GRAPHS OF LOGARITHMIC FUNCTIONS

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:
  1. In what quadrants is the graph of the function located? In what quadrants is the graph of the function located?
  2. 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 ?
  3. 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?
  4. Describe the relationship between the two graphs.
  5. 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?
  6. Describe what you can tell about the relationship between the graphs from just their equations.

  1. You can see that the both graphs are located in quadrants I and IV.
  2. You can see that neither of the graphs cross the y-axis; therefore, neither of the graphs has a y-intercept.

    3. 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.

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

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

  4. 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.
  5. 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).
  6. 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

Example 10: Graph the function and the function on the same rectangular coordinate system. and answer the following questions about each graph:
  1. In what quadrants is the graph of the function located? In what quadrants is the graph of the function located?
  2. 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 ?
  3. 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?
  4. Describe the relationship between the two graphs.
  5. 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?

  1. Both the graph of and the graph of are located in quadrants I and IV.
  2. Neither graph crosses the y-axis; therefore neither graph has a y-intercept.

    3. The graph of crosses the x-axis at 1.

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

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

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

  4. Both graphs have the same shape. The graph of shifts to the right and above 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.

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