Course 3

13-5 Exponential Functions13-5 Exponential Functions

Course 3

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Course 3

13-5 Exponential Functions

Warm UpWrite the rule for each linear function.

1.

2.

f(x) = -5x - 2

f(x) = 2x + 6

Course 3

13-5 Exponential Functions

Course 3

13-5 Exponential Functions

Problem of the Day

One point on the graph of the mystery linear function is (4, 4). No value of x gives a y-value of 3. What is the mystery function?y = 4

Course 3

13-5 Exponential Functions

Learn to identify and graph exponential functions.

Course 3

13-5 Exponential Functions

Vocabulary

exponential function

exponential growth

exponential decay

Course 3

13-5 Exponential Functions

A function rule that describes the pattern is f(x) = 15(4)x, where 15 is a1, the starting number, and 4 is r the common ratio. This type of function is an exponential function.

Course 3

13-5 Exponential Functions

Course 3

13-5 Exponential Functions

In an exponential function, the y-intercept is f(0) = a1. The expression rx is defined for all values of x, so the domain of f(x)= a1 rx is all real numbers.

Course 3

13-5 Exponential Functions

Create a table for the exponential function, and use it to graph the function.

f(x) = 3 2x

Additional Example 1A: Graphing Exponential Functions

3 20 = 3 1

3 21 = 3 2

3 22 = 3 4

3 2-2 = 3 14

3 2-1 = 3 12

x y

–2

–1

0

1

2

3 43 2

3

6

12

Course 3

13-5 Exponential Functions

Additional Example 1B: Graphing Exponential Functions

Create a table for the exponential function, and use it to graph the function.

f(x) = 2 3

x

x y

-2

-1

0

1

2

1

0.67

0.44…

2.25

1.5

Course 3

13-5 Exponential Functions

Create a table for the exponential function, and use it to graph the function.

f(x) = 2x

Check It Out: Example 1A

20

21

2-2

2-1

x y

–2

–1

0

1

2

1 41 2

1

2

4 22

Course 3

13-5 Exponential Functions

Create a table for the exponential function, and use it to graph the function.

f(x) = 2x+ 1

Check It Out: Example 1B

20 + 1

21 + 1

2-2 + 1

x y

–2

–1

0

1

2

5 43 2

2

3

5 22 + 1

2-1 + 1

Course 3

13-5 Exponential Functions

In the exponential function f(x) = a1 rx if r > 1, the output gets larger as the input gets larger. In this case, f is called an exponential growth function.

Course 3

13-5 Exponential Functions

Additional Example 2: Using an Exponential Growth Function

A bacterial culture contains 5000 bacteria, and the number of bacteria doubles each day. How many bacteria will be in the culture after a week?

Day Mon Tue Wed Thu

Number of days x 0 1 2 3

Number of bacteria f(x) 5000 10,000 20,000 40,000

Course 3

13-5 Exponential Functions

Additional Example 2 Continued

f(x) = 5000 rx

f(x) = 5000 2x

A week is 7 days so let x = 7.

f(7) = 5000 27 = 640,000

If the number of bacteria doubles each day, there will be 640,000 bacteria in the culture after a week.

f(0) = a1

The common ratio is 2.

f(x) = a1 rx Write the function.

Substitute 7 for x.

Course 3

13-5 Exponential Functions

Check It Out: Example 2

Robin invested $300 in an account that will double her balance every 4 years. Write an exponential function to calculate her account balance. What will her account balance be in 20 years?

Year 2003 2007 2011 2015

Number of 4 year intervals

0 1 2 3

Account balance f(x)

300 600 1200 2400

Course 3

13-5 Exponential Functions

Check It Out: Example 2 Continued

f(x) = 300 rx

f(x) = 300 2x

20 years will be x = 5.

f(5) = 300 25 = 9600

In 20 years, Robin will have a balance of $9600.

f(0) = a1

The common ratio is 2.

f(x) = a1 rx Write the function.

Substitute 5 for x.

Course 3

13-5 Exponential Functions

In the exponential function f(x) = a1 rx, if r < 1, the output gets smaller as x gets larger. In this case, f is called an exponential decay function.

Course 3

13-5 Exponential Functions

Additional Example 3: Using an Exponential Decay Function

Bohrium-267 has a half-life of 15 seconds. Find the amount that remains from a 16 mg sample of this substance after 2 minutes.

Seconds 0 15 30 45

Number of Half-lives x 0 1 2 3

Bohrium-267 f(x) (mg) 16 8 4 2

Course 3

13-5 Exponential Functions

Additional Example 3 Continued

f(x) = 16 rx

Since 2 minutes = 120 seconds, divide 120 seconds by 15 seconds to find the number of half-lives: x = 8.

There is 0.0625 mg of Bohrium-267 left after 2 minutes.

f(0) = a1

f(x) = a1 rx Write the function.

Substitute 8 for x.

f(x) = 16 1 2

x The common ratio is . 1

2

f(8) = 16 1 2

8

Course 3

13-5 Exponential Functions

Check It Out: Example 3

If an element has a half-life of 25 seconds. Find the amount that remains from a 8 mg sample of this substance after 3 minutes.

Seconds 0 25 50 75

Number of Half-lives x

0 1 2 3

Element (mg) 8 4 2 1

Course 3

13-5 Exponential Functions

Check It Out: Example 3 Continued

f(x) = 8 rx

Since 3 minutes = 180 seconds, divide 180 seconds by 25 seconds to find the number of half-lives: x = 7.2.

There is approximately 0.054 mg of the element left after 3 minutes.

f(0) = p

f(x) = a1 rx Write the function.

Substitute 7.2 for x.

f(x) = 8 1 2

x

f(7.2) = 8 1 2

7.2

The common ratio is . 1

2

Course 3

13-5 Exponential Functions

Lesson Quiz: Part I

1. Create a table for the exponential function

f(x) = , and use it to graph the

function.

3 1 2

x

x y

–2 12

–1 6

0 3

1

2 3 4

3 2

Course 3

13-5 Exponential Functions

Lesson Quiz: Part II

2. Linda invested $200 in an account that will

double her balance every 3 years. Write an

exponential function to calculate her account

balance. What will her balance be in 12 years?

f(x) = 200 2x, where x is the number of 3-year periods; $3200.