7-6
Exponential Functions
Vocabulary
Review
1. Circle each relation that is a function.
5(1, 3), (1, 5), (2, 7), (3, 9), (4, 11)6
5(0, 0), ( 21, 21), ( 22, 22), ( 23, 23)6
5(5, 5), (5, 4), (5, 3), (5, 2)6
5(2, 5), (3, 5), (4, 5), (5, 5)6
2. Cross out the equations that are NOT linear functions.
y 5 3x 1 2
3
y5x
y 5 22
x54
y 5 x2 1 5
Vocabulary Builder
exponential (adjective) ek spoh NEN shul
Main Idea: You can use exponential notation to write a repeated multiplication
(such as 8 3 8 3 8 3 8) using a base and an exponent (84) . An exponential
function has a variable as an exponent.
Use Your Vocabulary
Underline the correct word to complete each sentence.
3. In the expression 53 , the 3 is an exponential / exponent .
4. The expression 3x is an exponent / exponential expression.
5. Draw a line from each exponential expression in Column A to an equivalent
expression in Column B.
Column A
(3n) 21
1
Q4 R
2
2 3
Column B
1
25n2
8
n3
Qn R
4 22
(5n) 22
1
3n
Chapter 7
214
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Related Word: exponent (noun)
Key Concept Exponential Function
DefinitionAn exponential function is a function of the form y 5 a ? bx , where
a 2 0, b . 0, b 2 1, and x is a real number.
Examples
4
2
y
4
2
x
4
y  2x
1 x
2
y  2 
4 2 O
4
y  2x
O 2
4
y
4
4
x
y  1
x
2
If all of the x-values in a table have a constant difference and all of the y-values have a
constant ratio, then the table represents an exponential function.
Problem 1 Identifying Linear and Exponential Functions
Got It? Does the table represent a linear or an exponential function? Explain.
Difference:
For Exercises 6–8, use the table at the right.
6. Find the difference between each pair of consecutive x-values.
x
1
y 1
7. Find the difference between each pair of consecutive y-values.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
8. Find the ratio between each pair of consecutive y-values.
9. Underline the correct word or words to complete each sentence.
Difference:
There is / is not a constant difference between x-values.
Ratio:
2
3
4
1
3
5
There is / is not a constant difference between the y-values.
There is / is not a constant ratio between y-values.
The table represents a linear / exponential function.
Got It? Does the rule y 5 3 ? 6x represent a linear or an exponential function? Explain.
10. A linear function has the form y 5 mx 1 b. An exponential function has the form
y 5 a ? b x . The rule y 5 3 ? 6x seems to represent a linear / exponential function.
11. In the rule y 5 3 ? 6x , identify a and b.
a 5 b 5 12. Does the rule y 5 3 ? 6x meet the definition of an exponential function? Place
a ✓ in the box if the statement is correct. Place an ✗ in the box if it is incorrect.
a 2 0
b . 0
b21
13. Is y 5 3 ? 6x an exponential function? Explain.
215
Lesson 7-6
Problem 2 Evaluating an Exponential Function
Got It? An initial population of 20 rabbits triples every half year. The function
f (x) 5 20 ? 3x gives the population after x half-year periods. How many rabbits
will there be after 3 years?
14. In 3 years, there are half-year periods.
15. Complete the steps to evaluate the function.
Substitute the number of half-year periods.
5 20 ? Evaluate the power.
5
Simplify.
16. There will be rabbits after 3 years.
Problem 3 Graphing an Exponential Function
Got It? What is the graph of y 5 0.5 ?
y 5 Q 12 R
3x ?
y
y
4
O
4 x
–4
–4
x
y 5 22x
4
y
O
O
–4
y
4
y 5 2Q 12 R
y 5 2x
17. Use one of the functions in the box to label each graph.
4
x
–4
4 x
4 x
–4
–4
–4
O
4 x
–4
18. Complete the table. Then graph each ordered pair and draw a smooth curve.
HSM11_A1MC_0706_T13025 y
Ordered Pair
HSM11_A1MC_0706_T13028
HSM11_A1MC_0706_T13026
HSM11_A1MC_0706_T13027
x
y ∙ 0.5 ∙ 3x
(x, y)
5
–2
1
1
1
0.5 ∙ 3–2 = 2 ∙ 9 =
(–2,
1
4
)
3
–1
0.5 ∙ 3
1
=2∙
1
=
1
(–1,
2
)
1
1
0
0.5 ∙ 3
=2∙
=
(0,
)
1
0.5 ∙ 3
=2∙
1
=
(1,
)
x
–4
–2
O
2
4
HSM11_A1MC_0706_T91258
Chapter 7
216 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
f (x) 5 20 ? 3x 5 20 ? 3
Problem 4 Graphing an Exponential Model
Got It? Computer mapping software allows you to zoom in or out on an area
to view it in more detail. The function f (x) 5 100 ? 4x models the percent of the
original area the map shows after zooming out x times. Graph the function.
19. Complete the table of x and f (x) values. Then use the values to make a graph and
plot the points. Do not connect the points of the graph.
f(x) ∙ 100 ∙ 4x
x
0
100 ∙ 40 = 100 ⋅ 1 =
1
100 ∙
2
100 ∙
= 100 ∙
3
100 ∙
=
4
100 ∙
=
= 100 ∙
=
=
∙
y
(x, f(x))
30,000
(0,
)
(1,
)
(2,
25,000
20,000
)
15,000
10,000
=
5,000
=
O
1
2
3
x
Lesson Check • Do you UNDERSTAND?
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
HSM11_A1MC_0706_T91259
Reasoning Is y 5 (22) x an exponential function? Justify your answer.
20. An exponential function has the form y 5 a ? bx . What are the restrictions
on b?
HSM11_A1MC_0706_T91260
21. Is y 5 (22) x an exponential function? Justify your answer.
Math Success
Check off the vocabulary words that you understand.
exponent
exponential
exponential function
Rate how well you can graph exponential functions.
Need to
review
0
2
4
6
8
Now I
get it!
10
217
Lesson 7-6