5.2 Exponential Functions
Objective
• Graph and identify transformations of exponential functions
• Use exponential functions to solve application problems
Do Now Copy and complete the following table. Then
plot the points to graph each function. What is the domain
and range of each function?
x
F(x)=2x
F(x)=(1/2)x
-3
-2
-1
0
1
2
3
5.2 Exponential Functions
Objective
• Graph and identify transformations of exponential functions
• Use exponential functions to solve application problems
Do Now Copy and complete the following table. Then
plot the points to graph each function. What is the domain
and range of each function?
x
-3
-2
-1
0
1
2
3
F(x)=2x
0.125
.25
.5
1
2
4
8
8
4
4
1
.5
.25
.125
F(x)=(1/2)x
5.2 Exponential Functions
General exponential function form: f(x)=ax
The base of the function graphed here is 2:
F(x)=2x
When a>1:
•The graph is above the x-axis
•The y-intercept is 1
•f(x) is increasing
•This is called exponential growth
•f(x) approaches the negative x-axis as x approaches
-∞
The base of the function graphed here is 1/2: F(x)=(1/2)x
When 0<a<1:
•The graph is above the x-axis
•The y-intercept is 1
•f(x) is decreasing
•This is called exponential decay
•f(x) approaches the positive x-axis as x approaches +
∞
5.2 Exponential Functions
Consider other cases for the value of a
•What is the role a?
•When a=0, what is f(x)=ax?
•What about when a=1?
•What about when a<0?
5.2 Exponential Functions
Consider other cases for the value of a
•What is the role a?
•When a=0, what is f(x)=ax?
•It is a constant function
•What about when a=1?
•It is also a constant function
•What about when a<0?
•It is not defined
5.2 Exponential Functions
•Graphing Exploration Lab
a. Using a viewing window with -3≤ x ≤7 and -2≤ y ≤18,
graph each function below on the same screen, and
observe the behavior of each to the right of the yaxis.
f(x)=1.3x
g(x)=2x
h(x)=10x
As the graph continues to the right, which graph rises
least steeply? Most steeply?
How does the steepness of the graph of f(x)=ax to the
right of the y-axis seem to be related to the size of
the base a?
•Graphing Exploration Lab
b. Using the graphs of the same three functions in the
viewing window with -4≤ x ≤2 and -0.5≤ y ≤2,
observe the behavior to the left of the y-axis.
f(x)=1.3x
g(x)=2x
h(x)=10x
As the graph continues to the left, how does the size of
the base a seem to be related to how quickly the
graph of f(x)=ax falls toward the x-axis?
5.2 Exponential Functions
Varying the Value of “a”
This graph represents y=2x. A few of the most important
aspects of this graph include:
•the asymptotic behavior as x gets smaller and
smaller.
•the graph passes through the point (0,1). This is
somewhat like a "vertex" for a parabola.
y=2x
5.2 Exponential Functions
Varying the Value of “a”
The red graph represents y=5x. As you can see the graph
still passes through (0,1). The only significant change to
the graph is that the graph rises quicker than y=2x .
Generally speaking, the larger the base, the quicker
the graph rises.
y=2x
y=5x
5.2 Exponential Functions
Varying the Value of “a”
y=2x
y=(1/2)x
This graph represents y=(1/2)x. A fractional base causes the
graph to flip over the y-axis? This can be explained if you
recall some of the exponent laws.
Recall that (1/2) is equal to 2-1.
Thus (1/2)x can be rewritten as (2-1)x.
And using exponent laws this can be finally written as 2-x
which, using transformations, is the same as 2x reflected in
the x direction.
5.2 Exponential Functions
Transforming Exponential Functions
f ( x)  p(a
b ( x c )
)d
•What is the role of p, b, c and d?
• p is the vertical stretch factor and when p is negative, the
function is reflected over the x-axis
• b is the horizontal stretch factor, and when b is negative, the
function is reflected over the y-axis
• c shifts the function horizontally
• x – c shifts the graph right c units and x+c shifts the
graph left c units
• d shifts the function vertically
• d shifts the graph up d units and -d shifts the graph
down d units.
5.2 Exponential Functions
Translations of Functions
To graph the function y= -2x +3 we could do the following:
First, start
with the
basic graph
y=2x
Next, reflect
the graph
over the xaxis to get
y=-2x
Last, move
this graph up
3 to produce
the graph for
y=-2x +3
Notice how the "vertex" point of (0,1) has moved to (0,2)and the
asymptote has been translated 3 units up?
5.2 Exponential Functions
•Exponential functions are useful for modeling
situations in which a quantity increases or
decreases by a fixed factor.
•Exponential Growth
•Exponential Decay
5.2 Exponential Functions
Identify Exponential Growth and Decay
A. Determine whether
represents
exponential growth or decay.
Answer: The function represents exponential growth,
since the base,
, is greater than 1.
5.2 Exponential Functions
Identify Exponential Growth and Decay
B. Determine whether y = (0.7)x represents
exponential growth or decay.
Answer: The function represents exponential decay,
since the base, 0.7, is between 0 and 1.
5.2 Exponential Functions
A. What type of exponential function is represented
by y = (0.5)x?
A. exponential growth
B. exponential decay
C. both exponential
growth and
exponential decay
D. neither exponential
growth nor
exponential decay
1.
2.
3.
4.
A
B
C
D
5.2 Exponential Functions
B. What type of exponential function is represented
by
?
A. exponential growth
B. exponential decay
C. both exponential
growth and
exponential decay
D. neither exponential
growth nor
exponential decay
1.
2.
3.
4.
A
B
C
D
5.2 Exponential Functions
Applications
Example 1:
If you invest $10,000 in a certificate of deposit (CD) with an
interest rate of 4% a year, the value of your certificate is
given by f(x)=10,000(1.04)x where x is measured in years.
a. How much is the certificate worth in 5 years?
b. When will your certificate be worth $18,000?
If you invest $10,000 in a certificate of deposit (CD) with
an interest rate of 4% a year, the value of your certificate
is given by f(x)=10,000(1.04)x where x is measured in
years.
a. How much is the certificate worth in 5 years?
About $12,166.53
When will your certificate be worth $18,000? In about 15 yrs
Note: The graph here depicts
the solution found by
intersection method:
18,000 = 10,000(1.04)x
5.2 Exponential Functions
Example 2:
The projected population of Tokyo, Japan, in millions, from
2000 to 2015 can be approximated by the function
g(x)=26.4(1.0019)x where x=0 corresponds to the year
2000.
a. Estimate the population of Tokyo in 2015.
b. If the population continues to grow at is same rate after
2014, in what year will the population reach 30
millions?
Example 2:
The projected population of Tokyo, Japan, in millions,
from 2000 to 2015 can be approximated by the
function g(x)=26.4(1.0019)x where x corresponds to
the year 2000.
a. Estimate the population of Tokyo in 2015.
27.16mm
b. If the population continues to grow at is same rate after
2014, in what year will the population reach 30
millions? 2067
5.2 Exponential Functions
Example 3:
Geologists and archeologists use carbon-14 to
determine the age of organic substances, such as bones
or small plants and animals found embedded in rocks.
The amount from one kilogram of carbon-14 that
remains after x years can be approximated by
M(x)=0.99988x. Estimate the amount of carbon-14
remaining after 2000 years.
Example 3:
Geologists and archeologists use carbon-14 to
determine the age of organic substances, such as bones
or small plants and animals found embedded in rocks.
The amount from one kilogram of carbon-14 that
remains after x years can be approximated by
M(x)=0.99988x. Estimate the amount of carbon-14
remaining after 2000 years.
About .79kg
5.2 Exponential Functions
The number e and the Natural Exponential Function
There is an irrational number, denoted e, that arises
naturally in a variety of phenomena and plays a
central role in the mathematical description of the
physical universe. Its decimal expansion begins as:
e=2.718281828459045…
•Find the value of e1 on your calculator
5.2 Exponential Functions
Since the base e is a number between 2 and 3, the
graph of f(x)= ex has a steepness in between the
graph of f(x)= 2x and f(x)= 3x .
5.2 Exponential Functions
Example 4:
The population of Los Angeles since 1970, in
millions can be modeled by the function
P(t)=2.7831e0.0096t where t represents time in
years. Assuming that this model continues to
be appropriate, use it to:
a. estimate the population in 2010.
b. When will the population reach 5 million?
Example 4:
The population of Los Angeles since 1970, in
millions can be modeled by the function
P(t)=2.7831e0.0096t where t represents time
in years. Assuming that this model
continues to be appropriate, use it to:
a. estimate the population in 2010.
4.1 million
b. When will the population reach 5 million?
2031
5.2 Exponential Functions
Example 5:
The population of certain bacteria in a beaker at time t
hours is given by the function
p (t ) 
100, 000
1  50e
t
2
, where t  0
•Graph the function and find the upper limit on the
bacteria production.
Upper limit??
What’s that??
5.2 Exponential Functions
•The population of certain bacteria in a beaker at time
t hours is given by the function
p (t ) 
100, 000
1  50e
t
2
, where t  0
•Graph the function and find the upper limit on the
bacteria production. 100,000