Lesson 5.1 Skills Practice
Name
Date
Go for the Curve!
Comparing Linear and Exponential Functions
Vocabulary
Describe each type of account as simple interest or compound interest based on the scenario given.
Explain your reasoning.
1. Andrew deposits $300 into an account that earns 2% interest each year. After the first year, Andrew
has $306 in the account. After the second year, Andrew has $312 in the account, and after the third
year, Andrew has $318 in the account.
This is a simple interest account because the interest earned at the end of each year is a percent
of the original deposit amount.
2. Marilyn deposits $600 in an account that earns 1.5% interest each year. After the first year, Marilyn
has $609 in the account. After the second year, Marilyn has $618.14 in the account, and after the
third year, Marilyn has $627.41 in the account.
This is a compound interest account because the interest earned at the end of each year is a
percent of the account balance at the beginning of the year.
Problem Set
Write a function to represent each problem situation.
5
1. Nami deposits $500 into a simple interest account. The interest rate for the account is 3%. Write a
function that represents the balance in the account as a function of time t.
© 2012 Carnegie Learning
P(t) 5 P0 1 (P0 ? r)t
P(t) 5 500 1 (500 ? 0.03)t
P(t) 5 500 1 15t
2. Carmen deposits $1000 into a simple interest account. The interest rate for the account is 4%. Write
a function that represents the balance in the account as a function of time t.
P(t) 5 P0 1 (P0 ? r)t
P(t) 5 1000 1 (1000 ? 0.04)t
P(t) 5 1000 1 40t
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Lesson 5.1 Skills Practice
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3. Emilio deposits $250 into a simple interest account. The interest rate for the account is 2.5%. Write a
function that represents the balance in the account as a function of time t.
P(t) 5 P0 1 (P0 ? r)t
P(t) 5 250 1 (250 ? 0.025)t
P(t) 5 250 1 6.25t
4. Vance deposits $1500 into a simple interest account. The interest rate for the account is 5.5%. Write
a function that represents the balance in the account as a function of time t.
P(t) 5 P0 1 (P0 ? r)t
P(t) 5 1500 1 (1500 ? 0.055)t
P(t) 5 1500 1 82.5t
5. Perry deposits $175 into a simple interest account. The interest rate for the account is 4.25%. Write a
function that represents the balance in the account as a function of time t.
P(t) 5 P0 1 (P0 ? r)t
P(t) 5 175 1 (175 ? 0.0425)t
P(t) 5 175 1 7.4375t
6. Julian deposits $5000 into a simple interest account. The interest rate for the account is 2.75%. Write
a function that represents the balance in the account as a function of time t.
P(t) 5 P0 1 (P0 ? r)t
P(t) 5 5000 1 (5000 ? 0.0275)t
P(t) 5 5000 1 137.5t
Sherwin deposits $500 into a simple interest account. The interest rate for the account is 3.75%. The
function P(t) 5 500 1 18.75t represents the balance in the account as a function of time. Determine the
account balance after each given number of years.
7. 3 years
P(t) 5 500 1 18.75t
P(t) 5 500 1 18.75t
P(3) 5 500 1 18.75(3)
P(2) 5 500 1 18.75(2)
P(3) 5 556.25
P(2) 5 537.5
In 3 years, the account balance will
be $556.25.
In 2 years, the account balance will
be $537.50.
9. 10 years
382 8. 2 years
10. 15 years
P(t) 5 500 1 18.75t
P(t) 5 500 1 18.75t
P(10) 5 500 1 18.75(10)
P(15) 5 500 1 18.75(15)
P(10) 5 687.5
P(15) 5 781.25
In 10 years, the account balance
will be $687.50.
In 15 years, the account balance will
be $781.25.
© 2012 Carnegie Learning
5
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Lesson 5.1 Skills Practice
page 3
Name
Date
11. 50 years
12. 75 years
P(t) 5 500 1 18.75t
P(t) 5 500 1 18.75t
P(50) 5 500 1 18.75(50)
P(75) 5 500 1 18.75(75)
P(50) 5 1437.5
P(75) 5 1906.25
In 50 years, the account balance
will be $1437.50.
In 75 years, the account balance will
be $1906.25.
Hector deposits $400 into a simple interest account. The interest rate for the account is 5.25%. The
function P(t) 5 400 1 21t represents the balance in the account as a function of time. Determine the
number of years it will take for the account balance to reach each given amount.
13. $505
14. $610
P(t) 5 400 1 21t
P(t) 5 400 1 21t
505 5 400 1 21t
610 5 400 1 21t
105 5 21t
210 5 21t
55t
It will take 5 years for the account balance to reach $505.
15. $1450
It will take 10 years for the account balance
to reach $610.
16. $2500
P(t) 5 400 1 21t
P(t) 5 400 1 21t
1450 5 400 1 21t
2500 5 400 1 21t
1050 5 21t
2100 5 21t
50 5 t
It will take 50 years for the account
balance to reach $1450.
© 2012 Carnegie Learning
10 5 t
17. double the original deposit
It will take 100 years for the account balance
to reach $2500.
18. triple the original deposit
P(t) 5 400 1 21t
P(t) 5 400 1 21t
800 5 400 1 21t
1200 5 400 1 21t
400 5 21t
19 ¯ t
5
100 5 t
800 5 21t
38 ¯ t
It will take about 19 years for the account It will take about 38 years for the account
balance to reach $800. balance to reach $1200.
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Lesson 5.1 Skills Practice
page 4
Write a function to represent each problem situation.
19. Ronna deposits $500 into a compound interest account. The interest rate for the account is 4%.
P(t) 5 P0 ? (1 1 r)t
P(t) 5 500 ? (1 1 0.04)t
P(t) 5 500 ? 1.04t
20. Leon deposits $250 into a compound interest account. The interest rate for the account is 6%.
P(t) 5 P0 ? (1 1 r)t
P(t) 5 250 ? (1 1 0.06)t
P(t) 5 250 ? 1.06t
21. Chen deposits $1200 into a compound interest account. The interest rate for the account is 3.5%.
P(t) 5 P0 ? (1 1 r)t
P(t) 5 1200 ? (1 1 0.035)t
P(t) 5 1200 ? 1.035t
22. Serena deposits $2700 into a compound interest account. The interest rate for the account is 4.25%.
P(t) 5 P0 ? (1 1 r)t
P(t) 5 2700 ? (1 1 0.0425)t
P(t) 5 2700 ? 1.0425t
5
23. Shen deposits $300 into a compound interest account. The interest rate for the account is 1.75%.
P(t) 5 P0 ? (1 1 r)t
P(t) 5 300 ? 1.0175t
24. Lea deposits $450 into a compound interest account. The interest rate for the account is 5.5%.
P(t) 5 P0 ? (1 1 r)t
P(t) 5 450 ? (1 1 0.055)t
© 2012 Carnegie Learning
P(t) 5 300 ? (1 1 0.0175)t
P(t) 5 450 ? 1.055t
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Lesson 5.1 Skills Practice
page 5
Name
Date
Cisco deposits $500 into a compound interest account. The interest rate for the account is 3.25%. The
function P(t) 5 500 ? 1.0325t represents the balance in the account as a function of time. Determine the
account balance after each given number of years.
25. 2 years
P(t) 5 500 ? 1.0325
26. 4 years
P(t) 5 500 ? 1.0325t
t
P(2) 5 500 ? 1.03252
P(4) 5 500 ? 1.03254
P(2) ¯ 533.03
P(4) ¯ 568.24
In 2 years, the account balance will
be $533.03.
In 4 years, the account balance will
be $568.24.
27. 15 years
28. 20 years
P(t) 5 500 ? 1.0325
P(t) 5 500 ? 1.0325t
P(15) 5 500 ? 1.032515
P(20) 5 500 ? 1.032520
P(15) ¯ 807.83
P(20) ¯ 947.92
In 15 years, the account balance will
be $807.83.
In 20 years, the account balance will
be $947.92.
t
© 2012 Carnegie Learning
29. 50 years
30. 65 years
P(t) 5 500 ? 1.0325t
P(t) 5 500 ? 1.0325t
P(50) 5 500 ? 1.032550
P(65) 5 500 ? 1.032565
P(50) ¯ 2474.42
P(65) ¯ 3997.83
In 50 years, the account balance will
be 2474.42.
In 65 years, the account balance will
be $3997.83.
5
Mario deposits $1000 into a compound interest account. The interest rate for the account is 5%. The
function P(t) 5 1000 ? 1.05t represents the balance in the account as a function of time. Use a graphing
calculator to estimate the number of years it will take for the account balance to reach each given amount.
31. $1500
32. $4000
It will take about 8.3 years for the It will take about 28.4 years for the account
account balance to reach $1500. balance to reach $4000.
33. $6000
34. $10,000
It will take about 36.7 years for the It will take about 47.2 years for the account
account balance to reach $6000. balance to reach $10,000.
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Lesson 5.1 Skills Practice
35. double the original amount
page 6
36. triple the original amount
It will take about 14.2 years for the It will take about 22.5 years for the account
account balance to reach $2000. balance to reach $3000.
Use the simple and compound interest formula to complete each table. Round to the nearest cent.
37. Teresa has $300 to deposit into an account. The interest rate available for the account is 4%.
Quantity
Time
Simple Interest
Balance
Compound
Interest Balance
Units
years
dollars
dollars
t
300 1 12t
300 ? 1.04t
0
300.00
300.00
2
324.00
324.48
6
372.00
379.60
10
420.00
444.07
Expression
5
Quantity
Time
Simple Interest
Balance
Compound
Interest Balance
Units
years
dollars
dollars
t
700 1 42t
700 ? 1.06t
0
700.00
700.00
3
826.00
833.71
10
1120.00
1253.59
20
1540.00
2244.99
Expression
386 © 2012 Carnegie Learning
38. Ye has $700 to deposit into an account. The interest rate available for the account is 6%.
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Lesson 5.1 Skills Practice
page 7
Name
Date
39. Pablo has $1100 to deposit into an account. The interest rate available for the account is 3.5%.
Quantity
Time
Simple Interest
Balance
Compound
Interest Balance
Units
years
dollars
dollars
t
1100 1 38.5t
1100 ? 1.035t
0
1100.00
1100.00
5
1292.50
1306.45
10
1485.00
1551.66
30
2255.00
3087.47
Expression
40. Ty has $525 to deposit into an account. The interest rate available for the account is 2.5%.
Quantity
Time
Simple Interest
Balance
Compound
Interest Balance
Units
years
dollars
dollars
t
525 1 13.125t
525 ? 1.025t
0
525.00
525.00
10
656.25
672.04
20
787.50
860.27
50
1181.25
1804.48
© 2012 Carnegie Learning
Expression
5
Chapter 5 Skills Practice 8069_Skills_Ch05.indd 387
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Lesson 5.1 Skills Practice
page 8
41. Xavier has $2300 to deposit into an account. The interest rate available for the account is 3.75%.
Quantity
Time
Simple Interest
Balance
Compound
Interest Balance
Units
years
dollars
dollars
t
2300 1 86.25t
2300 ? 1.0375t
0
2300.00
2300.00
2
2472.50
2475.73
5
2731.25
2764.83
15
3593.75
3995.30
Expression
Quantity
Time
Simple Interest
Balance
Compound
Interest Balance
Units
years
dollars
dollars
t
100 1 6.25t
100 ? 1.0625t
0
100.00
100.00
5
131.25
135.41
15
193.75
248.28
30
287.50
616.41
Expression
5
388 © 2012 Carnegie Learning
42. Denisa has $100 to deposit into an account. The interest rate available for the account is 6.25%.
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Lesson 5.2 Skills Practice
Name
Date
Downtown and Uptown
Graphs of Exponential Functions
Vocabulary
Define the term in your own words.
1. horizontal asymptote
A horizontal asymptote is a horizontal line that the graph of a function gets closer and closer to
but never intersects.
Problem Set
Write a function that represents each population as a function of time.
1. Blueville has a population of 7000. Its population is increasing at a rate of 1.4%.
P(t) 5 P0 ? (1 1 r)t
P(t) 5 7000 ? (1 1 0.014)t
P(t) 5 7000 ? 1.014t
2. Youngstown has a population of 12,000. Its population is increasing at a rate of 1.2%.
P(t) 5 P0 ? (1 1 r)t
P(t) 5 12,000 ? (1 1 0.012)t
5
P(t) 5 12,000 ? 1.012t
© 2012 Carnegie Learning
3. Greenville has a population of 8000. Its population is decreasing at a rate of 1.75%.
P(t) 5 P0 ? (1 2 r)t
P(t) 5 8000 ? (1 2 0.0175)t
P(t) 5 8000 ? 0.9825t
4. North Park has a population of 14,000. Its population is decreasing at a rate of 3.1%.
P(t) 5 P0 ? (1 2 r)t
P(t) 5 14,000 ? (1 2 0.031)t
P(t) 5 14,000 ? 0.969t
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Lesson 5.2 Skills Practice
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5. West Lake has a population of 9500. Its population is increasing at a rate of 2.8%.
P(t) 5 P0 ? (1 1 r)t
P(t) 5 9500 ? (1 1 0.028)t
P(t) 5 9500 ? 0.028t
6. Springfield has a population of 11,500. Its population is decreasing at a rate of 1.25%.
P(t) 5 P0 ? (1 2 r)t
P(t) 5 11,500 ? (1 2 0.0125)t
P(t) 5 11,500 ? 0.9875t
Waynesburg has a population of 16,000. Its population is increasing at a rate of 1.5%. The function
P(t) 5 16,000 ? 1.015t represents the population as a function of time. Determine the population after each
given number of years. Round your answer to the nearest whole number.
7. 1 year
8. 3 years
t
P(t) 5 16,000 ? 1.015t
P(1) 5 16,000 ? 1.0151
P(3) 5 16,000 ? 1.0153
P(t) 5 16,000 ? 1.015
P(3) ¯ 16.731
The population after 1 year The population after 3 years
will be 16,240. will be about 16,731.
P(1) 5 16,240
9. 5 years
10. 10 years
P(t) 5 16,000 ? 1.015
P(t) 5 16,000 ? 1.015t
P(5) 5 16,000 ? 1.0155
P(10) 5 16,000 ? 1.01510
t
5
P(10) ¯ 18.569
The population after 5 years will be The population after 10 years will be
about 17,237. about 18,569.
11. 20 years
12. 50 years
P(t) 5 16,000 ? 1.015
P(t) 5 16,000 ? 1.015t
P(20) 5 16,000 ? 1.01520
P(50) 5 16,000 ? 1.01550
t
P(20) ¯ 21,550
P(50) ¯ 33,684
The population after 20 years will be The population after 50 years will be
about 21,550. about 33,684.
390 © 2012 Carnegie Learning
P(5) ¯ 17,237
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Lesson 5.2 Skills Practice
page 3
Name
Date
Morristown has a population of 18,000. Its population is decreasing at a rate of 1.2%. The function,
P(t) 5 18,000 ? 0.988t represents the population as a function of time. Use a graphing calculator to
estimate the number of years it will take for the population to reach each given amount.
13. 17,000
14. 15,000
It will take about 4.7 years for the population It will take about 15.1 years for the population
to reach 17,000. to reach 15,000.
15. half
16. one-third
It will take about 57.4 years for the It will take about 91.0 years for the population
population to reach 9000. to reach 6000.
17. 0
18. 10,000
The range of the function is all numbers It will take about 48.7 years for the population
greater than 0. The function never actually to reach 10,000.
reaches 0.
Complete each table and graph the function. Identify the x-intercept, y-intercept, asymptote, domain,
range, and interval(s) of increase or decrease for the function.
19. f(x) 5 2x
© 2012 Carnegie Learning
y
x
f(x)
22
​ 1  ​
4
21
__ 
__​ 1 ​  
0
1
1
2
2
4
2
4
5
3
2
1
0 1
24 23 22 21
21
2
3
4
x
22
23
24
x-intercept: none
y-intercept: (0, 1)
asymptote: y 5 0
domain: all real numbers
range: y . 0
interval(s) of increase or decrease: increasing over the entire domain
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Lesson 5.2 Skills Practice
page 4
20. f(x) 5 4x
y
x
f(x)
22
​  1  ​
16
21
___ 
__​ 1 ​  
4
0
1
1
4
2
16
16
12
8
4
0 4
216212 28 24
24
8
12 16
x
28
212
216
x-intercept: none
y-intercept: (0, 1)
asymptote: y 5 0
domain: all real numbers
range: y . 0
interval(s) of increase or decrease: increasing over the entire domain
x
21. f(x) 5 __
​ 1 ​ 
3
x
f(x)
22
9
21
3
0
1
1
​ 1  ​
3
2
__ 
__​ 1 ​  
9
8
6
4
2
0 2
28 26 24 22
22
4
6
8
24
26
28
x-intercept: none
y-intercept: (0, 1)
x
© 2012 Carnegie Learning
5
y
asymptote: y 5 0
domain: all real numbers
range: y . 0
interval(s) of increase or decrease: decreasing over the entire domain
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Lesson 5.2 Skills Practice
page 5
Name
Date
x
22. f(x) 5 __
​ 1 ​ 
4
y
x
f(x)
22
16
21
4
0
1
1
​ 1  ​
4
2
___
​  1  ​ 
__ 
16
16
12
8
4
0 4
216212 28 24
24
8
12 16
x
28
212
216
x-intercept: none
y-intercept: (0, 1)
asymptote: y 5 0
domain: all real numbers
range: y . 0
interval(s) of increase or decrease: decreasing over the entire domain
© 2012 Carnegie Learning
23. f(x) 5 22 ? 2x
y
x
f(x)
22
2​ 1 ​ 
6
2
4
21
21
2
__ 
0
22
1
24
2
28
5
8
0 2
28 26 24 22
22
4
6
8
x
24
26
28
x-intercept: none
y-intercept: (0, 22)
asymptote: y 5 0
domain: all real numbers
range: y , 0
interval(s) of increase or decrease: decreasing over the entire domain
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Lesson 5.2 Skills Practice
page 6
x
24. f(x) 5 22 ? __
​ 1 ​ 
2
y
x
f(x)
22
28
21
24
0
22
1
21
2
2​ 2 ​ 
__1 
8
6
4
2
0 2
28 26 24 22
22
4
6
8
x
24
26
28
x-intercept: none
y-intercept: (0, 22)
asymptote: y 5 0
domain: all real numbers
range: y , 0
interval(s) of increase or decrease: increasing over the entire domain
© 2012 Carnegie Learning
5
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Lesson 5.3 Skills Practice
Name
Date
Let the Transformations Begin!
Translations of Linear and Exponential Functions
Vocabulary
Match each definition to its corresponding term.
1. t he mapping, or movement, of all the points of a
figure in a plane according to a common operation
A basic function
B transformation
2. a
type of transformation that shifts the entire graph
left or right
B transformation
F horizontal translation
C vertical translation
3. a
function that can be described as the simplest
function of its type
A basic function
4. a
type of transformation that shifts the entire graph
up or down
D coordinate notation
C vertical translation
E argument of a function
5. the variable on which a function operates
E argument of a function
F horizontal translation
6. n
otation that uses ordered pairs to describe a
transformation on a coordinate plane
5
D coordinate notation
© 2012 Carnegie Learning
Problem Set
Rewrite each function g(x) in terms of the basic function f(x).
1. f(x) 5 x
2. f(x) 5 x
g(x) 5 x 1 4 g(x) 5 x 2 7
g(x) 5 f(x) 1 4 g(x) 5 f(x) 2 7
3. f(x) 5 x
4. f(x) 5 3x
g(x) 5 x 2 8 g(x) 5 3x 1 1
g(x) 5 f(x) 2 8 g(x) 5 f(x) 1 1
5. f(x) 5 3x
6. f(x) 5 4x
g(x) 5 3x 1 2 g(x) 5 4x 2 6
g(x) 5 f(x) 1 2 g(x) 5 f(x) 2 6
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Lesson 5.3 Skills Practice
page 2
Represent each vertical translation, g(x), using coordinate notation.
7. f(x) 5 x
8. f(x) 5 x
g(x) 5 x 1 8 g(x) 5 x 1 9
(x, y) → (x, y 1 8) (x, y) → (x, y 1 9)
9. f(x) 5 x
10. f(x) 5 4x
g(x) 5 x 2 4 g(x) 5 4x 2 1
(x, y) → (x, y 2 4) (x, y) → (x, y 2 1)
11. f(x) 5 4x
12. f(x) 5 3x
g(x) 5 4x 1 6 g(x) 5 3x 2 5
(x, y) → (x, y 1 6) (x, y) → (x, y 2 5)
Rewrite each function g(x) in terms of the basic function f(x).
13. f(x) 5 3x
14. f(x) 5 3x
g(x) 5 3(x 1 1) g(x) 5 3(x 1 5)
g(x) 5 3(x 1 1) 5 f(x 1 1) g(x) 5 3(x1 5) 5 f(x 1 5)
15. f(x) 5 2x
16. f(x) 5 2x
g(x) 5 2(x 2 1) g(x) 5 2(x 2 9)
g(x) 5 2(x 2 1) 5 f(x 2 1) g(x) 5 2(x 2 9) 5 f(x 2 9)
17. f(x) 5 2x
5
18. f(x) 5 2x
g(x) 5 2(x 2 3) g(x) 5 2(x 1 4)
g(x) 5 2(x 2 3) 5 f(x 2 3) g(x) 5 2(x 1 4) 5 f(x 1 4)
19. f(x) 5 3x
20. f(x) 5 3x
g(x) 5 3(x 2 2) g(x) 5 3(x 1 2)
(x, y) → (x 1 2, y) (x, y) → (x 2 2, y)
21. f(x) 5 4x
22. f(x) 5 4x
g(x) 5 4(x 1 1) g(x) 5 4(x 2 3)
© 2012 Carnegie Learning
Represent each horizontal translation, g(x), using coordinate notation.
(x, y) → (x 2 1, y) (x, y) → (x 1 3, y)
23. f(x) 5 3x
24. f(x) 5 3x
g(x) 5 3(x 2 1) g(x) 5 3(x 1 1)
(x, y) → (x 1 1, y) (x, y) → (x 2 1, y)
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Lesson 5.3 Skills Practice
page 3
Name
Date
Describe each graph in relation to its basic function.
25. Compare f(x) 5 (x) 1 b when b , 0 to the basic function h(x) 5 x.
The graph of f(x) is b units below the graph of h(x).
26. Compare f(x) 5 bx 2 c when c . 0 to the basic function h(x) 5 bx.
The graph of f(x) is c units to the right of the graph of h(x).
27. Compare f(x) 5 (x 2 b) when b . 0 to the basic function h(x) 5 x.
The graph of f(x) is b units to the right of h(x).
28. Compare f(x) 5 bx 2 c when c , 0 to the basic function h(x) 5 bx.
The graph of f(x) is c units to the left of the graph of h(x).
29. Compare f(x) 5 bx 1 k when k . 0 to the basic function h(x) 5 bx.
The graph of f(x) is k units up from the graph of h(x).
30. Compare f(x) 5 (x 2 b) when b , 0 to the basic function h(x) 5 x.
The graph of f(x) is b units to the left of h(x).
Each coordinate plane shows the graph of f(x). Sketch the graph of g(x).
31. g(x) 5 f(x) 1 2
32. g(x) 5 f(x) 1 4
© 2012 Carnegie Learning
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Lesson 5.3 Skills Practice
33. g(x) 5 f(x) 2 2
page 4
34. g(x) 5 f(x 2 3)
y
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8
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0 2
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35. g(x) 5 f(x 1 3)
36. g(x) 5 f(x 2 4)
y
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y
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28
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38. g(x) 5 f(x 1 5)
y
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x
© 2012 Carnegie Learning
37. g(x) 5 f(x) 1 5
28
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Lesson 5.3 Skills Practice
page 5
Name
Date
Write the equation of the function given each translation.
39. f(x) 5 x
40. f(x) 5 x
Vertical translation up 2 units Vertical translation down 5 units
g(x) 5 x 1 2 g(x) 5 x 2 5
41. f(x) 5 3x
42. f(x) 5 2x
Horizontal translation right 4 units Horizontal translation left 6 units
g(x) 5 3x 2 4 g(x) 5 2x 1 6
43. f(x) 5 3x
44. f(x) 5 4x
Vertical translation down 5 units Horizontal translation right 3 units
g(x) 5 3x 2 5 g(x) 5 4(x 2 3)
Each graph shows the function g(x) as a translation of the function f(x). Write the equation of g(x).
45.
46.
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
© 2012 Carnegie Learning
y
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
28
28
4
6
8
g(x) 5 x 2 3 g(x) 5 x 1 6
Chapter 5 Skills Practice 8069_Skills_Ch05.indd 399
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Lesson 5.3 Skills Practice
y
47.
page 6
y
48.
8
8
6
6
4
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2
2
0 2
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22
4
6
8
x
0 2
28 26 24 22
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24
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26
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6
8
4
6
8
x
g(x) 5 2x 1 2 g(x) 5 2x 2 4
49.
50.
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
5
y
4
6
8
x
0 2
28 26 24 22
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24
24
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x
© 2012 Carnegie Learning
g(x) 5 2x 2 3 g(x) 5 2x 1 5
400 Chapter 5 Skills Practice
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Lesson 5.4 Skills Practice
Name
Date
Take Some Time to Reflect
Reflections of Linear and Exponential Functions
Vocabulary
Define each term in your own words.
1. reflection
A reflection of a graph is a mirror image of the graph about a line.
2. line of reflection
The line that a graph is reflected about is called the line of reflection.
Problem Set
Rewrite each function g(x) in terms of the basic function f(x).
1. f(x) 5 ​3x​​
2. f(x) 5 ​3x​​
g(x) 5 2(​3x​​)
g(x) 5 ​32x
​ ​
© 2012 Carnegie Learning
g(x) 5 2f(x)
g(x) 5 f(2x)
3. f(x) 5 ​4x​​
4. f(x) 5 ​4x​​
g(x) 5 2(​4x​​)
g(x) 5 ​42x
​ ​
g(x) 5 2f(x)
g(x) 5 f(2x)
5. f(x) 5 ​2x​​1 4
6. f(x) 5 ​2x​​2 1
g(x) 5 ​22x
​ ​1 4
g(x) 5 2(​2x​​2 1)
g(x) 5 f(2x)
5
g(x) 5 2f(x)
Chapter 5 Skills Practice 8069_Skills_Ch05.indd 401
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Lesson 5.4 Skills Practice
page 2
Represent each reflection using coordinate notation. Identify whether g(x) is a reflection about a horizontal
line of reflection or a vertical line of reflection.
7. f(x) 5 ​2x​​
8. f(x) 5 ​2x​​
g(x) 5 2(​2x​​)
g(x) 5 ​22x
​ ​
(x, y) → (x, 2y)
(x, y) → (2x, y)
g(x) is a horizontal reflection about y 5 0.
g(x) is a vertical reflection about x 5 0.
9. f(x) 5 5x
10. f(x) 5 5x
g(x) 5 2(5x)
g(x) 5 5(2x)
(x, y) → (x, 2y)
(x, y) → (2x, y)
g(x) is a horizontal reflection about y 5 0.
g(x) is a vertical reflection about x 5 0.
11. f(x) 5 ​3x​​1 7
12. f(x) 5 ​4x​​2 3
g(x) 5 ​32x
​ ​1 7
g(x) 5 2(​4x​​2 3)
(x, y) → (2x, y)
(x, y) → (x, 2y)
g(x) is a vertical reflection about x 5 0.
g(x) is a horizontal reflection about y 5 0.
Each coordinate plane shows the graph of f(x). Sketch the graph of g(x).
13. g(x) 5 2f(x)
14. g(x) 5 f(2x)
5
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
4
8
x
0 2
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22
24
24
26
26
28
28
402 6
4
6
8
x
© 2012 Carnegie Learning
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Lesson 5.4 Skills Practice
page 3
Name
Date
15. g(x) 5 f(2x)
16. g(x) 5 2f(x)
y
y
8
8
6
6
4
4
2
2
0 2
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22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
17. g(x) 5 2f(x)
18. g(x) 5 f(2x)
y
8
4
6
8
x
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
© 2012 Carnegie Learning
6
28
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4
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0 2
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Chapter 5 Skills Practice 8069_Skills_Ch05.indd 403
5
x
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Lesson 5.4 Skills Practice
page 4
Write a function, g(x), to describe each reflection of f(x).
20. f(x) 5 ​4x​​
19. f(x) 5 ​3x​​
Reflection about the horizontal line y 5 0.
Reflection about the vertical line x 5 0.
g(x) 5 2​3​​
g(x) 5 ​42x
​ ​
x
21. f(x) 5 212x
22. f(x) 5 7x
Reflection about the vertical line x 5 0.
Reflection about the horizontal line y 5 0.
g(x) 5 12x
g(x) 5 27x
23. f(x) 5 ​2x​​1 9
24. f(x) 5 2​8x​​1 1
Reflection about the horizontal line y 5 0.
Reflection about the vertical line x 5 0.
g(x) 5 2(​2​​1 9)
g(x) 5 2​82x
​ ​1 1
x
Write an equation for g(x) given each transformation. Sketch the graph of g(x).
25. f(x) 5 ​5x​​
26. f(x) 5 ​5x​​
g(x) is a reflection of f(x) over the line y 5 0.
g(x) is a reflection of f(x) over the line x 5 0.
g(x) 5 2​5​​
g(x) 5 ​52x
​ ​
x
y
8
8
6
6
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4
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2
0 2
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0 2
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404 28
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© 2012 Carnegie Learning
5
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Lesson 5.4 Skills Practice
page 5
Name
Date
27. f(x) 5 ​3x​​
28. f(x) 5 ​4x​​
g(x) is a translation of f(x) up 2 units.
g(x) is a translation of f(x) right 3 units.
g(x) 5 ​3x​​1 2
g(x) 5 ​4x23
​ ​
y
y
8
8
6
6
4
4
2
2
0 2
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6
8
x
0 2
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24
24
26
26
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6
8
29. f(x) 5 ​4x​​
30. f(x) 5 ​3x​​
g(x) is a translation of f(x) down 4 units.
g(x) is a translation of f(x) left 5 units.
g(x) 5 ​4​​2 4
g(x) 5 ​3x15
​ ​
y
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
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4
6
8
x
5
0 2
28 26 24 22
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24
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26
26
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6
8
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28
28
x
28
x
© 2012 Carnegie Learning
4
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Lesson 5.4 Skills Practice
page 6
Identify the transformation required to transform f(x) to g(x) as shown in each graph.
32.
y
16
16
12
12
8
8
4
4
0 4
216 212 28 24
24
8
12 16
x
0 4
216212 28 24
24
28
28
212
212
216
216
33.
34.
y
16
12
12
8
8
4
4
8
12 16
x
406 0 4
216212 28 24
24
28
28
212
212
216
216
g(x) is a translation of f(x) up 10 units
or g(x) is a translation of f(x) left 5 units.
12 16
x
y
16
0 4
216 212 28 24
24
8
g(x) is a reflection of f(x) over the line y 5 0.
g(x) is a reflection of f(x) over the line x 5 0.
5
y
8
12 16
x
g(x) is a translation of f(x) left 6 units.
© 2012 Carnegie Learning
31.
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Lesson 5.4 Skills Practice
page 7
Name
Date
35.
36.
y
y
16
16
12
12
8
8
4
4
0 4
216212 28 24
24
8
12 16
x
0 4
216212 28 24
24
28
28
212
212
216
216
g(x) is a translation of f(x) right 4 units.
8
12 16
x
g(x) is a translation of f(x) down 8 units
or g(x) is a translation of f(x) right 4 units.
Identify the transformation required to transform each f(x) to g(x).
37. f(x) 5 ​8x​​
38. f(x) 5 ​9x​​
g(x) 5 2(​8x​​)
g(x) 5 ​92x
​ ​
g(x) is a reflection of f(x) over the line y 5 0.
39. f(x) 5 ​8x​​
40. f(x) 5 ​3x​​
g(x) 5 ​8​​2 5
g(x) 5 ​3​ ​
x
g(x) is a translation of f(x) down 5 units.
© 2012 Carnegie Learning
g(x) is a reflection of f(x) over the line x 5 0.
g(x) is a translation of f(x) right 1 unit.
41. f(x) 5 10x
42. f(x) 5 212x
g(x) 5 10x 1 2
g(x) 5 212(x 1 1)
g(x) is a translation of f(x) up 2 units.
5
x21
g(x) is a translation of f(x) left 1 unit.
Chapter 5 Skills Practice 8069_Skills_Ch05.indd 407
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© 2012 Carnegie Learning
5
408 Chapter 5 Skills Practice
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Lesson 5.5 Skills Practice
Name
Date
Radical! Because It’s Cliché!
Properties of Rational Exponents
Vocabulary
Match each definition to its corresponding term.
n
1. the number a in the expression 
​ a ​
 
A cube root
D radicand
2. the number b when ​b​3​5 a
B index
A cube root
1 ​ in the expression ​a__​ ​n1 ​ ​
3. the exponent __
​ n
C nth root
n
4. the number n in the expression 
​ a ​
 
D radicand
E rational exponent
B index
5. the number b when ​b​n​5 a
E rational exponent
C nth root
Problem Set
5
Write each expression as a single power.
​10​ ​ 
2. ​ ___
 ​
​10​4​
1. ___
​ ​10​8​ 
 ​
​10​ ​
5
0
___
 5 1​0​
​ 1​0​ ​ ​ 
1​0​ ​
© 2012 Carnegie Learning
5
​5 ​10​23​
528
8
​10​2​ 
3. ​ ___
 ​
​10​5​
​5 ​10​23​
225
5
3
10
​5 ​10​24​
024
4
__​ ​x​​ ​  5 ​x​
​x​ ​
4
9
​5 ​x25
​ ​
429
​y2​ ​
6. __
​  8  ​
​y​ ​
​ ​3​  ​ 
5
5. ​ ___
​510
​​
___
​ ​5​ ​  ​ 5 ​5​
​5​ ​
0
​x​4​  ​
4. ​ __
​x9​ ​
___
 5 1​0​
​ 1​0​ ​ ​ 
1​0​ ​
2
___
 5 1​0​
​ 1​0​ ​ ​ 
1​0​ ​
​5 ​527
​ ​
3210
__​ ​y​​ ​  5 ​y​
​y​ ​
2
8
​5 ​y26
​ ​
228
Chapter 5 Skills Practice 8069_Skills_Ch05.indd 409
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Lesson 5.5 Skills Practice
page 2
Evaluate each expression.
3
 
5
7. 
​ 216 ​
 5
8. 
​3 64 ​
 
​3 216 ​
56

 
9. 
​3 2125 ​
5
 
​3 2125 ​
5 25

 
11. 
​3 729 ​
5
 5 4
​3 64 ​

 
10. 
​3 2343 ​
5
 
​3 2343 ​
5 27

 
12. 
​3 28 ​
5
 
​3 729 ​
59

 
​3 28 ​
5 22

Evaluate each expression.
 5
13. 
​5 32 ​
 
14. 
​4 625 ​
5
 5 2
​5 32 ​

 
​4 625 ​
55

 
15. 
​6 729 ​
5
 
16. 
​5 21024 ​
5
 
​6 729 ​
53

 
17. 
​7 2128 ​
5
5
 
​7 2128 ​
5 22

 
​5 21024 ​
5 24

 
18. 
​5 2243 ​
5
 
​5 2243 ​
5 23

 
19. 
​4 15 ​
__1
 51​5​ ​4 ​ ​ 
​4 15 ​

 
21. 
​4 31 ​
__​ 1 ​  
 5 3​1​4​
​4 31 ​

23. 
​6 y ​
 
__​ 1 ​  
 5 ​5​3​
​3 5 ​

22. 
​3 x ​
 
​ x ​
 5

3
__​ 1 ​  
​x​3​
__
__1
​   ​  
​6 y ​
 5 ​y​6​

410  
20. 
​3 5 ​
© 2012 Carnegie Learning
Write each radical as a power.
24. ​√z ​ 
__
√
​ z ​ 5
1 ​  
​ __
z​ ​2​
Chapter 5 Skills Practice
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Lesson 5.5 Skills Practice
page 3
Name
Date
Write each power as a radical.
__​ 1 ​  
__​ 1 ​  
25. ​12​3​
3
 
​12​3​5 
​ 12 ​
__
​ 1 ​ 
27. ​18​4​
__1
4
 
​18​​ 4 ​  ​5 
​ 18 ​
__
​ 1 ​ 
29. ​d ​5​
1 ​  
​ __
5
 
​d ​5​5 
​ d ​
__
​ 1 ​ 
26. ​7​5​
__​ 1 ​  
5
 
​7​5​5 
​ 7 ​
__
​ 1 ​ 
28. a
​ ​2​
__​ 1 ​  
__
​ ​2​5 √
a
​ a ​ 
1 ​ 
​ __
30. c
​ ​6​
1 ​  
​ __
6
​ ​6​5 
c
​ c ​
 
Write each expression in radical form.
__
​ 2 ​ 
31. 5
​ ​3​
__​ 2 ​  
3 2
​ ​3​5 
5
​ ​5
​ ​ ​ 
__
​ 3 ​ 
33. ​18​4​
__3
4

​18​​ 4 ​  ​5 
​ 1​
83​ ​ ​ 
__
​ 4 ​ 
35. ​y​3​
© 2012 Carnegie Learning
__​ 4 ​  
3 
​y​3​5 
​ ​y4​ ​ ​ 
__
​ 2 ​ 
32. 8
​ ​5​
__​ 2 ​  
5 2
​ ​5​5 
8
​ ​8
​ ​ ​ 
__
​ 3 ​ 
34. x​ ​5​
__​ 3 ​  
5
3
x​ ​5​5
​ ​x
​ ​ ​ 
5
__
​ 5 ​ 
36. ​m​2​
__​ 5 ​  
___
​m​2​5 √
​ ​m5​ ​ ​ 
Chapter 5 Skills Practice 8069_Skills_Ch05.indd 411
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25/04/12 4:13 PM
Lesson 5.5 Skills Practice
page 4
Write each expression in rational exponent form.
4 3
​ ​ ​ 
37. 
​ ​6
4
5 4
38. 
​ ​8
​ ​ ​ 
__​ 3 ​  
3
​ ​6
​ ​ ​ 5 ​6​4​

5
__
3

39. 
​ 1​
22​ ​ ​ 
__​ 2 ​  
3

​ 1​
22​ ​ ​ 51​2​3​

4 
41. 
​ ​p7​ ​ ​ 
40. √
​ ​n5​ ​ ​ 
__
__5
​   ​  
√
​ ​n5​ ​ ​ 5 ​n​2​
5
__​ 7 ​  
7
​ ​p
​ ​ ​ 5 ​p​4​

4
__​ 4 ​  
4
​ ​8
​ ​ ​ 5 ​8​5​

3

42. 
​ ​m
​ ​ ​ 
5
3 ​  
​ __
3

​ ​m
​ ​ ​ 5 ​m​5​

© 2012 Carnegie Learning
5
412 Chapter 5 Skills Practice
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Lesson 5.6 Skills Practice
Name
Date
Checkmate!
Solving Exponential Functions
Problem Set
Complete each table. Write a function that represents the data in the table and explain how you
determined your expression.
2.
1.
x
f(x)
Expression
x
f(x)
Expression
0
1
​ 0​ ​
3
0
5
​ 0​ ​1 5
4
1
3
​ 1​ ​
3
1
9
​ 1​ ​1 5
4
2
9
​ 2​ ​
3
2
21
​ 2​ ​1 5
4
3
27
​ 3​ ​
3
3
69
​ 3​ ​1 5
4
4
81
​ 4​ ​
3
4
261
​ 4​ ​1 5
4
5
243
​ 5​ ​
3
5
1029
​ 5​ ​1 5
4
x
​ x​​
3
-----
x
​ ​x​1 5
4
-----
The exponents of the expressions in the
third column equal x. So, f(x) 5 ​4x​​1 5.
5
© 2012 Carnegie Learning
The exponents of the expressions in the
third column equal x. So, f(x) 5 3x.
Chapter 5 Skills Practice 8069_Skills_Ch05.indd 413
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Lesson 5.6 Skills Practice
3.
page 2
4.
x
f(x)
Expression
x
f(x)
Expression
0
21
2​20​ ​
22
1
2​ __ ​ 
2
2​221
​ ​
1
22
2​21​ ​
21
21
2​20​ ​
2
24
2​22​ ​
0
22
2​21​ ​
3
28
2​23​ ​
1
24
2​22​ ​
4
216
2​24​ ​
2
28
2​23​ ​
5
232
2​25​ ​
3
216
2​24​ ​
x
​22​x​
-----
x
2​2​x11​
-----
The exponents of the expressions in the
third column equal x. So, f(x) 5 2​2x​​.
6.
5
x
f(x)
Expression
x
f(x)
Expression
0
1
2​ ___  ​ 
25
2​522
​ ​
0
16
​ 4​ ​
2
1
1
2​ __  ​
5
2​521
​ ​
1
8
​ 3​ ​
2
2
21
2​5​0​
2
4
​ 2​ ​
2
3
25
2​51​ ​
3
2
​21​ ​
4
225
2​52​ ​
4
1
​ 0​ ​
2
5
2125
2​53​ ​
5
​ 1  ​
2
__ 
​ 21
2
​ ​
x
2​5​x22​
-----
x
​ 2x14
2
​ ​
-----
The exponents of the expressions in the
third column equal x 2 2. So, f(x) 5 2​5x22
​ ​.
414 © 2012 Carnegie Learning
5.
The exponents of the expressions in the
third column equal x 1 1. So, f(x) 5 2​2x11
​ ​.
The exponents of the expressions in the
third column equal 4 2 x. So, f(x) 5 ​22x14
​ ​.
Chapter 5 Skills Practice
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Lesson 5.6 Skills Practice
page 3
Name
Date
Graph each function.
8. f(x) 5 ​82x
​ ​
7. f(x) 5 ​3x​​
y
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
28
9. f(x) 5 5 ? ​22x
​ ​
10. f(x) 5 2 ? ​3x​​
y
8
x
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
© 2012 Carnegie Learning
6
28
4
6
8
x
5
0 2
28 26 24 22
22
24
24
26
26
28
4
4
6
8
x
28
Chapter 5 Skills Practice 8069_Skills_Ch05.indd 415
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Lesson 5.6 Skills Practice
11. f(x) 5 2​4x​​
page 4
12. f(x) 5 2​3x12
​ ​
y
y
8
8
6
6
4
4
2
2
0 2
28 26 24 22
22
4
6
8
x
0 2
28 26 24 22
22
24
24
26
26
6
8
x
28
28
4
Use the intersection feature of your graphing calculator to answer each question.
13. For the function f(x) 5 ​6x21
​ ​determine the value of x for which f(x) 5 7776.
For the function f(x) 5 ​6x21
​ ​, f(x) 5 7776 when x 5 6.
14. For the function f(x) 5 2​4x12
​ ​determine the value of x for which f(x) 5 24096.
For the function f(x) 5 ​4x12
​ ​, f(x) 5 24096 when x 5 4.
15. For the function f(x) 5 ​52x11
​ ​determine the value of x for which f(x) 5 625.
For the function f(x) 5 ​5​2x11​, f(x) 5 625 when x 5 23.
5
16. For the function f(x) 5 ​2​x14​determine the values of x for which f(x) , 128.
17. For the function f(x) 5 2​3x11
​ ​determine the values of x for which f(x) . 29.
For the function f(x) 5 2​3x11
​ ​, f(x) . 29 when x , 1.
18. For the function f(x) 5 ​5x12
​ ​determine the values of x for which f(x) 5 15,625.
For the function f(x) 5 ​5x12
​ ​, f(x) 5 15,625 when x 5 4.
416 © 2012 Carnegie Learning
For the function f(x) 5 ​2x14
​ ​, f(x) , 128 when x , 3.
Chapter 5 Skills Practice
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Lesson 5.6 Skills Practice
page 5
Name
Date
Solve each exponential equation for x.
19. 4
​ x​​5 256
20. 6
​ 3x
​ ​5 216
​ x​​5 256
4
​ ​3x​5 216
6
​ 4​ ​5 256
4
​ ​3​5 216
6
x54
3x 5 3
x51
21. 2
​ 52x
​ ​5 ___
​  1  ​ 
16
​252x
​ ​5 ​  1  ​
16
​224
​ ​5 ​  1  ​
16
5 2 x 5 24
___ 
___ 
x59
23. 4
​ ​x13​5 4
​ x13
4
​ ​5 4
​ 1​ ​5 4
4
x1351
x 5 22
22. 3
​ ​22x​5 ____
​  1   ​ 
729
​322x
​ ​5 ​  1   ​
729
26
​3​ ​5 ​  1   ​
729
22x 5 26
____ 
____ 
x53
____ 
____ 
___ 
24. ​  1
   ​5 625
​5x14
​ ​
​  1
   ​5 625
​5x14
​ ​
​  124  ​5 625
​5​ ​
5
x 1 4 5 24
© 2012 Carnegie Learning
x 5 28
​  1   ​ 
25. 2​6​x22​5 _______
21296
x22
2​6​ ​5 ​  1   ​
21296
24
2​6​ ​5 ​  1   ​
21296
x 2 2 5 24
_______ 
_______ 
x 5 22
____ 
____ 
__ 
​ 1 ​ 
26. ​  1
   ​5 __
​2x26
​ ​ 4
   ​5 ​ 1 ​ 
​  1
​2x26
​ ​ 4
​ 12  ​5 ​ 1 ​ 
​2​ ​ 4
__ 
__ 
x2652
x58
Chapter 5 Skills Practice 8069_Skills_Ch05.indd 417
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Lesson 5.6 Skills Practice
page 6
For each pair of expressions, determine whether the second expression is an equivalent form of the
first expression.
1 ​  (2)s
27. 2
​ s21
​ ​ ​ __
2
221 ? 2s
1 ​  (3)x
28. 3
​ x11
​ ​ ​ __
3
321 ? 3x
2s21
3x21
1 ​  (25)x
30. 5
​ 2x21
​ ​ ​ __
5
21
2x
5 ?5
29. 2
​ 2x11
​ ​ 2(4)x
21 ? (22)x
__​ 1 ​ ?   (5 )
5
__​ 1 ​   (25)
21 ? 22x
2 x
22x11
x
5
(  )
x
32. __
​ 1 ​ ​  __
​ 1 ​   ​ 223x21
2 8
​223x
​ ​? ​221
​ ​
31. ​4(64)​x​ 43x11
4 ?4
3x
1
__ 
( ( __ ) ) __ 
( __ ) __ 
__ ( __ )
(43)x ? 4
(​223
​ ​​)​x​? ​ 1 ​ 
2
3 x
1
​​ ​​ ​   ​   ​​ ​  ​​​? ​ 1 ​ 
2
2
x
1
1
​​ ​   ​   ​​​? ​   ​ 
8
2
x
1
1
​   ​​​  ​   ​   ​​​
28
64 ? 4
x
4(64)x
Write the exponential function represented by the table of values.
5
x
y
x
y
0
2
0
1
1
1
2
25
2
__
​ 1 ​ 
4
625
3
__
​ 1 ​ 
6
15625
2
4
f(x) 5 a ? bx
f(x) 5 a ? bx
f(x) 5 2 ? bx
f(x) 5 1 ? bx
1 5 2 ? b1
​ 1 ​ 5
  b
2
x
f(x) 5 2 ​ ​ 1 ​   ​
2
25 5 1 ? b2
__ 
( __ )
© 2012 Carnegie Learning
34.
33.
25 5 b2
55b
f(x) 5 5x
418 Chapter 5 Skills Practice
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Lesson 5.6 Skills Practice
page 7
Name
Date
35.
36.
x
y
x
y
0
1
0
21
1
__
​ 3 ​ 
2
24
2
___
​  9  ​ 
4
216
3
___
​ 27 ​ 
6
264
4
16
64
f(x) 5 a ? bx
f(x) 5 a ? bx
f(x) 5 1 ? bx
f(x) 5 21 ? bx
1
24 5 21 ? b2
__​ 3 ​  5 1 ? b
4
__​ 3 ​  5 b
4
f(x) 5 (​ __
​ 3 ​   )​
4
24 5 2b2
25b
x
f(x) 5 2(2)x
© 2012 Carnegie Learning
37.
38.
x
y
x
y
0
3
0
22
3
__
​ 1 ​ 
1
1
2​ __ ​ 
2
6
____
​  1   ​ 
2
1
2​ __ ​ 
8
9
_____
​  1   ​ 
3
1
2​ ___  ​ 
32
f(x) 5 a ? bx
f(x) 5 3 ? bx
1 5 3 ? b1
__​ 1 ​  5 b
3
( __ )
x
f(x) 5 3 ​ ​ 1 ​   ​
3
9
243
6561
f(x) 5 a ? bx
f(x) 5 22 ? bx
1
2​   ​ 5 22 ? b1
2
1
​ 2​   ​   ​​ 1 ​  5 ​ 1 ​  (22b)
2 2 2
​ 1  ​5 b
4
x
f(x) 5 22 ​ ​ 1 ​   ​
4
__ 
(  __ ) __  __ 
__ 
( __ )
Chapter 5 Skills Practice 8069_Skills_Ch05.indd 419
5
419
25/04/12 4:13 PM
© 2012 Carnegie Learning
5
420 Chapter 5 Skills Practice
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