,Solution manual for Calculus Early Transcendental
Functions, 8th Edition Ron Larson
Notes
1- All Chapters are step by step.
2- We have shown you 10 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:
, C H A P T E R 1
Preparation for Calculus
Section 1.1 Graphs and Models................................................................................. 2
Section 1.2 Linear Models and Rates of Change ................................................... 11
Section 1.3 Functions and Their Graphs ................................................................. 22
Section 1.4 Review of Trigonometric Functions .................................................... 36
Section 1.5 Inverse Functions.................................................................................. 45
Section 1.6 Exponential and Logarithmic Functions ............................................. 62
Review Exercises .......................................................................................................... 72
Problem Solving ........................................................................................................... 87
© 2024 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
Preparation for Calculus
Section 1.1 Graphs and Models
1. To find the x-intercepts of the graph of an equation, 8. y = 5 − 2 x
let y be zero and solve the equation for x. To find the 5
y-intercepts of the graph of an equation, let x be zero x −1 0 1 2 2
3 4
and solve the equation for y. y 7 5 3 1 0 −1 −3
2. Symmetry helps in sketching a graph because you need
only half as many points to plot. Answers will vary.
3. y = − 32 x + 3
x-intercept: ( 2, 0)
y-intercept: (0, 3)
Matches graph (b).
4. y = 9 − x2 9. y = 4 − x 2
x-intercepts: ( −3, 0), (3, 0) x −3 −2 0 2 3
y-intercept: (0, 3) y −5 0 4 0 −5
Matches graph (d).
5. y = 3 − x 2
x-intercepts: ( )(
3, 0 , − 3, 0 )
y-intercept: (0, 3)
Matches graph (a).
6. y = x3 − x
10. y = ( x − 3)
2
x-intercepts: (0, 0), ( −1, 0), (1, 0)
y-intercept: (0, 0) x 0 1 2 3 4 5 6
Matches graph (c). y 9 4 1 0 1 4 9
7. y = 1x +2
2
x −4 −2 0 2 4
y 0 1 2 3 4
2 © 2024 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 Graphs and Models 3
11. y = x + 1 3
15. y =
x
x −4 −3 −2 −1 0 1 2
x −3 −2 −1 0 1 2 3
y 3 2 1 0 1 2 3
y −1 − 32 −3 Undef. 3 3
2
1
12. y = x − 1
x −3 −2 −1 0 1 2 3 1
16. y =
y 2 1 0 −1 0 1 2 x + 2
x −6 −4 −3 −2 −1 0 2
y − 14 − 12 −1 Undef. 1 1
2
1
4
13. y = x −6
x 0 1 4 9 16
y −6 −5 −4 −3 −2 17. y = 5− x
(a) (2, y) = ( 2, 1.73) (y = 5−2 = 3 ≈ 1.73 )
14. y = x + 2
(b) ( x, 3) = ( −4, 3) (3 = 5 − ( −4) )
x −2 −1 0 2 7 14 18. y = x5 − 5 x
y 0 1 2 2 3 4
(a) (−0.5, y) = ( −0.5, 2.47)
(b) ( x, − 4) = ( −1.65, − 4) and ( x, − 4) = (1, − 4)
© 2024 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 Preparation for Calculus
19. y = 2 x − 5 2− x
25. y =
y-intercept: y = 2(0) − 5 = −5; (0, − 5) 5x + 1
x-intercept: 0 = 2 x − 5 2− 0
y -intercept: y = = 2; (0, 2)
5 = 2x 5(0) + 1
x = 5;
2 ( 52 , 0) x-intercept: 0 =
2− x
5x + 1
20. y = 4 x 2 + 3 0 = 2− x
x = 4; (4, 0)
y-intercept: y = 4(0) + 3 = 3; (0, 3)
2
x-intercept: 0 = 4 x 2 + 3 x 2 + 3x
26. y =
(3 x + 1)
2
−3 = 4 x 2
None. y cannot equal 0. 02 + 3(0)
y-intercept: y = 2
21. y = x 2 + x − 2 3(0) + 1
y = 0; (0, 0)
y-intercept: y = 02 + 0 − 2
y = −2; (0, − 2) x 2 + 3x
x-intercepts: 0 =
(3x + 1)
2
2
x-intercepts: 0 = x + x − 2
x( x + 3)
0 = ( x + 2)( x − 1) 0 =
(3x + 1)
2
x = −2, 1; ( −2, 0), (1, 0)
x = 0, − 3; (0, 0), ( −3, 0)
22. y 2 = x3 − 4 x
27. x 2 y − x 2 + 4 y = 0
y-intercept: y 2 = 03 − 4(0)
y-intercept: 02 ( y ) − 02 + 4 y = 0
y = 0; (0, 0)
y = 0; (0, 0)
x-intercepts: 0 = x3 − 4 x
x-intercept: x 2 (0) − x 2 + 4(0) = 0
0 = x( x − 2)( x + 2)
x = 0; (0, 0)
x = 0, ± 2; (0, 0), ( ± 2, 0)
28. y = 2 x − x2 + 1
23. y = x 16 − x 2
y-intercept: y = 2(0) − 02 + 1
y-intercept: y = 0 16 − 02 = 0; (0, 0)
y = −1; (0, −1)
x-intercepts: 0 = x 16 − x 2
x-intercept: 0 = 2x − x2 + 1
0 = x (4 − x)(4 + x)
2x = x2 + 1
x = 0, 4, − 4; (0, 0), ( 4, 0), ( − 4, 0)
4x2 = x2 + 1
24. y = ( x − 1) x2 + 1 3x 2 = 1
1
x2 =
y-intercept: y = (0 − 1) 02 + 1 3
y = −1; (0, −1) 3
x = ±
3
x-intercept: 0 = ( x − 1) x2 + 1
3 3
x = 1; (1, 0) x = ; , 0
3 3
Note: x = − 3 3 is an extraneous solution.
© 2024 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 Graphs and Models 5
29. Symmetric with respect to the y-axis because 41. y = 2 − 3 x
y = ( − x) − 6 = x − 6.
2 2
y = 2 − 3(0) = 2, y -intercept
30. y = 9 x − x 2 0 = 2 − 3( x) 3 x = 2 x = 2
3
, x-intercept
No symmetry with respect to either axis or the origin. Intercepts: (0, 2), ( 23 , 0)
31. Symmetric with respect to the x-axis because Symmetry: none
(− y )2 = y 2 = x3 − 8 x.
32. Symmetric with respect to the origin because
(− y ) = ( − x) + ( − x)
3
− y = − x3 − x
42. y = 2x +1
y = x3 + x. 3
33. Symmetric with respect to the origin because y = 2
3
( 0) + 1 = 1, y -intercept
( − x)( − y ) = xy = 4. 0 = 2x + 1 − 23 x = 1 x = − 32 , x-intercept
3
34. Symmetric with respect to the x-axis because Intercepts: (0, 1), − 32 , 0( )
x( − y ) = xy = −10.
2 2
Symmetry: none
35. y = 4 − x +3
No symmetry with respect to either axis or the origin.
36. Symmetric with respect to the origin because
(− x)(− y ) 4 − ( − x) = 0
2
−
43. y = 9 − x 2
2
xy − 4− x = 0.
y = 9 − (0) = 9, y -intercept
2
37. Symmetric with respect to the origin because 0 = 9 − x 2 x 2 = 9 x = ± 3, x-intercepts
−x
−y = Intercepts: (0, 9), (3, 0), ( −3, 0)
( − x)
2
+1
y = 9 − (− x) = 9 − x 2
2
x
y = 2 .
x +1 Symmetry: y-axis
38. Symmetric with respect to the origin because
( − x)
5
−y =
− ( − x)
2
4
− x5 44. y = 2 x 2 + x = x( 2 x + 1)
−y =
4 − x2 y = 0( 2(0) + 1) = 0, y -intercept
x5
y = . 0 = x( 2 x + 1) x = 0, − 12 , x-intercepts
4 − x2
39. y = x 3 + x is symmetric with respect to the y-axis
Intercepts: (0, 0), − 12 , 0( )
Symmetry: none
because y = ( − x) + ( − x) = −( x3 + x) = x3 + x .
3
40. y − x = 3 is symmetric with respect to the x-axis
because
−y − x = 3
y − x = 3.
© 2024 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 Preparation for Calculus
45. y = x3 + 2 48. y = 25 − x 2
y = 03 + 2 = 2, y -intercept y = 25 − 02 = 25 = 5, y -intercept
3
0 = x + 2 x = −2 x = −
3 3
2, x-intercept
25 − x 2 = 0
Intercepts: − ( 3
)
2, 0 , (0, 2) 25 − x 2 = 0
Symmetry: none (5 + x)(5 − x) = 0
x = ± 5, x -intercept
Intercepts: (0, 5), (5, 0), ( −5, 0)
25 − ( − x) =
2
y = 25 − x 2
Symmetry: y-axis
46. y = x 3 − 4 x
y = 03 − 4(0) = 0, y -intercept
x3 − 4 x = 0
x ( x 2 − 4) = 0
x( x + 2)( x − 2) = 0 49. x = y 3
x = 0, ± 2, x -intercepts y 3 = 0 y = 0, y -intercept
Intercepts: (0, 0), ( 2, 0), ( −2, 0) x = 0, x-intercept
Intercept: (0, 0)
y = ( − x ) − 4( − x ) = − x 3 + 4 x = −( x 3 − 4 x)
3
− x = (− y ) − x = − y 3
3
Symmetry: origin
Symmetry: origin
50. x = y 4 − 16
y 4 − 16 = 0
( y2 − 4)( y 2 + 4) = 0
(y − 2)( y + 2)( y 2 + 4) = 0
y = ± 2, y -intercepts
47. y = x x +5 x = 0 4 − 16 = −16, x -intercept
y = 0 0 + 5 = 0, y -intercept Intercepts: (0, 2), (0, − 2), ( −16, 0)
x x + 5 = 0 x = 0, − 5, x-intercepts
Symmetry: x-axis because x = ( − y ) − 16 = y 4 − 16
4
Intercepts: (0, 0), ( −5, 0)
Symmetry: none
© 2024 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 Graphs and Models 7
8 54. y = 6 − x
51. y =
x
y = 6 − 0 = 6 = 6, y -intercept
8
y = Undefined no y -intercept
0 6− x = 0
8 6− x = 0
= 0 No solution no x-intercept
x 6 = x, x-intercept
Intercepts: none
Intercepts: (0, 6), (6, 0)
8 8
−y = y = Symmetry: none
−x x
Symmetry: origin
55. x 2 + y 2 = 9
y2 = 9 − x2
10
52. y = y = ± 9 − x2
x2 + 1
10 y = ± 9 − 0 = ±3, y -intercepts
y = 2
= 10, y -intercept
0 +1 ± 9− x 2
= 0
10 9− x = 0 2
2
= 0 No solution no x-intercepts
x +1
9 = x2
Intercept: (0, 10) ±3 = x, x-intercepts
10 10
y = = 2 Intercepts: ( ±3, 0), (0, ± 3)
( − x)
2
+1 x +1
Symmetry: y-axis ( − x)2 + y2 = 9 x2 + y2 = 9
x2 + (− y) = 9 x 2 + y 2 = 9
2
(− x)2 + (− y) = 9 x 2 + y 2 = 9
2
Symmetry: x-axis, y-axis, origin
53. y = 6 − x
y = 6 − 0 = 6, y -intercept
6− x = 0
6 = x
x = ± 6, x-intercepts
Intercepts: (0, 6), ( −6, 0), (6, 0)
y = 6 − −x = 6 − x
Symmetry: y-axis
© 2024 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
, 8 Chapter 1 Preparation for Calculus
4 − x2 58. 3x − 4 y 2 = 8
56. x 2 + 4 y 2 = 4 y = ±
2 − 4 y 2 = − 3x + 8
4 − 02 4 y2 = 3
x − 2
y = ± = ± = ±1, y -intercepts 4
2 2 3
y = ± 4
x − 2
x + 4(0) = 4
2 2
y = ± 0− 2
x2 = 4
No real solution No y -intercepts
x = ± 2, x -intercepts
Intercepts: ( −2, 0), ( 2, 0), (0, −1), (0, 1) ± 3
4
x − 2 = 0
3
( − x)2 + 4( − y ) = 4 x 2 + 4 y 2 = 4
2
4
x − 2 = 0
x = 83 , x-intercept
Symmetry: origin and both axes
Intercept: ( 83 , 0)
Symmetry: x-axis
59. x + y = 8 y = 8− x
4x − y = 7 y = 4x − 7
8 − x = 4x − 7
15 = 5 x
57. 3 y 2 − x = 9
3 = x
3y2 = x + 9
The corresponding y-value is y = 5.
y2 = 1x +3
3 Point of intersection: (3, 5)
y = ± 1x +3
3
3x + 4
y = ± 0+3 = ± 3, y -intercepts 60. 3x − 2 y = − 4 y =
2
1 − 4 x − 10
± 3
x +3 = 0 4 x + 2 y = −10 y =
2
1
3
x +3 = 0 3x + 4 − 4 x − 10
=
x = − 9, x-intercept 2 2
3 x + 4 = − 4 x − 10
Intercepts: 0, ( )(
3 , 0, − )
3 , ( − 9, 0)
7 x = −14
3( − y ) − x = 3 y − x = 9
2 2 x = −2
The corresponding y-value is y = −1.
Symmetry: x-axis
Point of intersection: ( −2, −1)
61. x 2 + y = 15 y = − x 2 + 15
− 3 x + y = 11 y = 3 x + 11
− x 2 + 15 = 3 x + 11
0 = x 2 + 3x − 4
0 = ( x + 4)( x − 1)
x = − 4, 1
The corresponding y-values are y = −1 (for x = − 4)
and y = 14 (for x = 1).
Points of intersection: ( − 4, −1), (1, 14)
© 2024 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Functions, 8th Edition Ron Larson
Notes
1- All Chapters are step by step.
2- We have shown you 10 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:
, C H A P T E R 1
Preparation for Calculus
Section 1.1 Graphs and Models................................................................................. 2
Section 1.2 Linear Models and Rates of Change ................................................... 11
Section 1.3 Functions and Their Graphs ................................................................. 22
Section 1.4 Review of Trigonometric Functions .................................................... 36
Section 1.5 Inverse Functions.................................................................................. 45
Section 1.6 Exponential and Logarithmic Functions ............................................. 62
Review Exercises .......................................................................................................... 72
Problem Solving ........................................................................................................... 87
© 2024 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
Preparation for Calculus
Section 1.1 Graphs and Models
1. To find the x-intercepts of the graph of an equation, 8. y = 5 − 2 x
let y be zero and solve the equation for x. To find the 5
y-intercepts of the graph of an equation, let x be zero x −1 0 1 2 2
3 4
and solve the equation for y. y 7 5 3 1 0 −1 −3
2. Symmetry helps in sketching a graph because you need
only half as many points to plot. Answers will vary.
3. y = − 32 x + 3
x-intercept: ( 2, 0)
y-intercept: (0, 3)
Matches graph (b).
4. y = 9 − x2 9. y = 4 − x 2
x-intercepts: ( −3, 0), (3, 0) x −3 −2 0 2 3
y-intercept: (0, 3) y −5 0 4 0 −5
Matches graph (d).
5. y = 3 − x 2
x-intercepts: ( )(
3, 0 , − 3, 0 )
y-intercept: (0, 3)
Matches graph (a).
6. y = x3 − x
10. y = ( x − 3)
2
x-intercepts: (0, 0), ( −1, 0), (1, 0)
y-intercept: (0, 0) x 0 1 2 3 4 5 6
Matches graph (c). y 9 4 1 0 1 4 9
7. y = 1x +2
2
x −4 −2 0 2 4
y 0 1 2 3 4
2 © 2024 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 Graphs and Models 3
11. y = x + 1 3
15. y =
x
x −4 −3 −2 −1 0 1 2
x −3 −2 −1 0 1 2 3
y 3 2 1 0 1 2 3
y −1 − 32 −3 Undef. 3 3
2
1
12. y = x − 1
x −3 −2 −1 0 1 2 3 1
16. y =
y 2 1 0 −1 0 1 2 x + 2
x −6 −4 −3 −2 −1 0 2
y − 14 − 12 −1 Undef. 1 1
2
1
4
13. y = x −6
x 0 1 4 9 16
y −6 −5 −4 −3 −2 17. y = 5− x
(a) (2, y) = ( 2, 1.73) (y = 5−2 = 3 ≈ 1.73 )
14. y = x + 2
(b) ( x, 3) = ( −4, 3) (3 = 5 − ( −4) )
x −2 −1 0 2 7 14 18. y = x5 − 5 x
y 0 1 2 2 3 4
(a) (−0.5, y) = ( −0.5, 2.47)
(b) ( x, − 4) = ( −1.65, − 4) and ( x, − 4) = (1, − 4)
© 2024 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 Preparation for Calculus
19. y = 2 x − 5 2− x
25. y =
y-intercept: y = 2(0) − 5 = −5; (0, − 5) 5x + 1
x-intercept: 0 = 2 x − 5 2− 0
y -intercept: y = = 2; (0, 2)
5 = 2x 5(0) + 1
x = 5;
2 ( 52 , 0) x-intercept: 0 =
2− x
5x + 1
20. y = 4 x 2 + 3 0 = 2− x
x = 4; (4, 0)
y-intercept: y = 4(0) + 3 = 3; (0, 3)
2
x-intercept: 0 = 4 x 2 + 3 x 2 + 3x
26. y =
(3 x + 1)
2
−3 = 4 x 2
None. y cannot equal 0. 02 + 3(0)
y-intercept: y = 2
21. y = x 2 + x − 2 3(0) + 1
y = 0; (0, 0)
y-intercept: y = 02 + 0 − 2
y = −2; (0, − 2) x 2 + 3x
x-intercepts: 0 =
(3x + 1)
2
2
x-intercepts: 0 = x + x − 2
x( x + 3)
0 = ( x + 2)( x − 1) 0 =
(3x + 1)
2
x = −2, 1; ( −2, 0), (1, 0)
x = 0, − 3; (0, 0), ( −3, 0)
22. y 2 = x3 − 4 x
27. x 2 y − x 2 + 4 y = 0
y-intercept: y 2 = 03 − 4(0)
y-intercept: 02 ( y ) − 02 + 4 y = 0
y = 0; (0, 0)
y = 0; (0, 0)
x-intercepts: 0 = x3 − 4 x
x-intercept: x 2 (0) − x 2 + 4(0) = 0
0 = x( x − 2)( x + 2)
x = 0; (0, 0)
x = 0, ± 2; (0, 0), ( ± 2, 0)
28. y = 2 x − x2 + 1
23. y = x 16 − x 2
y-intercept: y = 2(0) − 02 + 1
y-intercept: y = 0 16 − 02 = 0; (0, 0)
y = −1; (0, −1)
x-intercepts: 0 = x 16 − x 2
x-intercept: 0 = 2x − x2 + 1
0 = x (4 − x)(4 + x)
2x = x2 + 1
x = 0, 4, − 4; (0, 0), ( 4, 0), ( − 4, 0)
4x2 = x2 + 1
24. y = ( x − 1) x2 + 1 3x 2 = 1
1
x2 =
y-intercept: y = (0 − 1) 02 + 1 3
y = −1; (0, −1) 3
x = ±
3
x-intercept: 0 = ( x − 1) x2 + 1
3 3
x = 1; (1, 0) x = ; , 0
3 3
Note: x = − 3 3 is an extraneous solution.
© 2024 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 Graphs and Models 5
29. Symmetric with respect to the y-axis because 41. y = 2 − 3 x
y = ( − x) − 6 = x − 6.
2 2
y = 2 − 3(0) = 2, y -intercept
30. y = 9 x − x 2 0 = 2 − 3( x) 3 x = 2 x = 2
3
, x-intercept
No symmetry with respect to either axis or the origin. Intercepts: (0, 2), ( 23 , 0)
31. Symmetric with respect to the x-axis because Symmetry: none
(− y )2 = y 2 = x3 − 8 x.
32. Symmetric with respect to the origin because
(− y ) = ( − x) + ( − x)
3
− y = − x3 − x
42. y = 2x +1
y = x3 + x. 3
33. Symmetric with respect to the origin because y = 2
3
( 0) + 1 = 1, y -intercept
( − x)( − y ) = xy = 4. 0 = 2x + 1 − 23 x = 1 x = − 32 , x-intercept
3
34. Symmetric with respect to the x-axis because Intercepts: (0, 1), − 32 , 0( )
x( − y ) = xy = −10.
2 2
Symmetry: none
35. y = 4 − x +3
No symmetry with respect to either axis or the origin.
36. Symmetric with respect to the origin because
(− x)(− y ) 4 − ( − x) = 0
2
−
43. y = 9 − x 2
2
xy − 4− x = 0.
y = 9 − (0) = 9, y -intercept
2
37. Symmetric with respect to the origin because 0 = 9 − x 2 x 2 = 9 x = ± 3, x-intercepts
−x
−y = Intercepts: (0, 9), (3, 0), ( −3, 0)
( − x)
2
+1
y = 9 − (− x) = 9 − x 2
2
x
y = 2 .
x +1 Symmetry: y-axis
38. Symmetric with respect to the origin because
( − x)
5
−y =
− ( − x)
2
4
− x5 44. y = 2 x 2 + x = x( 2 x + 1)
−y =
4 − x2 y = 0( 2(0) + 1) = 0, y -intercept
x5
y = . 0 = x( 2 x + 1) x = 0, − 12 , x-intercepts
4 − x2
39. y = x 3 + x is symmetric with respect to the y-axis
Intercepts: (0, 0), − 12 , 0( )
Symmetry: none
because y = ( − x) + ( − x) = −( x3 + x) = x3 + x .
3
40. y − x = 3 is symmetric with respect to the x-axis
because
−y − x = 3
y − x = 3.
© 2024 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 Preparation for Calculus
45. y = x3 + 2 48. y = 25 − x 2
y = 03 + 2 = 2, y -intercept y = 25 − 02 = 25 = 5, y -intercept
3
0 = x + 2 x = −2 x = −
3 3
2, x-intercept
25 − x 2 = 0
Intercepts: − ( 3
)
2, 0 , (0, 2) 25 − x 2 = 0
Symmetry: none (5 + x)(5 − x) = 0
x = ± 5, x -intercept
Intercepts: (0, 5), (5, 0), ( −5, 0)
25 − ( − x) =
2
y = 25 − x 2
Symmetry: y-axis
46. y = x 3 − 4 x
y = 03 − 4(0) = 0, y -intercept
x3 − 4 x = 0
x ( x 2 − 4) = 0
x( x + 2)( x − 2) = 0 49. x = y 3
x = 0, ± 2, x -intercepts y 3 = 0 y = 0, y -intercept
Intercepts: (0, 0), ( 2, 0), ( −2, 0) x = 0, x-intercept
Intercept: (0, 0)
y = ( − x ) − 4( − x ) = − x 3 + 4 x = −( x 3 − 4 x)
3
− x = (− y ) − x = − y 3
3
Symmetry: origin
Symmetry: origin
50. x = y 4 − 16
y 4 − 16 = 0
( y2 − 4)( y 2 + 4) = 0
(y − 2)( y + 2)( y 2 + 4) = 0
y = ± 2, y -intercepts
47. y = x x +5 x = 0 4 − 16 = −16, x -intercept
y = 0 0 + 5 = 0, y -intercept Intercepts: (0, 2), (0, − 2), ( −16, 0)
x x + 5 = 0 x = 0, − 5, x-intercepts
Symmetry: x-axis because x = ( − y ) − 16 = y 4 − 16
4
Intercepts: (0, 0), ( −5, 0)
Symmetry: none
© 2024 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 Graphs and Models 7
8 54. y = 6 − x
51. y =
x
y = 6 − 0 = 6 = 6, y -intercept
8
y = Undefined no y -intercept
0 6− x = 0
8 6− x = 0
= 0 No solution no x-intercept
x 6 = x, x-intercept
Intercepts: none
Intercepts: (0, 6), (6, 0)
8 8
−y = y = Symmetry: none
−x x
Symmetry: origin
55. x 2 + y 2 = 9
y2 = 9 − x2
10
52. y = y = ± 9 − x2
x2 + 1
10 y = ± 9 − 0 = ±3, y -intercepts
y = 2
= 10, y -intercept
0 +1 ± 9− x 2
= 0
10 9− x = 0 2
2
= 0 No solution no x-intercepts
x +1
9 = x2
Intercept: (0, 10) ±3 = x, x-intercepts
10 10
y = = 2 Intercepts: ( ±3, 0), (0, ± 3)
( − x)
2
+1 x +1
Symmetry: y-axis ( − x)2 + y2 = 9 x2 + y2 = 9
x2 + (− y) = 9 x 2 + y 2 = 9
2
(− x)2 + (− y) = 9 x 2 + y 2 = 9
2
Symmetry: x-axis, y-axis, origin
53. y = 6 − x
y = 6 − 0 = 6, y -intercept
6− x = 0
6 = x
x = ± 6, x-intercepts
Intercepts: (0, 6), ( −6, 0), (6, 0)
y = 6 − −x = 6 − x
Symmetry: y-axis
© 2024 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
, 8 Chapter 1 Preparation for Calculus
4 − x2 58. 3x − 4 y 2 = 8
56. x 2 + 4 y 2 = 4 y = ±
2 − 4 y 2 = − 3x + 8
4 − 02 4 y2 = 3
x − 2
y = ± = ± = ±1, y -intercepts 4
2 2 3
y = ± 4
x − 2
x + 4(0) = 4
2 2
y = ± 0− 2
x2 = 4
No real solution No y -intercepts
x = ± 2, x -intercepts
Intercepts: ( −2, 0), ( 2, 0), (0, −1), (0, 1) ± 3
4
x − 2 = 0
3
( − x)2 + 4( − y ) = 4 x 2 + 4 y 2 = 4
2
4
x − 2 = 0
x = 83 , x-intercept
Symmetry: origin and both axes
Intercept: ( 83 , 0)
Symmetry: x-axis
59. x + y = 8 y = 8− x
4x − y = 7 y = 4x − 7
8 − x = 4x − 7
15 = 5 x
57. 3 y 2 − x = 9
3 = x
3y2 = x + 9
The corresponding y-value is y = 5.
y2 = 1x +3
3 Point of intersection: (3, 5)
y = ± 1x +3
3
3x + 4
y = ± 0+3 = ± 3, y -intercepts 60. 3x − 2 y = − 4 y =
2
1 − 4 x − 10
± 3
x +3 = 0 4 x + 2 y = −10 y =
2
1
3
x +3 = 0 3x + 4 − 4 x − 10
=
x = − 9, x-intercept 2 2
3 x + 4 = − 4 x − 10
Intercepts: 0, ( )(
3 , 0, − )
3 , ( − 9, 0)
7 x = −14
3( − y ) − x = 3 y − x = 9
2 2 x = −2
The corresponding y-value is y = −1.
Symmetry: x-axis
Point of intersection: ( −2, −1)
61. x 2 + y = 15 y = − x 2 + 15
− 3 x + y = 11 y = 3 x + 11
− x 2 + 15 = 3 x + 11
0 = x 2 + 3x − 4
0 = ( x + 4)( x − 1)
x = − 4, 1
The corresponding y-values are y = −1 (for x = − 4)
and y = 14 (for x = 1).
Points of intersection: ( − 4, −1), (1, 14)
© 2024 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
