Graphing Exponential Functions - Algebra II
Card 0 of 12
Determine whether each function represents exponential decay or growth.
Determine whether each function represents exponential decay or growth.
a)
This is exponential decay since the base, 
, is between 
and 
.
b)
This is exponential growth since the base, 
, is greater than 
.
a)
This is exponential decay since the base,
b)
This is exponential growth since the base,
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Give the 
-intercept of the graph of the equation 
.
Give the
Set 
and solve for 
We need not work further. It is impossible to raise a positive number 2 to any real power to obtain a negative number. Therefore, the equation has no solution, and the graph of 
has no 
-intercept.
Set
We need not work further. It is impossible to raise a positive number 2 to any real power to obtain a negative number. Therefore, the equation has no solution, and the graph of
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What is/are the asymptote(s) of the graph of the function 
?
What is/are the asymptote(s) of the graph of the function
An exponential function of the form
has as its one and only asymptote the horizontal line 
.
Since we define 
as
,
then 
,
and the only asymptote is the line of the equation 
.
An exponential function of the form
has as its one and only asymptote the horizontal line
Since we define
then
and the only asymptote is the line of the equation
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Match each function with its graph.
1. 
2. 
3. 
a.
b.
c.
Match each function with its graph.
1.
2.
3.
a.
b.
c.
For 
, our base is greater than 
so we have exponential growth, meaning the function is increasing. Also, when 
, we know that 
since 
. The only graph that fits these conditions is 
.
For 
, we have exponential growth again but when 
, 
. This is shown on graph 
.
For 
, we have exponential decay so the graph must be decreasing. Also, when 
, 
. This is shown on graph 
.
For
For
For
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In 2010, the population of trout in a lake was 416. It has increased to 521 in 2015.
Write an exponential function of the form 
that could be used to model the fish population of the lake. Write the function in terms of 
, the number of years since 2010.
In 2010, the population of trout in a lake was 416. It has increased to 521 in 2015.
Write an exponential function of the form
We need to determine the constants 
and 
. Since 
in 2010 (when 
), then 
and 
To get 
, we find that when 
, 
. Then 
.
Using a calculator, 
, so 
.
Then our model equation for the fish population is 
We need to determine the constants
To get
Using a calculator,
Then our model equation for the fish population is
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An exponential funtion 
is graphed on the figure below to model some data that shows exponential decay. At 
, 
is at half of its initial value (value when 
). Find the exponential equation of the form 
that fits the data in the graph, i.e. find the constants 
and 
.
An exponential funtion
To determine the constant 
, we look at the graph to find the initial value of 
, (when 
) and find it to be 
. We can then plug this into our equation 
and we get 
. Since 
, we find that 
.
To find 
, we use the fact that when 
, 
is one half of the initial value 
. Plugging this into our equation with 
now known gives us 
. To solve for 
, we make use the fact that the natural log is the inverse function of 
, so that
.
We can write our equation as 
and take the natural log of both sides to get:
or 
.
Then 
.
Our model equation is 
.
To determine the constant
To find
We can write our equation as
Then
Our model equation is
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What is the 
-intercept of the graph 
?
What is the
The 
-intercept of any graph describes the 
-value of the point on the graph with a 
-value of 
.
Thus, to find the 
-intercept substitute 
.
In this case, you will get,
.
The
Thus, to find the
In this case, you will get,
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What is the 
-intercept of 
?
What is the
The 
-intercept of a graph is the point on the graph where the 
-value is 
.
Thus, to find the 
-intercept, substitute 
and solve for 
.
Thus, we get:
The
Thus, to find the
Thus, we get:
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What is the 
-intercept of 
?
What is the
The 
-intercept of any function describes the point where 
.
Substituting this in to our funciton, we get:
The
Substituting this in to our funciton, we get:
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Which of the following functions represents exponential decay?
Which of the following functions represents exponential decay?
Exponential decay describes a function that decreases by a factor every time 
increases by 
.
These can be recognizable by those functions with a base which is between 
and 
.
The general equation for exponential decay is,
where the base is represented by 
and 
.
Thus, we are looking for a fractional base.
The only function that has a fractional base is,
.
Exponential decay describes a function that decreases by a factor every time
These can be recognizable by those functions with a base which is between
The general equation for exponential decay is,
Thus, we are looking for a fractional base.
The only function that has a fractional base is,
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Does the function 
have any 
-intercepts?
Does the function
The 
-intercept of a function is where 
. Thus, we are looking for the 
-value which makes 
.
If we try to solve this equation for 
we get an error.
To bring the exponent down we will need to take the natural log of both sides.
Since the natural log of zero does not exist, there is no exponent which makes this equation true.
Thus, there is no 
-intercept for this function.
The
If we try to solve this equation for
To bring the exponent down we will need to take the natural log of both sides.
Since the natural log of zero does not exist, there is no exponent which makes this equation true.
Thus, there is no
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Which of the following correctly describes the graph of an exponential function with a base of three?
Which of the following correctly describes the graph of an exponential function with a base of three?
Exponential functions with a base greater than one are models of exponential growth. Thus, we know that our function will increase and not decrease. Remembering the graph of an exponential function, we can determine that the graph will begin gradually, almost like a flat line. Then, as 
increases, 
begins to increase very quickly.
Exponential functions with a base greater than one are models of exponential growth. Thus, we know that our function will increase and not decrease. Remembering the graph of an exponential function, we can determine that the graph will begin gradually, almost like a flat line. Then, as
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