GRAPHS OF LOGARITHMIC FUNCTIONS
In this section we will illustrate, interpret, and discuss the graphs of logarithmic functions.
Reflection across the y-axis: The graph of f(x) versus the graph of f(-x)
Normally, if there is a minus sign (-) before an x in the argument of a function, it indicates that there is a reflection across the y-axis. For example, the graph of f(-x) is a reflection of the graph of f(x) across the y-axis. The graph of f(6-x) involves the reflection of the graph of f(x) over the y-axis and then the rightward shift of the reflected graph six units.
Graph the function and the function on the same rectangular coordinate system and answer the following questions about each graph:
- What is the domain of f(x)? What is the domain of g(x)?
- In what quadrants is the graph of the function located? In what quadrants is the graph of the function located?
- 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 ?
- 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?
- Describe the relationship between the two graphs.
- Describe the difference in the two equations.
- How would you physically move the graph of so that it is superimposed on the graph of ?
- Where would the point (1, 0) be located after such a move?
- The domain of f(x) is the set of all positive real numbers, and the domain main of g(x) is the set of all negative numbers.
- The graph of f(x) is located in quadrants I and IV, where the x values are positive. The graph of g(x) is located in quadrants II and III, where the x values are negative.
- You can see that neither of the graphs cross the y-axis; therefore, neither of the graphs has a y-intercept. The graph of f(x) crosses the x-axis at (1, 0) and the graph of g(x) crosses the x-axis at (-1, 0).
- The point , rounded to (2, 0.3) for graphing purposes, is located on the graph of .
The point , rounded to (-2, 0.3) for graphing purposes, is located on the graph of .
- Note that both points have the same y-coordinate and their x-coordinates differ by a minus sign.
- 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.
- For x values that differ by a minus sign, the function values are the same.
- 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 itís 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 (1, 0) on the graph of would be located at (- 1, 0) on the graph of .
If you would like to review another example, click on Example.
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