Transcript exponential function

13-5 Exponential Functions
Warm Up
Write the rule for each linear function.
1.
f(x) = –5x – 2
2.
f(x) = 2x + 6
13-5 Exponential Functions
Learn to identify and graph exponential
functions.
13-5 Exponential Functions
Vocabulary
exponential function
exponential growth
exponential decay
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.
13-5 Exponential Functions
13-5 Exponential Functions
In an exponential function, the y-intercept is
f(0) = a. The expression rx is defined for all
values of x, so the domain of f(x)= a  rx is
all real numbers.
13-5 Exponential Functions
Additional Example 1A: Graphing Exponential
Functions
Create a table for the exponential function,
and use it to graph the function.
f(x) = 3
x

2x
y
1
3
4
3
2
3

2-2 = 3
3

2-1
0
3
3

20 = 3

1
1
6
3

21 = 3

2
2
12
3

22 = 3

4
–2
–1
4
1
= 3 2
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
y
-2
2.25
-1
1.5
0
1
1
0.67
2
0.44…
x
13-5 Exponential Functions
In the function f(x) = a  rx if r > 1 and
a > 0, the output gets larger as the
input gets larger. In this case, f is called
an exponential growth function.
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
Number of days x
Mon
Tue
Wed
Thu
0
1
2
3
Number of bacteria
5000 10,000 20,000 40,000
f(x)
13-5 Exponential Functions
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.
13-5 Exponential Functions
In the exponential function f(x) = a  rx, if 0 < r < 1
and a > 0, the output gets smaller as x gets larger. In
this case, f is called an exponential decay function.
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
16
8
4
2
Bohrium-267
f(x) (mg)
13-5 Exponential Functions
Additional Example 3 Continued
f(x) = a1  rx
f(x) = 16

f(x) = 16

Write the function.
rx
f(0) = a1
1
2
x
The common ratio
is 1 .
2
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 0.0625 mg of Bohrium-267 left after 2
minutes.
13-5 Exponential Functions
Check It Out: Example 1A
Create a table for the exponential function,
and use it to graph the function.
f(x) = 2x
x
–2
–1
y
1
4
1
2
2-2
2-1
0
1
20
1
2
21
2
4
22
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
13-5 Exponential Functions
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.
13-5 Exponential Functions
Check It Out: Example 3
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
Number of
Half-lives x
Element
(mg)
0
25
50
75
0
1
2
3
8
4
2
1
13-5 Exponential Functions
Check It Out: Example 3 Continued
f(x) = a1  rx
f(x) = 8
f(x) = 8

Write the function.
rx

f(0) = p
1
2
x
The common ratio
is 1 .
2
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 7.2
2
Substitute 7.2 for x.
There is approximately 0.054 mg of the element left
after 3 minutes.
13-5 Exponential Functions
Exit Ticket
13-5 Exponential Functions
Exit Ticket
1. Identify a table and the correct graph for the
given exponential function.
A.
B.
13-5 Exponential Functions
Exit Ticket
2. Peter invested $1000 in an account that will
double his balance every 5 years. What will his
balance be in 25 years?
A. $5,000
B. $25,000
C. $32,000
D. $100,000