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.
Vertical Shifts: A vertical shift takes place when a function is shifted up or down. The following examples look at the difference between the graph of f(x) and the graph of f(x) + C.
Example 3: Graph the function 
and graph 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 of the graph of
the function
? What is the x-intercept and the y-intercept
of 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?
What is the difference between the two points.
4.Describe the relationship between the two graphs.
5.How would you 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.
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1.You can see that the both graphs are located in quadrants I and II. Therefore, both function values will always be positive.
2.You can see that from the graph that neither of the graphs crosses the x-axis; therefore, neither of the graphs has an x-intercept.
Note that the graph of f(x) crosses the y-axis at 1 because 
; the graph of g(x) crosses the y-axis at 4 because
.
3.The point 
, rounded to (2, 7.4) for graphing purposes, is
located on the graph of
.
The point
, rounded to (2, 10.4) for graphing purposes, is
located on the graph of
. For each x-coordinate, the y-coordinates
differ by 3.
4.Both graphs have the same shape. The graph of
is nothing
more than the graph of
shifted up three units.
5.Shift (move) the graph of
up 3 units. Every point on the
graph of
would be moved up 3 units. Therefore, the point (0,
1) would wind up at 
or 
after the move.
6.The only difference between the two equations is the + 3.
The equations could be rewritten as follows:
since
. This
means that for every value of x, the function g(x) will always
be 3 units larger than the function f(x).
Example 4: Graph the function
and the function
on the same
rectangular coordinate system. and 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 of the graph of
the function
?
What is the x-intercept and the y-intercept
of 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 shift (move) the graph of
so that
it is superimposed on the graph of
. Where would the point
(0, 1) on the graph of
be located after the move?
6.What is the difference between the two equations?
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1.The graph of
is located in quadrants I and II. The graph
of
is located in quadrants I, III, and IV. This means that
f(x) will always be positive, and g(x) can be positive, negative,
and zero.
2.The graph of
does not cross the x-axis. There is no x value
such that
will ever equal 0.
The graph of
crosses the x-axis at 1.60943791243, rounded
to 1.6 for graphing purposes:
Let
and solve for x. Add 5 to both sides of the equation
and we have 
. Take the natural logarithms of both sides of
:
or
3.The point
, rounded to (2, 7.4) for graphing purposes, is
located on the graph of
.
4.The point
, rounded to (2, 2.4) for graphing purposes, is
located on the graph of
. For each x-coordinate, the y-coordinates
differ by 5.
5.Both graphs have the same shape. The graph of
is nothing more than the graph of
shifted down 5 units.
6.Shift (move) the graph of
down five units so that it is
superimposed on the graph of
. When we move the graph of
down 5 units so that it is superimposed on the graph of .
, every
point on the graph of
is shifted down 5 units. Therefore,
the point (0, 1) will be shifted down to 
or 
.
7.The difference between the two equations is the - 5
because
. The only substitution we made was to replace
with f(x)
in the equation
. If we knew that the point (c, d) is located
on the graph of f(x), we can conclude that the point (c, d -
5) is located on the graph of g(x). For every value of x, the
function g(x) will always be 5 units less than the function f(x).
If you would like to review another example, click on Example.
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Author: Nancy Marcus