5-2: Exponential Functions
© 2007 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Graph and identify
transformations of
ex...
Definition
For all real numbers a, ,1a ≠
there is an exponential
function with base a whose
domain is all real numbers
and...
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=
Important Idea
The graph of looks
like:
( ) x
f x a=
1a > 0 1a< <
Exponential
Grow
Exponential
Decay
(0,1)
Important Idea
A good
investment will
grow
exponentially
due to
compounding. years
$
Exponential Grow
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 ...
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=...
Solution
Exponential Decay
1
2
x
y
 
=  
 
1
2
x
y
 
=  
 
(0,1)
Exponential
Grow
Compare:
Exponential
Decay
Important Idea
2xy =
Important Idea
For 0<a<1: y=ax
models
exponential decay
For a>1: y=ax
models
exponential growth
Example
Graph on the same axes
and describe behavior:
y=2x
y=4x
y=8x
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
 
=  ÷...
Solution
1
1
2
x
y
 
=  ÷
 
2
1
4
x
y
 
=  ÷
 
3
1
8
x
y
 
=  ÷
 
Try This
Graph on
the same
axes and
describe
behavior:
1 2x
y =
3
2 2x
y +
=
3
3 2 4x
y −
= −
Solutions
1 2x
y =
3
2 2x
y +
=
3
3 2 4x
y −
= −
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 g...
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 g...
Try This
( ) 5000(1.08)x
f x =
How would you change the
equation from the last
problem, ,
if your
investment was $6000 and...
Definition
The natural exponential
function is a variation of
( ) x
f x a= and is written
( ) x
f x e= . e ≈ 2.718.
Important Idea
The
number e is
located on
your
calculator
in 2 places
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 ...
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 ...
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. o...
Important Idea
In real world applications,
most things cannot grow
forever as suggested by the
exponential growth model.
T...
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 ...
Lesson Close
We will examine other
applications of
exponential functions in
future lessons.
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  • (a) P(35)=314.3 ; (b) approx 2065
  • Hprec5.2

    1. 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. 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. 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. 4. Important Idea The graph of looks like: ( ) x f x a= 1a > 0 1a< < Exponential Grow Exponential Decay (0,1)
    5. 5. Important Idea A good investment will grow exponentially due to compounding. years $ Exponential Grow
    6. 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. 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. 8. Solution Exponential Decay 1 2 x y   =    
    9. 9. 1 2 x y   =     (0,1) Exponential Grow Compare: Exponential Decay Important Idea 2xy =
    10. 10. Important Idea For 0<a<1: y=ax models exponential decay For a>1: y=ax models exponential growth
    11. 11. Example Graph on the same axes and describe behavior: y=2x y=4x y=8x
    12. 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. 13. Solution 1 1 2 x y   =  ÷   2 1 4 x y   =  ÷   3 1 8 x y   =  ÷  
    14. 14. Try This Graph on the same axes and describe behavior: 1 2x y = 3 2 2x y + = 3 3 2 4x y − = −
    15. 15. Solutions 1 2x y = 3 2 2x y + = 3 3 2 4x y − = −
    16. 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. 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. 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. 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. 20. Important Idea The number e is located on your calculator in 2 places
    21. 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. 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. 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. 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. 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. 26. Lesson Close We will examine other applications of exponential functions in future lessons.
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