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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.