1. Exponential Functions
13-5
Exponential Functions
Course 3
Warm Up
Problem of the Day
Lesson Presentation

2. Exponential Functions
Course 3
13-5
Exponential Functions
Warm Up
Write the rule for each linear function. 1. 2.
f(x) = -5x - 2
f(x) = 2x + 6

3. 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

4. Learn to identify and graph exponential functions.

5. Vocabulary
exponential function
exponential growth
exponential decay

6. 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.

8. In an exponential function, the y-intercept is f(0) = a1
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.

9. Additional Example 1A: Graphing Exponential Functions
Create a table for the exponential function, and use it to graph the function.
f(x) = 3  2x x y –2 –1 1 2
3 4
3  2-2 = 3  1 4
3  2-1 = 3  1 2
3 2 3
3  20 = 3  1 6
3  21 = 3  2 12
3  22 = 3  4

10. 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 1 2
2.25
1.5 1
0.67
0.44…

11. Check It Out: Example 1A
Create a table for the exponential function, and use it to graph the function.
f(x) = 2x x y –2 –1 1 2
1 4
2-2
1 2
2-1 1 20 2 21 4 22

12. Check It Out: Example 1B
Create a table for the exponential function, and use it to graph the function.
f(x) = 2x+ 1 x y –2 –1 1 2
5 4
3 2 2
20 + 1 3
21 + 1 5
22 + 1

13. 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.

14. 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 1 2 3
Number of bacteria f(x)
5000
10,000
20,000
40,000

15. Additional Example 2 Continued
f(x) = a1  rx Write the function.
f(x) = 5000  rx
f(0) = a1
f(x) = 5000  2x
The common ratio is 2.
A week is 7 days so let x = 7.
f(7) = 5000  27 = 640,000
Substitute 7 for x.
If the number of bacteria doubles each day, there will be 640,000 bacteria in the culture after a week.

16. 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 1 2 3
Account balance f(x)
300
600
1200
2400

17. Check It Out: Example 2 Continued
f(x) = a1  rx Write the function.
f(x) = 300  rx
f(0) = a1
f(x) = 300  2x
The common ratio is 2.
20 years will be x = 5.
f(5) = 300  25 = 9600
Substitute 5 for x.
In 20 years, Robin will have a balance of $9600.

18. 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.

19. 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 15 30 45
Number of Half-lives x 1 2 3
Bohrium-267 f(x) (mg) 16 8 4

20. Additional Example 3 Continued
f(x) = a1  rx Write the function.
f(x) = 16  rx
f(0) = a1
The common ratio is
1 2
f(x) = 16 
1 2 x
Since 2 minutes = 120 seconds, divide 120 seconds by 15 seconds to find the number of half-lives: x = 8.
f(8) = 16 
1 2 8
Substitute 8 for x.
There is mg of Bohrium-267 left after 2 minutes.

21. 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 25 50 75
Number of Half-lives x 1 2 3
Element (mg) 8 4

22. Check It Out: Example 3 Continued
f(x) = a1  rx Write the function.
f(x) = 8  rx
f(0) = p
The common ratio is
1 2
f(x) = 8 
1 2 x
Since 3 minutes = 180 seconds, divide 180 seconds by 25 seconds to find the number of half-lives: x = 7.2.
f(7.2) = 8 
1 2
7.2
Substitute 7.2 for x.
There is approximately mg of the element left after 3 minutes.

23. 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 3 1 2
3 4
3 2

24. 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.