**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.

**Example 2:**

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 .
- 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 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.

The point , rounded to (-2, 0.3) for graphing purposes, is located on the graph of .

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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