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.
Reflection across the y-axis: The graph of f(x) versus the graph of f(-x)
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 .
4.What do these two points have in common?
5.Describe the relationship between the two graphs.
6.Describe the difference in the two equations.
7.How would you physically move the graph of so that it is superimposed on the graph of ?
Where would the point , rounded to (1, 2.7) for graphing purposes, be located after such a move?
8.What is the difference between the two equations?
1.You can see that the both graphs are located in quadrants I and II. Therefore, the value of both functions is positive.
2.You can see that neither of the graphs cross the x-axis; therefore, neither of the graphs has an x-intercept. You can see that both of the graphs cross the y-axis at 1; therefore, both graphs have a y-intercept of 1.
3.The point , rounded to (2, 7.4) for graphing purposes, is located on the graph of .
The point , rounded to (-2, 7.4) for graphing purposes, is located on the graph of .
4.Note that both points have the same y-coordinate and their x-coordinates differ by a minus sign.
5.The graphs are mirror images of each other over the y-axis. This is another way of saying that the graphs are symmetric to each other with respect to the y-axis. The shapes are the same. The graph of is a reflection over the y-axis of the graph of . This is also the definition of an even function.
6.Since , substitute f(x) for in the equation . If we rewrite as before the substitution, we have . For every x, the function values are reciprocals of one another. For x values that differ by a minus sign, the function values are the same.
7.Mentally fold the graph of over the y-axis so that it is superimposed on the graph of . In the move, every point is moved to the left twice it's distance from the y-axis. In other words, if a point (a, b) is located on in quadrant I, the point would be a units from the y-axis. When you fold the graph of over the y-axis, the point (a, b) would be located in quadrant II at (- a, b). The distance between + a and - a is 2a.
After the move, the point (2, 7.4) on the graph of would be located at (- 2, 7.4) on the graph of
If you would like to review another example, click on Example.
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Author: Nancy Marcus