SOLUTIONS
, 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 Fitting Models to Data..........................................................................34
Section 1.5 Inverse Functions..................................................................................37
Section 1.6 Exponential and Logarithmic Functions .............................................54
Review Exercises ..........................................................................................................63
Problem Solving ...........................................................................................................73
© 2015 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. y = − 32 x + 3 7. y = 4 − x 2
x-intercept: (2, 0)
x −3 −2 0 2 3
y-intercept: (0, 3)
y −5 0 4 0 −5
Matches graph (b).
y
2. y = 9 − x2 6
x-intercepts: ( −3, 0), (3, 0) (0, 4)
2
y-intercept: (0, 3) (− 2, 0) (2, 0)
x
−6 −4 4 6
Matches graph (d). −2
(−3, − 5) −4 (3, − 5)
3. y = 3 − x 2 −6
x-intercepts: ( )(
3, 0 , − 3, 0 )
8. y = ( x − 3)
2
y-intercept: (0, 3)
Matches graph (a).
x 0 1 2 3 4 5 6
4. y = x − x 3
y 9 4 1 0 1 4 9
x-intercepts: (0, 0), ( −1, 0), (1, 0) y
y-intercept: (0, 0) 10
(0, 9) (6, 9)
8
Matches graph (c).
6
4 (1, 4) (5, 4)
5. y = 1x
2
+ 2 (2, 1)
2
(4, 1)
x
x −4 −2 0 2 4 −6 −4 −2
−2
2 4 6
(3, 0)
y 0 1 2 3 4
9. y = x + 2
y
6 x −5 −4 −3 −2 −1 0 1
(4, 4)
4 (2, 3) y 3 2 1 0 1 2 3
(0, 2)
(− 2, 1) y
x
−4 −2 2 4 6
(−4, 0) −2
4
(−5, 3)
(1, 3)
6. y = 5 − 2 x (− 4, 2) 2 (0, 2)
(− 3, 1) (− 1, 1)
x
x −1 0 1 2 5
2
3 4 −6 −4 (− 2, 0) 2
−2
y 7 5 3 1 0 −1 −3
y
8
(− 1, 7)
(0, 5)
4
(1, 3)
2
(2, 1)
x
−6 −4 −2 (3, −1)
−2
−4
( (
5,0
2 (4, −3)
2 © 2015 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
10. y = x − 1 1
14. y =
x+ 2
x −3 −2 −1 0 1 2 3
x −6 −4 −3 −2 −1 0 2
y 2 1 0 −1 0 1 2
y − 14 − 12 −1 Undef. 1 1
2
1
4
y
y
4
3 5
4
(− 3, 2) 2 (3, 2) 3 (0, 12 )
2
(− 2, 1) (2, 1) (−1, 1)
(2, 14 )
x x
−3 −2 1 2 3 −1 1 2 3
(− 1, 0) −1 (1, 0) (− 6, − 14 ) −2
−2 (0, − 1) (− 4, − 12 ) −3
−4
(−3, − 1)
−5
11. y = x −6
15. y = 5− x
x 0 1 4 9 16
5
y −6 −5 −4 −3 −2 (− 4.00, 3)
(2, 1.73)
y
−6 6
2
x −3
−4 4 8 12 16
(y = )
−2
(2, y) = ( 2, 1.73)
(9, −3)
(16, −2) (a) 5−2 = 3 ≈ 1.73
−4 (4, −4)
(3 = )
(1, −5)
( x, 3) = ( −4, 3) 5 − ( − 4)
−6
(0, −6) (b)
−8
16. y = x 5 − 5 x
12. y = x + 2
6
−2 −1
(−0.5, 2.47)
x 0 2 7 14
−9 9
y 0 1 2 2 3 4
(1, − 4)
y
−6
5
4 (14, 4)
(a) (−0.5, y ) = ( −0.5, 2.47)
3
(− 1, 1)
(7, 3) (b) ( x , − 4) = ( −1.65, − 4) and ( x, − 4) = (1, − 4)
2 (2, 2)
(0, 2)
17. y = 2 x − 5
y-intercept: y = 2(0) − 5 = −5; (0, − 5)
x
(− 2, 0) 5 10 15 20
3 x-intercept: 0 = 2 x − 5
13. y =
x 5 = 2x
x −3 −2 −1 0 1 2 3 x = 5;
2 ( 52 , 0)
y −1 − 32 −3 Undef. 3 3
2
1 18. y = 4 x 2 + 3
y-intercept: y = 4(0) + 3 = 3; (0, 3)
2
y
x-intercept: 0 = 4 x 2 + 3
3
(1, 3)
2 (2, 32 (
1
(3, 1) −3 = 4 x 2
(− 3, −1)
x None. y cannot equal 0.
−3 −2 −1 1 2 3
−1
−2 (−2, − 32 (
(− 1, −3)
© 2015 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.................................................................................2
Section 1.2 Linear Models and Rates of Change....................................................11
Section 1.3 Functions and Their Graphs.................................................................22
Section 1.4 Fitting Models to Data..........................................................................34
Section 1.5 Inverse Functions..................................................................................37
Section 1.6 Exponential and Logarithmic Functions .............................................54
Review Exercises ..........................................................................................................63
Problem Solving ...........................................................................................................73
© 2015 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. y = − 32 x + 3 7. y = 4 − x 2
x-intercept: (2, 0)
x −3 −2 0 2 3
y-intercept: (0, 3)
y −5 0 4 0 −5
Matches graph (b).
y
2. y = 9 − x2 6
x-intercepts: ( −3, 0), (3, 0) (0, 4)
2
y-intercept: (0, 3) (− 2, 0) (2, 0)
x
−6 −4 4 6
Matches graph (d). −2
(−3, − 5) −4 (3, − 5)
3. y = 3 − x 2 −6
x-intercepts: ( )(
3, 0 , − 3, 0 )
8. y = ( x − 3)
2
y-intercept: (0, 3)
Matches graph (a).
x 0 1 2 3 4 5 6
4. y = x − x 3
y 9 4 1 0 1 4 9
x-intercepts: (0, 0), ( −1, 0), (1, 0) y
y-intercept: (0, 0) 10
(0, 9) (6, 9)
8
Matches graph (c).
6
4 (1, 4) (5, 4)
5. y = 1x
2
+ 2 (2, 1)
2
(4, 1)
x
x −4 −2 0 2 4 −6 −4 −2
−2
2 4 6
(3, 0)
y 0 1 2 3 4
9. y = x + 2
y
6 x −5 −4 −3 −2 −1 0 1
(4, 4)
4 (2, 3) y 3 2 1 0 1 2 3
(0, 2)
(− 2, 1) y
x
−4 −2 2 4 6
(−4, 0) −2
4
(−5, 3)
(1, 3)
6. y = 5 − 2 x (− 4, 2) 2 (0, 2)
(− 3, 1) (− 1, 1)
x
x −1 0 1 2 5
2
3 4 −6 −4 (− 2, 0) 2
−2
y 7 5 3 1 0 −1 −3
y
8
(− 1, 7)
(0, 5)
4
(1, 3)
2
(2, 1)
x
−6 −4 −2 (3, −1)
−2
−4
( (
5,0
2 (4, −3)
2 © 2015 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
10. y = x − 1 1
14. y =
x+ 2
x −3 −2 −1 0 1 2 3
x −6 −4 −3 −2 −1 0 2
y 2 1 0 −1 0 1 2
y − 14 − 12 −1 Undef. 1 1
2
1
4
y
y
4
3 5
4
(− 3, 2) 2 (3, 2) 3 (0, 12 )
2
(− 2, 1) (2, 1) (−1, 1)
(2, 14 )
x x
−3 −2 1 2 3 −1 1 2 3
(− 1, 0) −1 (1, 0) (− 6, − 14 ) −2
−2 (0, − 1) (− 4, − 12 ) −3
−4
(−3, − 1)
−5
11. y = x −6
15. y = 5− x
x 0 1 4 9 16
5
y −6 −5 −4 −3 −2 (− 4.00, 3)
(2, 1.73)
y
−6 6
2
x −3
−4 4 8 12 16
(y = )
−2
(2, y) = ( 2, 1.73)
(9, −3)
(16, −2) (a) 5−2 = 3 ≈ 1.73
−4 (4, −4)
(3 = )
(1, −5)
( x, 3) = ( −4, 3) 5 − ( − 4)
−6
(0, −6) (b)
−8
16. y = x 5 − 5 x
12. y = x + 2
6
−2 −1
(−0.5, 2.47)
x 0 2 7 14
−9 9
y 0 1 2 2 3 4
(1, − 4)
y
−6
5
4 (14, 4)
(a) (−0.5, y ) = ( −0.5, 2.47)
3
(− 1, 1)
(7, 3) (b) ( x , − 4) = ( −1.65, − 4) and ( x, − 4) = (1, − 4)
2 (2, 2)
(0, 2)
17. y = 2 x − 5
y-intercept: y = 2(0) − 5 = −5; (0, − 5)
x
(− 2, 0) 5 10 15 20
3 x-intercept: 0 = 2 x − 5
13. y =
x 5 = 2x
x −3 −2 −1 0 1 2 3 x = 5;
2 ( 52 , 0)
y −1 − 32 −3 Undef. 3 3
2
1 18. y = 4 x 2 + 3
y-intercept: y = 4(0) + 3 = 3; (0, 3)
2
y
x-intercept: 0 = 4 x 2 + 3
3
(1, 3)
2 (2, 32 (
1
(3, 1) −3 = 4 x 2
(− 3, −1)
x None. y cannot equal 0.
−3 −2 −1 1 2 3
−1
−2 (−2, − 32 (
(− 1, −3)
© 2015 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
