculus, 11th Edition Ron
Larson
Notes
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2- We have shown you 10 or five pages.
3- The file contains all Appendix and Excel
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4- We have all what you need, we make
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, CONTENTS
Part I Solutions to All Exercises .................................................................... 1
Chapter 1 Functions and Their Graphs................................................................... 1
Chapter 2 Polynomial and Rational Functions .................................................. 117
Chapter 3 Exponential and Logarithmic Functions ........................................... 233
Chapter 4 Trigonometry ...................................................................................... 303
Chapter 5 Analytic Trigonometry....................................................................... 401
Chapter 6 Additional Topics in Trigonometry ................................................... 485
Chapter 7 Systems of Equations and Inequalities .............................................. 567
Chapter 8 Matrices and Determinants ................................................................ 681
Chapter 9 Sequences, Series, and Probability .................................................... 778
Chapter 10 Topics in Analytic Geometry............................................................. 858
Appendix A Review of Fundamental Concepts of Algebra ................................ 1004
Solutions to Checkpoints ............................................................... 1051
Solutions to Practice Tests ............................................................ 1237
Part II Solutions to Chapter and Cumulative Tests............................... 1261
,
, C H A P T E R 1
Functions and Their Graphs
Section 1.1 Rectangular Coordinates ........................................................................ 2
Section 1.2 Graphs of Equations ............................................................................... 7
Section 1.3 Linear Equations in Two Variables .....................................................17
Section 1.4 Functions ...............................................................................................31
Section 1.5 Analyzing Graphs of Functions ...........................................................40
Section 1.6 A Library of Parent Functions ............................................................. 52
Section 1.7 Transformations of Functions .............................................................. 57
Section 1.8 Combinations of Functions: Composite Functions ............................. 69
Section 1.9 Inverse Functions..................................................................................77
Section 1.10 Mathematical Modeling and Variation ................................................88
Review Exercises ..........................................................................................................96
Problem Solving .........................................................................................................109
Practice Test .............................................................................................................115
© 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
,C H A P T E R 1
Functions and Their Graphs
Section 1.1 Rectangular Coordinates
1. Cartesian 16. x < 0 and y < 0 in Quadrant III.
2. Midpoint Formula 17. x = − 4 and y > 0 in Quadrant II.
3. The x-axis is the horizontal real number line. 18. x < 0 and y = 7 in Quadrant II.
Matches (c).
19. x + y = 0, x ≠ 0, y ≠ 0 means
4. The y-axis is the vertical real number line. x = − y or y = − x. This occurs in Quadrant II or IV.
Matches (f).
5. The origin is the point of intersection of the vertical
20. ( x, y ), xy > 0 means x and y have the same signs.
and horizontal axes. This occurs in Quadrant I or III.
Matches (a).
21.
6. The quadrants are four regions of the coordinate plane.
Matches (d).
7. An x-coordinate is the directed distance from the
y-axis. Matches (e).
8. A y-coordinate is the directed distance from the x-axis.
Matches (b).
9. A: ( 2, 6), B: ( −6, − 2), C: ( 4, − 4), D: ( −3, 2)
22.
10. A: ( 32 , − 4); B: (0, − 2); C: (−3, 52 ); D: (−6, 0)
11.
( x2 − x1 ) + ( y2 − y1 )
2 2
23. d =
(3 − (−2))
2
+ ( −6 − 6)
2
=
12. = (5)
2
+ ( −12)
2
= 25 + 144
= 13 units
( x2 − x1 ) + ( y2 − y1 )
2 2
24. d =
= (0 − 8)2 + ( 20 − 5)
2
(−8) + (15)
2 2
=
13. ( − 3, 4)
= 64 + 225
14. ( −12, 0) = 289
15. x > 0 and y < 0 in Quadrant IV. = 17 units
2 © 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
, Section 1.1 Rectangular Coordinates 3
30. (a) The distance between ( −1, 1) and (9, 1) is 10.
25. d = ( x2 − x1 ) + ( y2 − y1 )
2 2
The distance between (9, 1) and (9, 4) is 3.
(−5 − 1) + ( −1 − 4)
2 2
=
The distance between ( −1, 1) and (9, 4) is
(−6) + ( −5)
2 2
=
(9 − (−1))
2
+ ( 4 − 1)
2
= 36 + 25 = 100 + 9 = 109.
=
( )
2
61 units (b) 10 2 + 32 = 109 = 109
26. d = ( x2 − x1 ) + ( y2 − y1 )
2 2
(4 − 2) + (0 − 1)
2 2
31. d1 = = 4+1 = 5
(3 − 1) + ( −2 − 3)
2 2
=
(4 + 1) + (0 + 5)
2 2
d2 = = 25 + 25 = 50
(2) + ( −5)
2 2
=
(2 + 1) + (1 + 5)
2 2
d3 = = 9 + 36 = 45
= 4 + 25
( 5) ( ) ( )
2 2 2
+ 45 = 50
= 29 units
(3 − (−1))
2
+ (5 − 3)
2
27. d = ( x2 − x1 ) + ( y2 − y1 )
2 2
32. d1 = = 16 + 4 = 20
(5 − 3) + (1 − 5)
2 2 2 2
1 4 d2 = = 4 + 16 = 20
= 2 − + −1 −
2 3
(5 − (−1))
2
+ (1 − 3)
2
d3 = = 36 + 4 = 40
2 2
3 7
= + −
( ) ( ) ( )
2 2 2
2 3 20 + 20 = 40
9 49
= +
(1 − 3) + ( −3 − 2)
2 2
4 9 33. d1 = = 4 + 25 = 29
277
(3 + 2) + ( 2 − 4)
2 2
= d2 = = 25 + 4 = 29
36
(1 + 2) + ( −3 − 4)
2 2
277 d3 = = 9 + 49 = 58
= units
6 d1 = d 2
28. d = ( x2 − x1 ) + ( y2 − y1 )
2 2
34. d1 = ( 4 − 2)
2
+ (9 − 3) =
2
4 + 36 = 40
− 9.5) + (8.2 − ( −2.6))
2
(−3.9
2
= d2 = ( −2 − 4 )
2
+ ( 7 − 9) =
2
36 + 4 = 40
(−13.4) + (10.8)
2 2
= d3 = ( 2 − ( −2 ) )
2
+ (3 − 7 ) =
2
16 + 16 = 32
= 179.56 + 116.64 d1 = d 2
= 296.2
35. (a)
29. (a) (1, 0), (13, 5)
Distance = (13 − 1)2 + (5 − 0)
2
= 122 + 52 = 169 = 13
(13, 5), (13, 0)
Distance = 5 − 0 = 5 = 5
(1, 0), (13, 0) (b) d = (5 − (−3)) 2 + (6 − 6)2 = 64 = 8
Distance = 1 − 13 = −12 = 12 6 + 6 5 + (−3)
(c) , = (6, 1)
2 2
(b) 52 + 122 = 25 + 144 = 169 = 132
© 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
,4 Chapter 1 Functions and Their Graphs
36. (a) 40. (a)
(b) d = (4 − 4) 2 + (8 − 1)2 = 49 = 7
(2 − 10) + (10 − 2)
2 2
(b) d =
1 + 8 4 + 4 9
(c) , = , 4 = 64 + 64 = 8 2
2 2 2
2 + 10 10 + 2
37. (a)
(c) , = (6, 6)
2 2
41. (a)
(9 − 1) + (7 − 1)
2 2
(b) d = = 64 + 36 = 10
9 + 1 7 + 1 (b) d = (−16.8 − 5.6)
2
+ (12.3 − 4.9)
2
(c) , = (5, 4)
2 2
= 501.76 + 54.76 = 556.52
38. (a) −16.8 + 5.6 12.3 + 4.9
(c) , = ( −5.6, 8.6)
2 2
42. (a)
(1 − 6) + (12 − 0) =
2 2
(b) d = 25 + 144 = 13
1 + 6 12 + 0 7
(c) , = , 6 2 2
2 2 2 1 5 4
(b) d = + + 1 −
2 2 3
39. (a)
1 82
= 9+ =
9 3
−(5 2) + (1 2) ( 4 3) + 1 7
(c) , = −1,
2 2 6
(5 + 1) + ( 4 − 2) + =
2 2
(b) d = 36 + 4 = 2 10
−1 + 5 2 + 4
(c) , = ( 2, 3)
2 2
© 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
, Section 1.1 Rectangular Coordinates 5
48. ( −3 + 6, 6 − 3) = (3, 3)
43. d = (42 − 18) + (50 − 12)
2 2
(−5 + 6, 3 − 3) = (1, 0)
= 242 + 382
(−3 + 6, 0 − 3) = (3, − 3)
= 2020
(−1 + 6, 3 − 3) = (5, 0)
= 2 505
≈ 45 49. ( −7 + 4, − 2 + 8) = ( −3, 6)
The pass is about 45 yards. ( −2 + 4, 2 + 8) = ( 2, 10)
44. d = 1202 + 1502 ( −2 + 4, − 4 + 8) = ( 2, 4)
= 36,900 ( −7 + 4, − 4 + 8) = ( −3, 4)
= 30 41 50. (5 − 10, 8 − 6) = ( −5, 2)
≈ 192.09
(3 − 10, 6 − 6) = ( −7, 0)
The plane flies about 192 kilometers.
(7 − 10, 6 − 6) = ( −3, 0)
x + x2 y1 + y2
45. midpoint = 1 , 51. True. Because x < 0 and y > 0, 2 x < 0 and
2 2
− 3 y < 0, which is located in Quadrant III.
2016 + 2018 485.9 + 514.4
= ,
2 2 52. False. The Midpoint Formula would be used 15 times.
= ( 2017, 500.15)
53. True. Two sides of the triangle have lengths 149 and
In 2017, the sales were about $500.15 billion.
the third side has a length of 18.
x + x2 y1 + y2
46. midpoint = 1 , . In this problem, you
2 2 54. False. The polygon could be a rhombus. For example,
2017 + 2019 6.16 + y2 consider the points ( 4, 0), (0, 6), ( − 4, 0), and (0, − 6).
have , = ( 2018, 7.57).
2 2
55. The y-coordinate of a point on the x-axis is 0. The
6.16 + y2 x-coordinates of a point on the y-axis is 0.
So, = 7.57 y2 = 8.98.
2
56. The x- and y-coordinates are switched. For example,
In 2019, the earnings per share were about $8.98.
(3, 2) should be (2, 3).
47. ( − 2 + 2, − 4 + 5) = (0, 1)
x1 + x2 y + y2
(2 + 2, − 3 + 5) = ( 4, 2) 57. Because xm = and ym = 1 we have:
2 2
(−1 + 2, −1 + 5) = (1, 4) 2 xm = x1 + x2 2 ym = y1 + y2
2 xm − x1 = x2 2 ym − y1 = y2
So, ( x2 , y2 ) = ( 2 xm − x1 , 2 ym − y1 ).
x + x2 y1 + y2
58. The midpoint of the given line segment is 1 , .
2 2
x + x2 y + y2
x + x2 y1 + y2 x1 + 1 y1 + 1 = 3x1 + x2 , 3 y1 + y2 .
The midpoint between ( x1 , y1 ) and 1 , is 2 , 2
2 2 4 4
2 2
x + x2 y1 + y2 x1 + x2 y + y2
+ x2 1 + y2 x + 3 x2 y1 + 3 y2
The midpoint between 1 , and ( x2 , y2 ) is 2 , 2
= 1 , .
2 2 4 4
2 2
3 x + x2 3 y1 + y2 x1 + x2 y1 + y2 x1 + 3x2 y1 + 3 y2
So, the three points are 1 , , , , and , .
4 4 2 2 4 4
© 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
, 6 Chapter 1 Functions and Their Graphs
59. Use the Midpoint Formula to prove the diagonals of the
( x1 − x1 ) + ( y2 − y1 )
2 2
62. (a) distance =
parallelogram bisect each other.
( y2 − y1 )
2
b + a c + 0 a + b c = = y2 − y1
, = ,
2 2 2 2
( x2 − x1 ) + ( y1 − y1 )
2 2
(b) distance =
a + b + 0 c + 0 a + b c
, = ,
( x2 − x1 )
2
2 2 2 2 = = x2 − x1
60. (a) Because ( x0 , y0 ) lies in Quadrant II, ( x0 , − y0 ) Answers will vary.
must lie in Quadrant III. Matches (ii). 63. 4 x − 6
(b) Because ( x0 , y0 ) lies in Quadrant II, ( −2 x0 , y0 ) (a) 4( −1) − 6 = −4 − 6 = −10
must lie in Quadrant I. Matches (iii).
(b) 4(0) − 6 = 0 − 6 = −6
(
(c) Because ( x0 , y0 ) lies in Quadrant II, x0 , 12 y0 must )
lie in Quadrant II. Matches (iv). 64. 9 − 7x
(d) Because ( x0 , y0 ) lies in Quadrant II, ( − x0 , − y0 ) (a) 9 − 7( −3) = 9 + 21 = 30
must lie in Quadrant IV. Matches (i). (b) 9 − 7(3) = 9 − 21 = −12
61. (a) First Set
65. (a) When x = −3, 2 x 3 = 2( −3) = 2( −27) = −54.
3
d ( A, B ) = (2 − 2 ) + ( 3 − 6)
2 2
= 9 = 3
(b) When x = 0, 2 x3 = 2(0) = 0.
3
d ( B, C ) = (2 − 6) + (6 − 3)
2 2
= 16 + 9 = 5
−3 3
d ( A, C ) = (2 − 6) + (3 − 3)
2 2
= 16 = 4 66. (a) When x = 0, −3x − 4 = = − . Division by
x4 0
Because 32 + 42 = 52 , A, B, and C are the 0 is undefined.
vertices of a right triangle. −3 −3 3
(b) When x = −2, −3x − 4 = = = − .
Second Set x4 ( )
−2
4
16
d ( A, B ) = (8 − 5) + (3 − 2 )
2 2
= 10
67. ( − x) − 2 = x 2 − 2
2
d ( B, C ) = ( 5 − 2) + ( 2 − 1)
2 2
= 10
68. ( − x) − ( − x) + 2 = − x 3 − x 2 + 2
3 2
d ( A, C ) = (8 − 2) + (3 − 1)
2 2
= 40
69. − x 2 + ( − x) = − x 2 + x 2 = 0
2
A, B, and C are the vertices of an isosceles triangle
or are collinear: 10 + 10 = 2 10 = 40.
70. − x3 + ( − x) + ( − x) − x 4 = − x3 − x3 + x 4 − x 4
3 4
(b)
= −2 x3
71. (a) P = R − C
= 135 x − (93 x + 35,000)
= 42 x − 35,000
(b) P = 42(5000) − 35,000 = $175,000
First set: Not collinear 1
72. (a) Time to copy one page: min
Second set: Collinear. 50
(c) A set of three points is collinear when the sum of x
(b) Time to copy x pages: min
two distances among the points is exactly equal to 50
the third distance. 120 12
(c) Time to copy 120 pages: = = 2.4 min
50 5
10,000
(d) Time to copy 10,000 pages: = 200 min
50
© 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Larson
Notes
1- The file is chapter after chapter.
2- We have shown you 10 or five pages.
3- The file contains all Appendix and Excel
sheet if it exists.
4- We have all what you need, we make
update at every time. There are many
new editions waiting you.
5- If you think you purchased the wrong file
You can contact us at every time, we can
replace it with true one.
Our email:
, CONTENTS
Part I Solutions to All Exercises .................................................................... 1
Chapter 1 Functions and Their Graphs................................................................... 1
Chapter 2 Polynomial and Rational Functions .................................................. 117
Chapter 3 Exponential and Logarithmic Functions ........................................... 233
Chapter 4 Trigonometry ...................................................................................... 303
Chapter 5 Analytic Trigonometry....................................................................... 401
Chapter 6 Additional Topics in Trigonometry ................................................... 485
Chapter 7 Systems of Equations and Inequalities .............................................. 567
Chapter 8 Matrices and Determinants ................................................................ 681
Chapter 9 Sequences, Series, and Probability .................................................... 778
Chapter 10 Topics in Analytic Geometry............................................................. 858
Appendix A Review of Fundamental Concepts of Algebra ................................ 1004
Solutions to Checkpoints ............................................................... 1051
Solutions to Practice Tests ............................................................ 1237
Part II Solutions to Chapter and Cumulative Tests............................... 1261
,
, C H A P T E R 1
Functions and Their Graphs
Section 1.1 Rectangular Coordinates ........................................................................ 2
Section 1.2 Graphs of Equations ............................................................................... 7
Section 1.3 Linear Equations in Two Variables .....................................................17
Section 1.4 Functions ...............................................................................................31
Section 1.5 Analyzing Graphs of Functions ...........................................................40
Section 1.6 A Library of Parent Functions ............................................................. 52
Section 1.7 Transformations of Functions .............................................................. 57
Section 1.8 Combinations of Functions: Composite Functions ............................. 69
Section 1.9 Inverse Functions..................................................................................77
Section 1.10 Mathematical Modeling and Variation ................................................88
Review Exercises ..........................................................................................................96
Problem Solving .........................................................................................................109
Practice Test .............................................................................................................115
© 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
,C H A P T E R 1
Functions and Their Graphs
Section 1.1 Rectangular Coordinates
1. Cartesian 16. x < 0 and y < 0 in Quadrant III.
2. Midpoint Formula 17. x = − 4 and y > 0 in Quadrant II.
3. The x-axis is the horizontal real number line. 18. x < 0 and y = 7 in Quadrant II.
Matches (c).
19. x + y = 0, x ≠ 0, y ≠ 0 means
4. The y-axis is the vertical real number line. x = − y or y = − x. This occurs in Quadrant II or IV.
Matches (f).
5. The origin is the point of intersection of the vertical
20. ( x, y ), xy > 0 means x and y have the same signs.
and horizontal axes. This occurs in Quadrant I or III.
Matches (a).
21.
6. The quadrants are four regions of the coordinate plane.
Matches (d).
7. An x-coordinate is the directed distance from the
y-axis. Matches (e).
8. A y-coordinate is the directed distance from the x-axis.
Matches (b).
9. A: ( 2, 6), B: ( −6, − 2), C: ( 4, − 4), D: ( −3, 2)
22.
10. A: ( 32 , − 4); B: (0, − 2); C: (−3, 52 ); D: (−6, 0)
11.
( x2 − x1 ) + ( y2 − y1 )
2 2
23. d =
(3 − (−2))
2
+ ( −6 − 6)
2
=
12. = (5)
2
+ ( −12)
2
= 25 + 144
= 13 units
( x2 − x1 ) + ( y2 − y1 )
2 2
24. d =
= (0 − 8)2 + ( 20 − 5)
2
(−8) + (15)
2 2
=
13. ( − 3, 4)
= 64 + 225
14. ( −12, 0) = 289
15. x > 0 and y < 0 in Quadrant IV. = 17 units
2 © 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
, Section 1.1 Rectangular Coordinates 3
30. (a) The distance between ( −1, 1) and (9, 1) is 10.
25. d = ( x2 − x1 ) + ( y2 − y1 )
2 2
The distance between (9, 1) and (9, 4) is 3.
(−5 − 1) + ( −1 − 4)
2 2
=
The distance between ( −1, 1) and (9, 4) is
(−6) + ( −5)
2 2
=
(9 − (−1))
2
+ ( 4 − 1)
2
= 36 + 25 = 100 + 9 = 109.
=
( )
2
61 units (b) 10 2 + 32 = 109 = 109
26. d = ( x2 − x1 ) + ( y2 − y1 )
2 2
(4 − 2) + (0 − 1)
2 2
31. d1 = = 4+1 = 5
(3 − 1) + ( −2 − 3)
2 2
=
(4 + 1) + (0 + 5)
2 2
d2 = = 25 + 25 = 50
(2) + ( −5)
2 2
=
(2 + 1) + (1 + 5)
2 2
d3 = = 9 + 36 = 45
= 4 + 25
( 5) ( ) ( )
2 2 2
+ 45 = 50
= 29 units
(3 − (−1))
2
+ (5 − 3)
2
27. d = ( x2 − x1 ) + ( y2 − y1 )
2 2
32. d1 = = 16 + 4 = 20
(5 − 3) + (1 − 5)
2 2 2 2
1 4 d2 = = 4 + 16 = 20
= 2 − + −1 −
2 3
(5 − (−1))
2
+ (1 − 3)
2
d3 = = 36 + 4 = 40
2 2
3 7
= + −
( ) ( ) ( )
2 2 2
2 3 20 + 20 = 40
9 49
= +
(1 − 3) + ( −3 − 2)
2 2
4 9 33. d1 = = 4 + 25 = 29
277
(3 + 2) + ( 2 − 4)
2 2
= d2 = = 25 + 4 = 29
36
(1 + 2) + ( −3 − 4)
2 2
277 d3 = = 9 + 49 = 58
= units
6 d1 = d 2
28. d = ( x2 − x1 ) + ( y2 − y1 )
2 2
34. d1 = ( 4 − 2)
2
+ (9 − 3) =
2
4 + 36 = 40
− 9.5) + (8.2 − ( −2.6))
2
(−3.9
2
= d2 = ( −2 − 4 )
2
+ ( 7 − 9) =
2
36 + 4 = 40
(−13.4) + (10.8)
2 2
= d3 = ( 2 − ( −2 ) )
2
+ (3 − 7 ) =
2
16 + 16 = 32
= 179.56 + 116.64 d1 = d 2
= 296.2
35. (a)
29. (a) (1, 0), (13, 5)
Distance = (13 − 1)2 + (5 − 0)
2
= 122 + 52 = 169 = 13
(13, 5), (13, 0)
Distance = 5 − 0 = 5 = 5
(1, 0), (13, 0) (b) d = (5 − (−3)) 2 + (6 − 6)2 = 64 = 8
Distance = 1 − 13 = −12 = 12 6 + 6 5 + (−3)
(c) , = (6, 1)
2 2
(b) 52 + 122 = 25 + 144 = 169 = 132
© 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
,4 Chapter 1 Functions and Their Graphs
36. (a) 40. (a)
(b) d = (4 − 4) 2 + (8 − 1)2 = 49 = 7
(2 − 10) + (10 − 2)
2 2
(b) d =
1 + 8 4 + 4 9
(c) , = , 4 = 64 + 64 = 8 2
2 2 2
2 + 10 10 + 2
37. (a)
(c) , = (6, 6)
2 2
41. (a)
(9 − 1) + (7 − 1)
2 2
(b) d = = 64 + 36 = 10
9 + 1 7 + 1 (b) d = (−16.8 − 5.6)
2
+ (12.3 − 4.9)
2
(c) , = (5, 4)
2 2
= 501.76 + 54.76 = 556.52
38. (a) −16.8 + 5.6 12.3 + 4.9
(c) , = ( −5.6, 8.6)
2 2
42. (a)
(1 − 6) + (12 − 0) =
2 2
(b) d = 25 + 144 = 13
1 + 6 12 + 0 7
(c) , = , 6 2 2
2 2 2 1 5 4
(b) d = + + 1 −
2 2 3
39. (a)
1 82
= 9+ =
9 3
−(5 2) + (1 2) ( 4 3) + 1 7
(c) , = −1,
2 2 6
(5 + 1) + ( 4 − 2) + =
2 2
(b) d = 36 + 4 = 2 10
−1 + 5 2 + 4
(c) , = ( 2, 3)
2 2
© 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
, Section 1.1 Rectangular Coordinates 5
48. ( −3 + 6, 6 − 3) = (3, 3)
43. d = (42 − 18) + (50 − 12)
2 2
(−5 + 6, 3 − 3) = (1, 0)
= 242 + 382
(−3 + 6, 0 − 3) = (3, − 3)
= 2020
(−1 + 6, 3 − 3) = (5, 0)
= 2 505
≈ 45 49. ( −7 + 4, − 2 + 8) = ( −3, 6)
The pass is about 45 yards. ( −2 + 4, 2 + 8) = ( 2, 10)
44. d = 1202 + 1502 ( −2 + 4, − 4 + 8) = ( 2, 4)
= 36,900 ( −7 + 4, − 4 + 8) = ( −3, 4)
= 30 41 50. (5 − 10, 8 − 6) = ( −5, 2)
≈ 192.09
(3 − 10, 6 − 6) = ( −7, 0)
The plane flies about 192 kilometers.
(7 − 10, 6 − 6) = ( −3, 0)
x + x2 y1 + y2
45. midpoint = 1 , 51. True. Because x < 0 and y > 0, 2 x < 0 and
2 2
− 3 y < 0, which is located in Quadrant III.
2016 + 2018 485.9 + 514.4
= ,
2 2 52. False. The Midpoint Formula would be used 15 times.
= ( 2017, 500.15)
53. True. Two sides of the triangle have lengths 149 and
In 2017, the sales were about $500.15 billion.
the third side has a length of 18.
x + x2 y1 + y2
46. midpoint = 1 , . In this problem, you
2 2 54. False. The polygon could be a rhombus. For example,
2017 + 2019 6.16 + y2 consider the points ( 4, 0), (0, 6), ( − 4, 0), and (0, − 6).
have , = ( 2018, 7.57).
2 2
55. The y-coordinate of a point on the x-axis is 0. The
6.16 + y2 x-coordinates of a point on the y-axis is 0.
So, = 7.57 y2 = 8.98.
2
56. The x- and y-coordinates are switched. For example,
In 2019, the earnings per share were about $8.98.
(3, 2) should be (2, 3).
47. ( − 2 + 2, − 4 + 5) = (0, 1)
x1 + x2 y + y2
(2 + 2, − 3 + 5) = ( 4, 2) 57. Because xm = and ym = 1 we have:
2 2
(−1 + 2, −1 + 5) = (1, 4) 2 xm = x1 + x2 2 ym = y1 + y2
2 xm − x1 = x2 2 ym − y1 = y2
So, ( x2 , y2 ) = ( 2 xm − x1 , 2 ym − y1 ).
x + x2 y1 + y2
58. The midpoint of the given line segment is 1 , .
2 2
x + x2 y + y2
x + x2 y1 + y2 x1 + 1 y1 + 1 = 3x1 + x2 , 3 y1 + y2 .
The midpoint between ( x1 , y1 ) and 1 , is 2 , 2
2 2 4 4
2 2
x + x2 y1 + y2 x1 + x2 y + y2
+ x2 1 + y2 x + 3 x2 y1 + 3 y2
The midpoint between 1 , and ( x2 , y2 ) is 2 , 2
= 1 , .
2 2 4 4
2 2
3 x + x2 3 y1 + y2 x1 + x2 y1 + y2 x1 + 3x2 y1 + 3 y2
So, the three points are 1 , , , , and , .
4 4 2 2 4 4
© 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
, 6 Chapter 1 Functions and Their Graphs
59. Use the Midpoint Formula to prove the diagonals of the
( x1 − x1 ) + ( y2 − y1 )
2 2
62. (a) distance =
parallelogram bisect each other.
( y2 − y1 )
2
b + a c + 0 a + b c = = y2 − y1
, = ,
2 2 2 2
( x2 − x1 ) + ( y1 − y1 )
2 2
(b) distance =
a + b + 0 c + 0 a + b c
, = ,
( x2 − x1 )
2
2 2 2 2 = = x2 − x1
60. (a) Because ( x0 , y0 ) lies in Quadrant II, ( x0 , − y0 ) Answers will vary.
must lie in Quadrant III. Matches (ii). 63. 4 x − 6
(b) Because ( x0 , y0 ) lies in Quadrant II, ( −2 x0 , y0 ) (a) 4( −1) − 6 = −4 − 6 = −10
must lie in Quadrant I. Matches (iii).
(b) 4(0) − 6 = 0 − 6 = −6
(
(c) Because ( x0 , y0 ) lies in Quadrant II, x0 , 12 y0 must )
lie in Quadrant II. Matches (iv). 64. 9 − 7x
(d) Because ( x0 , y0 ) lies in Quadrant II, ( − x0 , − y0 ) (a) 9 − 7( −3) = 9 + 21 = 30
must lie in Quadrant IV. Matches (i). (b) 9 − 7(3) = 9 − 21 = −12
61. (a) First Set
65. (a) When x = −3, 2 x 3 = 2( −3) = 2( −27) = −54.
3
d ( A, B ) = (2 − 2 ) + ( 3 − 6)
2 2
= 9 = 3
(b) When x = 0, 2 x3 = 2(0) = 0.
3
d ( B, C ) = (2 − 6) + (6 − 3)
2 2
= 16 + 9 = 5
−3 3
d ( A, C ) = (2 − 6) + (3 − 3)
2 2
= 16 = 4 66. (a) When x = 0, −3x − 4 = = − . Division by
x4 0
Because 32 + 42 = 52 , A, B, and C are the 0 is undefined.
vertices of a right triangle. −3 −3 3
(b) When x = −2, −3x − 4 = = = − .
Second Set x4 ( )
−2
4
16
d ( A, B ) = (8 − 5) + (3 − 2 )
2 2
= 10
67. ( − x) − 2 = x 2 − 2
2
d ( B, C ) = ( 5 − 2) + ( 2 − 1)
2 2
= 10
68. ( − x) − ( − x) + 2 = − x 3 − x 2 + 2
3 2
d ( A, C ) = (8 − 2) + (3 − 1)
2 2
= 40
69. − x 2 + ( − x) = − x 2 + x 2 = 0
2
A, B, and C are the vertices of an isosceles triangle
or are collinear: 10 + 10 = 2 10 = 40.
70. − x3 + ( − x) + ( − x) − x 4 = − x3 − x3 + x 4 − x 4
3 4
(b)
= −2 x3
71. (a) P = R − C
= 135 x − (93 x + 35,000)
= 42 x − 35,000
(b) P = 42(5000) − 35,000 = $175,000
First set: Not collinear 1
72. (a) Time to copy one page: min
Second set: Collinear. 50
(c) A set of three points is collinear when the sum of x
(b) Time to copy x pages: min
two distances among the points is exactly equal to 50
the third distance. 120 12
(c) Time to copy 120 pages: = = 2.4 min
50 5
10,000
(d) Time to copy 10,000 pages: = 200 min
50
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