GRAPHS OF LOGARITHMIC FUNCTIONS

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

Reflection across the y-axis and horizontal shift: The next examples discuss the difference between the graph of f(x) and the graph of f(C - x)

Example 7: Graph the function and the function on the same rectangular coordinate system. and answer the following questions about each graph:

  1. Describe the graphs of both functions.
  2. State the domains of both functions
  3. In what quadrants is the graph of the function located? In what quadrants is the graph of the function located?
  4. What is the x-intercept and the y-intercept of the graph of the function ? What is the x-intercept and the y-intercept of the graph of the function ?
  5. Find the point (2, f(2)) on the graph of and find (5, g(5)) on the graph of . What do these two points have in common?
  6. Describe the relationship between the two graphs.
  7. How would you shift (move) the graph of so that it would be superimposed on the graph of ? Where would the point (1, 0) on be located after the move?

  1. The graph of f(x) is reflected over the y-axis and then shifted to the right 7 units to become the graph of g(x).
  2. The domain of f(x) is the set of all positive real numbers. After just the reflection, the domain changes to the set of all negative real numbers. After the shift to the right 7 units, the domain is the set of real numbers less than 7.
  3. You can see that the graph of f(x) is located in quadrants I and IV. This verifies that the domain of the function is the set of positive real numbers. The graph of the function g(x) is located in quadrants I, III, and IV to the left of the line x = 7. This verifies that the domain of g(x) is the set of real numbers less than 7.
  4. The graph of f(x) crosses the x-axis at x =1 because f(x) equals 0 when .
  5. The graph of g(x) crosses the x-axis at x = 6 because g(x) equals 0 when .

  6. The point , rounded to (2, 0.3) for graphing purposes, is located on the graph of . The point , rounded to (5, 0.3) for graphing purposes, is located on the graph of .
  7. Both graphs have the same shape. There is a reflection of across the y-axis because the graphs are pointing in different directions. There is also a horizontal shift to the right 7 units after the reflection.
  8. From the equation, you might be tempted to say that the graph of is nothing more than the graph of shifted to the right 7 units. Good thing that you resisted that temptation because you would be wrong.

The minus in front of the x indicates that the graph of will be reflected over the y-axis. The 7 means that there will a horizontal shift of 7 units; but which way?

You can answer the questions in two ways: One, rewrite by factoring out - 1 from the argument: . Now you can see that the shift is to the right 7 units. Two, set the argument equal to 0: when x = 7. Since the 7 is positive, you know that the shift is to the right.

The point (1, 0) on the graph of is reflected over the y-axis. After the reflection, the point is shifted to (-1,0). The graph is then shifted right to (-1 + 7, 0) or (6, 1).

Example 8: Graph the function and the function on the same rectangular coordinate system. Answer the following questions about each graph:

  1. Describe the graphs of f(x) and g(x).
  2. State the domains of f(x) and g(x)
  3. In what quadrants is the graph of the function located? In what quadrants is the graph of the function located?
  4. 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 ?
  5. Find the point (2, f(2)) on the graph of and find (-12, g(-12)) on the graph of . What do these two points have in common?
  6. Describe the relationship between the two graphs.
  7. Describe how you would move the graph of so that it is superimposed on the graph of . Where would the point (1, 0) on the graph of wind up on after the move?
  8. Describe how you can determine the movement of the f(x) to g(x) from the two equations.

  1. When the graph of f(x) is reflected over the y-axis and then shifted to the left 10 units, you have the graph of g(x).
  2. The domain of f(x) is the set of all positive real numbers. The domain of g(x) is the subset of all real numbers where . This indicates the domain of g(x) is the set of real numbers less than - 10.
  3. The graph of is located in quadrants I and II. This verifies that the domain of f(x) is the set of positive real numbers. The graph of is located in quadrants II and III to the left of x = -10. This verifies that the domain of g(x) is the set of real numbers less than - 10.
  4. The graph of f(x) crosses the x-axis at x =1 because f(x) equals 0 when . The graph of f(x) never crosses the y-axis.
  5. The graph of g(x) crosses the x-axis at x = - 11 because . The graph of g(x) never crosses the y-axis.

  6. The point , rounded to (2, 0.3) for graphing purposes, is located on the graph of .
  7. The point , rounded to (-12, 0.3) for graphing purposes, is located on the graph of .

  8. Both graphs have the same shape. The graph of appears to open in a direction opposite to the direction of the graph of , indicating a reflection across the y-axis.

There also appears to be some kind of horizontal shift. The minus sign in front of the x in the exponents indicates that there is a reflection across the y-axis. Rewrite the equation by factoring out a - 1 from the exponent to read .

From the equation, you can see that the horizontal shift will be to the left 10 units and there will be a reflection over the y-axis.

Therefore, reflect the graph of over the y-axis and then shift (move) the reflected graph left 10 units. The point (1, 0) is shifted to (-1, 0) with the reflection and then to (-1 - 10, 0) or (- 11, 0).

Suppose the point (a, b) was located on the graph of f(x) to the right of the y-axis. Where would that point be located after the move? After the reflection, the point would be located at (- a, b). After the shift to the left 10 units, the point would be located at ( -a - 10, b).

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

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Author: Nancy Marcus

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