**In this section we will illustrate, interpret, and discuss the graphs of logarithmic functions. We will also illustrate how you can use graphs to HELP you solve logarithmic problems.**

(Reflection across the x-axis):

Whenever the minus sign (-) is in front of the function notation, it indicates a reflection across the x-axis. For example, the graph of - f(x) is a reflection of the graph of f(x) across the x-axis. The graph of 3 - g(x) involves the reflection of the graph of g(x) across the x-axis and the upward shift of the reflected graph 3 units.

**Example 1:**

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 both functions?
- 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 ?
- 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.
- How would you move the graph of so that it would be superimposed on the graph of ? Where would the point (1, 0) on be located after such a move?

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 ?

The graph to the right of the y-axis is the graph of the function , and the graph on the left to the left of the y-axis is the graph of the function .

- The domain of both functions is the set of positive real numbers.
- You can see that the graphs of both functions are located in quadrants I and IV to the right of the y-axis. This verifies that the domain of both functions is the set of positive real numbers.
- The graphs of both functions cross the x-axis at x = 1. Since neither of the graphs cross the y-axis, there is no y-intercept.
- The point is located on the graph of . The point is located on the graph of .
- Note that both points have the same x-coordinate and the y-coordinate’s differ by a minus sign.
- Mentally fold the coordinate system at the x-axis. Note that the graph of the function is superimposed on the graph of the function .
- Fold the graph of over the x-axis so that it would be superimposed on the graph of . Every point on the graph of would be shifted up or down twice it’s distance from the x-axis. For example, the point (a, 8) is located 8 units up from the x-axis. If we shifted the point (a, 8) down 16 units, it would wind up at (a, - 8) units. If the point (b, -11) is located on the graph of , it would be shifted up 22 units to (b, 11). Since the point (1, 0) is on the x-axis, the point would not move.

This means both graphs are symmetric to each other with respect to the x-axis. What exactly does that mean? Well for one thing, it means if there is a point (a, b) on the graph of , we know that the point (a, - b) is located on the graph of .

The shapes are the same. The graph of is a reflection over the x-axis of the graph of .

**If you would like to review examples on the following, click on Example:**

Reflection over the y-axis: The graph of f(x) versus the graph of f(-x). Example.

Vertical shifts: The graph of f(x) versus the graph of f(x) + C. Example.

Horizontal shifts: The graph of f(x) versus the graph of f(x + C) Example.

Combination horizontal shift and reflection across the y-axis: The graph of f(x) versus the graph of f(- x + C) or f(C - x) Example.

Combination horizontal and vertical shifts: The graph of f(x) versus the graph of f(x + A) + B Example.

Combination horizontal and vertical shifts and reflections: The graph of f(x) versus the graphs of -f(x) + C. f(-x) + C, f(x + C) + D. Example.

Stretch and Shrink: The graph of f(x) versus the graph of C(x). Example.

Stretch and Shrink: The graph of f(x) versus the graph of f(Cx). Example.

Combination of stretch, shrink, reflection, horizontal, and vertical shifts: Example.

Solving an equation from a graph: Example.

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

Contact us

Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA

users online during the last hour