**In this section we will illustrate, interpret, and discuss the graphs of logarithmic functions. **

Horizontal and vertical shifts: The next examples discuss the difference between the graph of f(x) and the graph of f(x + A) + B.

**Example 9:** Graph the function and the function on the same rectangular coordinate system. and answer the following questions about each graph:

- 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 (8, g(8)) on the graph of . What do these two points have in common?
- Describe the relationship between the two graphs.
- How would you physically shift (move) the graph of so that it would be superimposed on the graph of ? After you move the graph, where would the point (1, 0) be located?
- Describe what you can tell about the relationship between the graphs from just their equations.

- You can see that the both graphs are located in quadrants I and IV.
- You can see that neither of the graphs cross the y-axis; therefore, neither of the graphs has a y-intercept.
- The point , rounded to (2, 0.3) for graphing purposes, is located on the graph of .
- Both graphs have the same shape. It appears that the graph of is the result of shifting the graph of to the right and upward.
- After we move the graph of to the right 6 units and up 5 units, it is superimposed on the graph of . The point (1, 0) on the graph of would first be shifted to the right 6 units and up 5 units to (1 + 6, 0 + 5) or (7, 5).
- Since the arguments differ in each equation by a constant, there will be a horizontal shift. By setting x - 6 to 0, you can tell that the shift is to the right 6 units. You can also tell that the equations differ by a constant. This means there will also be vertical shift of 5 units up

Notice that the graph of f(x) crosses the x-axis at 1 because . The graph of g(x) crosses the x-axis at 6.00001 because when x = 6.00001.

The point , rounded to (8, 5.3) for graphing purposes, is located on the graph of .

**Example 10: **Graph the function and the function on the same rectangular coordinate system. and answer the following questions about each graph:

- 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 (10, g(10)) on the graph of . What do these two points have in common?
- Describe the relationship between the two graphs.
- Describe how you would physically move (shift) the graph of so that it would be superimposed on the graph of . Where would the point (1, 0) on the graph of wind up on after the move?

- Both the graph of
- Neither graph crosses the y-axis; therefore neither graph has a y-intercept.
- The point , rounded to (2, 0.3) for graphing purposes, is located on the graph of .
- Both graphs have the same shape. The graph of shifts to the right and above the graph of .

The graph of crosses the x-axis at 1.

The graph of crosses the x-axis at 1008 because when x = 1008.

The point , rounded to (2, -2.7) for graphing purposes. is located on the graph of .

Whenever the arguments differ by a constant, there is a horizontal shift in the graphs. Whenever the constant terms differ in the equations, there is a vertical shift in the graphs.

From the equation, you can see that the horizontal shift is to the right 8 units, and the vertical shift is down 3 units. It does no make any difference what you do first.

Therefore, shift the graph of to the right 8 units and down 3 units. The point (1, 0) would be moved to the right 8 units and down 3 units to (1 + 8, 0 - 3) or (9, - 3).

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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