Transcript Lesson 6 – Exponential Functions

12-6 Exponential Functions
Learn to identify and graph exponential
functions.
Pre-Algebra
12-6 Exponential Functions
An exponential function has the form
f(x) = p  ax, where a > 0 and a ≠ 1. If the
input values are the set of whole numbers,
the output values form a geometric
sequence. The y-intercept is f(0) = p. The
expression ax is defined for all values of x, so
the domain of f(x) = p  ax is all real
numbers.
Pre-Algebra
12-6 Exponential Functions
Additional Example 1A: Graphing an Exponential
Function
Create a table for the exponential function, and
use it to graph the function.
A. f(x) = 3
x

2x
3

2-2 = 3
3

2-1
f(x)
1
4
1
= 3 2
–1
3
4
3
2
0
3
3

20 = 3

1
1
6
3

21 = 3

2
2
12
3

22 = 3

4
–2
Pre-Algebra

12-6 Exponential Functions
Additional Example 1B: Graphing an Exponential
Function
Create a table for the exponential function, and
use it to graph the function.
B. f(x) = 4
f(x)
x
–2
16
–1
8
0
1
2
4
2
1
Pre-Algebra
4
4
4
4
4






1
2
1
2
1
2
1
2
1
2
1
2
x
–2
=4

4
=4

2
=4

1
=4

=4

–1
0
1
2
1
2
1
4
12-6 Exponential Functions
Try This: Example 1A
Create a table for the exponential function, and
use it to graph the function.
A. f(x) = 2x
x
f(x)
–1
1
4
1
2
0
1
20
1
2
21
2
4
22
–2
Pre-Algebra
2-2
2-1
12-6 Exponential Functions
Try This: Example 1B
Create a table for the exponential function, and
use it to graph the function.
B. f(x) = 2x+ 1
x
f(x)
–1
5
4
3
2
0
2
20 + 1
1
3
21 + 1
2
5
22 + 1
–2
Pre-Algebra
2-2 + 1
2-1 + 1
12-6 Exponential Functions
If a > 1, the output f(x) gets larger as the
input x gets larger. In this case, f is called an
exponential growth function.
Pre-Algebra
12-6 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
Pre-Algebra
12-6 Exponential Functions
Additional Example 2 Continued
f(x) = p

ax
f(x) = 5000

ax
f(0) = p
f(x) = 5000

2x
f(1) = 5000  a1
= 10,000, so a = 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.
Pre-Algebra
12-6 Exponential Functions
Try This: Example 2
Quinn 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
Every 4
years x
Account
balance f(x)
Pre-Algebra
2003
2007
2011
2015
0
1
2
3
300
600
1200
2400
12-6 Exponential Functions
Try This: Example 2 Continued
f(x) = p

ax
f(x) = 300

ax
f(x) = 300
 2x
f(0) = p
f(1) = 300
so a = 2.

a1 = 300
20 years will be x = 5.
f(5) = 300

25 = 9600
Substitute 5 for x.
In 20 years, Quinn will have a balance of $9600.
Pre-Algebra
12-6 Exponential Functions
In the exponential function f(x) = p  ax, if a < 1,
the output gets smaller as x gets larger. In this
case, f is called an exponential decay function.
Pre-Algebra
12-6 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)
Pre-Algebra
12-6 Exponential Functions
Additional Example 3 Continued
f(x) = p

f(x) = 16
f(x) = 16
ax

ax

1
2
f(0) = p
x
f(1) = 16  a1 = 8
so a = 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.
Pre-Algebra
12-6 Exponential Functions
Try This: 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
8
4
2
1
Element (mg)
Pre-Algebra
12-6 Exponential Functions
Try This: Example 3 Continued
f(x) = p
ax

f(x) = 8
f(x) = 8

ax

f(0) = p
1
2
x
f(1) = 8  a1 = 4
so a = 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.
Pre-Algebra