Lesson 10-5

Exponential Functions

Level Number of Pliers Power of 2

Original 1 20

First 1 (2) = 2 21

Second 2 (2) = 4 22

Third 2 (2)(2) = 8 23

Fourth 2 (2)(2)(2) = 16 24

Fifth 2(2)(2)(2)(2)= 32 25

Sixth 2(2)(2)(2)(2)(2) = 64 26

Seventh 2(2)(2)(2)(2)(2)(2) = 128 27

Eighth 2(2)(2)(2)(2)(2)(2)(2) =256 28

The number of pliers on each level is given in the table below.

Key Concept

An exponential function is a function that can be described by an equation of the form y = ax, where a 0 and a 1.

The type of function in which the variable is the exponent, is called an exponential function.

x 4 x y

-2 4-2

-1 4-1

0 40 1

1 41 4

2 42 16

3 43 64

€

1

16

€

1

4

Graph y = 4x. State the y-intercept.

Graph an exponential function with a 1.

o x

y

4

8

12

16

18

2 4 6 8-2-4-6-8

Graph the ordered pairs and connect the points with a smooth curve. The y-intercept is 1. Notice that the y value change little for small values of x, but they increase quickly as values of x become greater.Use the graph to determine the approximate value of 41.8.The graph represents all the real values of x and their corresponding values of y for y = 4x. So the value of y is about 12 when x = 1.8. Use a calculator to confirm this value. 41,8 = 12.12573252.

x 3 x y

-2

-1

0

1

2

3

Graph y = 3x. State the y-intercept.

Graph an exponential function with a 1.

o x

y

4

8

12

16

18

2 4 6 8-2-4-6-8

Use the graph to determine the approximate value of 31.5.

x y

-3 8

-2 4

-1 2

0 1

1

2€

1

2

€

1

4

Graph . State the y-intercept.

Graph an exponential function with 0 a 1.

o x

y

4

8

12

16

18

2 4 6 8-2-4-6-8

Graph the ordered pairs and connect the points with a smooth curve. The y-intercept is 1. Notice that the y value decrease less rapidly as x increases.

€

y =1

2

⎛

⎝ ⎜

⎞

⎠ ⎟x

€

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟x

€

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟−3

€

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟−2

€

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟−1

€

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟0

€

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟

1

€

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟2

Use the graph to determine the approximate value of .The value of y is about 5.5 when x = -2.5. Use a calculator to confirm this value.

€

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟−2.5

€

1

2

⎛

⎝ ⎜

⎞

⎠ ⎟−2.5

≈ 5.6568549

Graph . State the y-intercept.

o x

y

4

8

12

16

18

2 4 6 8-2-4-6-8

€

1

4

⎛

⎝ ⎜

⎞

⎠ ⎟x

Use the graph to determine the approximate value of .

€

1

4

⎛

⎝ ⎜

⎞

⎠ ⎟−1.5

x y

-2

-1

0

1

2

3

€

1

4

⎛

⎝ ⎜

⎞

⎠ ⎟x

Depreciation: People joke that the value of a car decreases as soon as it is driven off the dealer’s lot. The function V= 25,000 0.82t models the depreciation of the value of a new car that originally cost $25,000. V represents the value of the car and t represents the time in years from the time the car was purchased.

What is the value of the car after one year?

What is the value of the car after five years?

x 0 10 20 30

y 10 25 40 55

Determine whether each set of data displays exponential behavior.

x 0 10 20 30

y 10 25 65.5 156.25 Exponential 2.5

Linear 15

x y

o x

y

4

8

12

16

18

2 4 6 8-2-4-6-8