CHAPTER P
Preparation for Calculus
Section P.1 Graphs and Models................................................................................. 2
Section P.2 Linear Models and Rates of Change.................................................... 11
Section P.3 Functions and Their Graphs .................................................................22
Section P.4 Review of Trigonometric Functions ....................................................35
Review Exercises ..........................................................................................................44
Problem Solving ...........................................................................................................55
© 2023 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 P
Preparation for Calculus
Section P.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
x-intercepts: (0, 0), ( −1, 0), (1, 0) 10. y = ( x − 3)
2
y-intercept: (0, 0)
x 0 1 2 3 4 5 6
Matches graph (c).
y 9 4 1 0 1 4 9
1
7. y = 2
x + 2
x −4 −2 0 2 4
y 0 1 2 3 4
2 © 2023 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 P.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 =
x + 2
y 2 1 0 −1 0 1 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 )
(b) ( x, 3) = ( −4, 3) (3 = 5 − ( −4) )
14. y = x + 2
18. y = x5 − 5 x
x −2 −1 0 2 7 14
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)
© 2023 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 P 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
3(0) + 1
21. y = x 2 + x − 2
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 = x 3 − 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
y-intercept: y = (0 − 1) 02 + 1 x2 =
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.
© 2023 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Preparation for Calculus
Section P.1 Graphs and Models................................................................................. 2
Section P.2 Linear Models and Rates of Change.................................................... 11
Section P.3 Functions and Their Graphs .................................................................22
Section P.4 Review of Trigonometric Functions ....................................................35
Review Exercises ..........................................................................................................44
Problem Solving ...........................................................................................................55
© 2023 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 P
Preparation for Calculus
Section P.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
x-intercepts: (0, 0), ( −1, 0), (1, 0) 10. y = ( x − 3)
2
y-intercept: (0, 0)
x 0 1 2 3 4 5 6
Matches graph (c).
y 9 4 1 0 1 4 9
1
7. y = 2
x + 2
x −4 −2 0 2 4
y 0 1 2 3 4
2 © 2023 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 P.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 =
x + 2
y 2 1 0 −1 0 1 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 )
(b) ( x, 3) = ( −4, 3) (3 = 5 − ( −4) )
14. y = x + 2
18. y = x5 − 5 x
x −2 −1 0 2 7 14
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)
© 2023 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 P 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
3(0) + 1
21. y = x 2 + x − 2
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 = x 3 − 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
y-intercept: y = (0 − 1) 02 + 1 x2 =
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.
© 2023 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
