This document discusses exponential functions including: - The definition and graphs of exponential functions with bases a > 0 and 0 < a < 1, representing exponential growth and decay. - How exponential functions are used to model real-world situations like investments, population growth, radioactive decay. - The natural exponential function with base e and its applications. - Limitations of the exponential growth model and how the logistic model accounts for limited future growth.

1. 5-2: Exponential Functions
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Graph and identify
transformations of
exponential functions.
•Use exponential functions to
solve application problems.

2. Definition
For all real numbers a, ,1a ≠
there is an exponential
function with base a whose
domain is all real numbers
and whose rule is:
( ) x
f x a=

3. Try This
How does the exponential
function differ from
other functions such as
( ) x
f x a=
2
( )f x x= and ?3
( )f x x=

4. Important Idea
The graph of looks
like:
( ) x
f x a=
1a > 0 1a< <
Exponential
Grow
Exponential
Decay
(0,1)

5. Important Idea
A good
investment will
grow
exponentially
due to
compounding. years
$
Exponential Grow

6. Example
Graph y=2x
using a table
of values...
•What happens to y as x → ∞
•What happens to y as x → −∞
•What is the value of y
when x=0.

7. Try This
Graph using your
calculator
What happens to y as x → ∞
What happens to y as x → −∞
What is the value of y when
x=0.
1
2
x
y
 
=  
 

8. Solution
Exponential Decay
1
2
x
y
 
=  
 

9. 1
2
x
y
 
=  
 
(0,1)
Exponential
Grow
Compare:
Exponential
Decay
Important Idea
2xy =

10. Important Idea
For 0<a<1: y=ax
models
exponential decay
For a>1: y=ax
models
exponential growth

11. Example
Graph on the same axes
and describe behavior:
y=2x
y=4x
y=8x

12. Try This
2
1
4
x
y
 
=  ÷
 
Graph on
the same
axes and
describe
behavior:
1
1
2
x
y
 
=  ÷
 
3
1
8
x
y
 
=  ÷
 

13. Solution
1
1
2
x
y
 
=  ÷
 
2
1
4
x
y
 
=  ÷
 
3
1
8
x
y
 
=  ÷
 

14. Try This
Graph on
the same
axes and
describe
behavior:
1 2x
y =
3
2 2x
y +
=
3
3 2 4x
y −
= −

15. Solutions
1 2x
y =
3
2 2x
y +
=
3
3 2 4x
y −
= −

16. Example
If you invest $5000 in a stock
that increases at an average
rate of 8% per year, then the
value of your stock is given
by the function:
( ) 5000(1.08)x
f x =
where x is measured in years.
What is your investment worth
in 10 years?

17. Example
If you invest $5000 in a stock
that increases at an average
rate of 8% per year, then the
value of your stock is given
by the function:
( ) 5000(1.08)x
f x =
where x is measured in years.
When will your investment be
worth $15000?

18. Try This
( ) 5000(1.08)x
f x =
How would you change the
equation from the last
problem, ,
if your
investment was $6000 and
your investment increases at
10% per year?
( ) 6000(1.10)x
f x =

19. Definition
The natural exponential
function is a variation of
( ) x
f x a= and is written
( ) x
f x e= . e ≈ 2.718.

20. Important Idea
The
number e is
located on
your
calculator
in 2 places

21. Example
If the population of the U.S.
continues to grow as it has
since 1980, then the pop. (in
millions) in year t where t=0
corresponds to 1980 is given
by: .0093
( ) 227 t
p t e=
a. Estimate the population
in 2015.

22. Example
If the population of the U.S.
continues to grow as it has
since 1980, then the pop. (in
millions) in year t where t=0
corresponds to 1980 is given
by: .0093
( ) 227 t
p t e=
b. when will the population
reach 500 million?

23. Try This
The amount of 1 kg. of
plutonium that remains after
t years is ( ) .99997t
m t =
How much of the original 1
kg. of plutonium remains
after 10,000 years?
.74 kg

24. Important Idea
In real world applications,
most things cannot grow
forever as suggested by the
exponential growth model.
The Logistic Model is
designed to model situations
that have limited future
growth.

25. Example
The population of certain
bacteria in a beaker at time t
hours is given by
2
100,000
( )
1 50
t
p t
e
−
=
+
Graph and find the upper
limit on the bacteria
population.

26. Lesson Close
We will examine other
applications of
exponential functions in
future lessons.