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 and vertical shifts: The next examples discuss the difference between the graph of f(x) and the graph of f(x + A) + B.
Example 9: 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 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 physically shift (move) the graph of
so that
it would be superimposed on the graph of
? After you move the
graph, where would the point (0, 1) be located?
6.Describe what you can tell about the relationship between the graphs from just their equations.
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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 because 
. The graph of g(x) crosses the y-axis at 5.00247875218 because 
.
3.The point
, rounded to (2, 7.4) for graphing purposes, is
located on the graph of
. The point
, rounded to (2, 5) for
graphing purposes, is located on the graph of
.
4.Both graphs have the same shape. It appears from the graph,
that the graph of
is a result of shifting the graph of
to
the right and upward.
5.After we move the graph of
to the right 6 units and up 5
units, it is superimposed on the graph of
. The point (0,
1) on the graph of
would first be shifted to the right 6 units
and up 5 units to (0 + 6, 1 + 5) or (6, 6).
6.Since the exponents differ in each equation by a constant, there will be a horizontal shift. By setting x - 6 to 0, you can tell that the shift is to the right 6 units. You can also tell that the equations differ by a constant. This means there will also be vertical shift of 5 units up
Example 10: 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 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 physically move (shift) the graph of
so that it would be superimposed on the graph of
. Where
would the point (0, 1) on the graph of
wind up on after the
move?
|
1.The graph of
is located in quadrants I and II. The graph
of
is also located in quadrants I, III, and IV.
2.The graph of
does not cross the x-axis because there is
no value of x that would cause
to equal zero.
The graph of
crosses the x-axis at -9.09861228867 because
that is the solution when we set 
:
3.The graph of
crosses the y-axis at 1, and the graph of
crosses the y-axis at -2.99966453737 because
.
4.The point
, rounded to (2, 7.4) for graphing purposes, is
located on the graph of
.
The point 
, rounded to (2, -3) for graphing purposes. is located
on the graph of
.
5.Both graphs have the same shape. The graph of
appears to
the right and above the graph of
.
Whenever the exponents differ by a constant, there is a horizontal shift in the graphs. Whenever the constant terms differ in the equations, there is a vertical shift in the graphs.
From the equation, you can see that the horizontal shift is to the right 8 units, and the vertical shift is down 3 units. It does no make any difference what you do first.
Therefore, shift the graph of
to the right 8 units and down
3 units. The point (0, 1) would be moved to the right 8 units
and down 3 units to (0 + 8, 1 - 3) or (8, - 2).
If you would like to review another example, click on Example.
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