GRAPHS OF EXPONENTIAL FUNCTIONS GRAPHS OF EXPONENTIAL FUNCTIONS 
GRAPHS OF EXPONENTIAL FUNCTIONS
By Nancy Marcus
In this section we will illustrate, interpret, and discuss the graphs of exponential functions. We will also illustrate how you can use graphs to HELP you solve exponential problems and check your answers.
Horizontal Shifts: This is where the graph is shifted to the right or to the left on the rectangular coordinate axis. The following examples discuss the difference between the graph of f(x) and f(x + C).
Example 5: Graph the function
and the graph function
on the
same rectangular coordinate system. Answer the following questions
about each graph:
1.In what quadrants is the graph of the function
located?
In what quadrants is the graph of the function
located?
2.What is the x-intercept and the y-intercept on the graph of
the function
? What is the x-intercept and the y-intercept
on the graph of the function
?
3.Find the point (2, f(2)) on the graph of
and find (2, g(2))
on the graph of
. What do these two points have in common?
4.Describe the relationship between the two graphs.
5.How would you shift (move) the graph of
so that it is superimposed
on the graph of
? After you move the graph, where would the
point (0, 1) on .
be located?
6.Describe the difference between the two equations.
|
1.You can see that the both graphs are located in quadrants I and II. This means that both function values will always be positive.
2.You can see that neither of the graphs cross the x-axis; therefore, neither of the graphs has an x-intercept. Notice that the graph of f(x) crosses the y-axis at 1. This is because .
The graph of g(x) crosses the y-axis at 0.00673794699909 (just above the x-axis). This is because .
3.The point
, rounded to (2, 7.4) for graphing purposes, is
located on the graph of
.
The point
, rounded to (2, 0) for graphing purposes, is located
on the graph of
, just above the x-axis.
4.Both graphs have the same shape. The graph of
is nothing
more than the graph of
shifted to the right 6 units.
5.Shift (move) the graph of
to the right 6 units. Every point
on the graph of
is thus moved to the right 6 units. The point
(0, 1) is shifted to the right 6 units to
or
.
6.The only difference in the two equations is that the exponent in the g(x) equation is 6 units less than the exponent in the f(x) equation.
Example 6: Graph the function and the function on the same rectangular coordinate system. Answer the following questions about each graph:
1.In what quadrants is the graph of the function
located?
In what quadrants is the graph of the function located?
2.What is the x-intercept and the y-intercept on the graph of
the function
? What is the x-intercept and the y-intercept
on the graph of the function
?
3.Find the point (2, f(2)) on the graph of
and find (2, g(2))
on the graph of
. What do these two points have in common?
4.Describe the relationship between the two graphs.
5.Describe how you would move the graph of
so that it is
superimposed on the graph of
. Where would the point (0, 1)
on the graph of
wind up on after the move?
6.Describe the difference between the two equations.
|
1.The graph of
is located in quadrants I and II. The graph
of
is also located in quadrants I and II. This means that
both functions values will always be positive.
2.Neither graphs cross the x-axis; therefore, neither graph has an x-intercept.
The graph of
crosses the y-axis at 1, and the graph of
crosses the y-axis at 54.5981500331, rounded to (0, 54.6) for
graphing purposes, because
.
3.The point
, rounded to (2, 7.4) for graphing purposes, is
located on the graph of
. The point
, rounded to (2, 403.4)
for graphing purposes, is located on the graph of
.
4.Both graphs have the same shape. The graph of
is nothing
more than the graph of
shifted to the left 4 units.
5.Shift (move) the graph of
to the left 4 units. It will
then be superimposed on the graph of
. When you move the graph
of
to the left 4 units, every point on the graph of
is shifted
to the left 4 units. Therefore, the point (0, 1) will be shifted
left to
or
.
6.The only difference in the two equations is in their exponents. The exponent in the g(x) equation is 4 units greater than the exponent in the f(x) equation.
Hint: Many students have a hard time determining whether the shift is to the left or to the right. One easy way to determine the answer is to set the exponent equal to zero and solve. If the answer is negative, the shift is to the left. If the answer is positive, the shift is to the right.
For example, in the equation
, the shift would be to the right
because
x-10=0
when x = 10. In the equation
, the shift would be
to the left because
x+10=0
when x = -10.
If you would like to review another example, click on Example.
S.O.S MATHematics home page 