**In this section we will illustrate, interpret, and discuss the graphs of logarithmic functions. Recall that whenever there is a minus sign in front of the logarithmic term, it means that there is a reflection across the x-axis. Recall also that whenever there is a minus sign in front of the x in the argument, it means that there is a reflection across the y-axis. **

Horizontal and Vertical Shifts and Reflection Across the x-axis: The following examples discuss the difference between the graph of f(x) and the graph of -f(x + A) + B

**Example 11:** Graph the function and the function on the same rectangular coordinate system. and answer the following questions about each graph:

- In what quadrants in 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 (4, g(4)) on the graph of . What do these two points have in common?
- Describe the relationship between the two graphs.
- How would you moved the graph of so that it would be superimposed on the graph of ? When you moved the graph, where would the point (1, 0) on be after the move?

- You can see that the both graphs are located in quadrants I and IV.
- You can see that neither of the graphs crosses the y-axis; therefore, neither of the graphs has a y-intercept. Notice that the graph of f(x) crosses the x-axis at 1 because . The graph of g(x) crosses the x-axis at 22.085537 because when x = 22,085537.
- The point is located on the graph of . The point is located on the graph of .
- Both graphs have the same shape. It appears that the graph of is the result of reflecting the graph over the x-axis, then shifting the graph to the right and upward.
- After we reflect the graph across the x-axis, the point (1,0) stays the same. After the graph is then moved to the right 2 units and up 3 units, it would be superimposed on the graph of . The point (1, 0) on the graph of would first be shifted to the right 2 units and up 3 units to (1 + 2, 0 + 3) or (3, 3).

**Example 12: **Graph the function and the function on the same rectangular coordinate system. Answer the following questions about each graph:

- In what quadrants in the graph of the function located? In what quadrants is the graph of the function located?
- State the domain of both functions.
- 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 (6, g(6)) on the graph of
- What do these two points have in common?
- Describe the relationship between the two graphs.
- Describe how you would move the graph of moved 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?

- The graph of
- The domain of is the set of all positive real numbers. The graph of is the set of real numbers less than 8.
- The graph of
- The point is located on the graph of . The point is located on the graph of .
- The point (6, g(6)) is located to the right and down from the point (2, f(2)).
- Both graphs have the same shape. The graph of opens to the left whereas the graph of opens to the right. There are horizontal and vertical shifts.
- Rewrite the equation by factoring out -1 from the argument to read .

The graph of crosses the x-axis at -12.085537 because when x = -12.085537. It crosses the y-axis at -0.920558 because .

Whenever there is a minus in front of the x, it means there is a reflection across the y-axis. Whenever something is added or subtracted from the x in the argument, it means there is a horizontal shift. Whenever a constant is added to the logarithmic term, it means there is a vertical shift.

From the equation, you can see that there is a reflection across the y-axis (minus sign in front of the x), a horizontal shift will be to the right 8 units, and a vertical shift down 3 units.

Therefore, reflect the graph of over the y-axis and then shift (move) the reflected graph to the right 8 units and down 3 units. The point (1, 0) would be located (-1, 0) after the reflection and (1 + 8, 0 - 3) or (9, - 3) after the horizontal and vertical shifts..

If you would like to review another example, click on Example.

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