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GRAPHS OF EXPONENTIAL FUNCTIONS
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By Nancy Marcus
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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.
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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).
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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. *

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

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