Chapter 6: Logarithmic Functions

6.2: Transformations of the Logarithmic Function

Learning Objectives

For the logarithmic function [latex]f(x)=\log_b{x}[/latex],

  • Perform vertical and horizontal shifts
  • Perform vertical compressions and stretches
  • Perform reflections across the [latex]x[/latex]-axis
  • Perform reflections across the [latex]y[/latex]-axis
  • Determine the transformations of the logarithmic function [latex]f(x)=a\log_b{(x-h)}+k[/latex]
  • Determine the equation of a function given the transformations
  • Determine what happens to the vertical asymptote as transformations are made

Vertical Shifts

If we shift the graph of the logarithmic function [latex]f(x)=\log_2{x}[/latex] up 5 units, all of the points on the graph increase their [latex]y[/latex]-coordinates by 5, but their [latex]x[/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=\log_2{x}[/latex] after it has been shifted up 5 units transforms to [latex]f(x)=\log_2{x}+5[/latex]. The vertical asymptote at [latex]x=0[/latex] remains the same. Table 1 shows the changes to specific values of this function, which are replicated on the graph in figure 1.

[latex]x[/latex] [latex]\log_2{x}[/latex] [latex]\log_2{x}+5[/latex]

Figure 1. Shifting the graph of [latex]f(x)=\log_2{x}[/latex] up 5 units.
[latex]\dfrac{1}{8}[/latex] [latex]-3[/latex] [latex]2[/latex]
[latex]\dfrac{1}{4}[/latex] [latex]-2[/latex] [latex]3[/latex]
[latex]\dfrac{1}{2}[/latex] [latex]-1[/latex] [latex]4[/latex]
[latex]1[/latex] [latex]0[/latex] [latex]5[/latex]
[latex]2[/latex] [latex]1[/latex] [latex]6[/latex]
[latex]4[/latex] [latex]2[/latex] [latex]7[/latex]
[latex]8[/latex] [latex]3[/latex] [latex]8[/latex]
Table 1. [latex]f(x)=\log_2{x}[/latex] is transformed to [latex]f(x)=\log_2{x}+5[/latex].

If we shift the graph of the function [latex]f(x)=\log_2{x}[/latex] down 6 units, all of the points on the graph decrease their [latex]y[/latex]-coordinates by 6, but their [latex]x[/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=\log_2{x}[/latex] after it has been shifted down 6 units transforms to [latex]f(x)=\log_2{x}-6[/latex].The vertical asymptote at [latex]x=0[/latex] remains the same. Table 2 shows the changes to specific values of this function, which are replicated on the graph in figure 2.

[latex]x[/latex] [latex]\log_2{x}[/latex] [latex]\log_2{x}-6[/latex]

Figure 2. Shifting the graph of [latex]f(x)=\log_2{x}[/latex] down 6 units.
[latex]\dfrac{1}{8}[/latex] [latex]-3[/latex] [latex]-9[/latex]
[latex]\dfrac{1}{4}[/latex] [latex]-2[/latex] [latex]-8[/latex]
[latex]\dfrac{1}{2}[/latex] [latex]-1[/latex] [latex]-7[/latex]
[latex]1[/latex] [latex]0[/latex] [latex]-6[/latex]
[latex]2[/latex] [latex]1[/latex] [latex]-5[/latex]
[latex]4[/latex] [latex]2[/latex] [latex]-4[/latex]
[latex]8[/latex] [latex]3[/latex] [latex]-3[/latex]
Table 2. [latex]f(x)=\log_2{x}[/latex] is transformed to [latex]f(x)=\log_2{x}-6[/latex].

Notice that vertical shifts up or down do not change the vertical asymptote.

Move the red dots in manipulation 1 to change the values of [latex]b[/latex] and [latex]k[/latex]. Pay attention to what happens to the graph and the relationship between the value of [latex]k[/latex] and the transformed function.

Manipulation 1. Vertical shifts

Vertical shifts

We can represent a vertical shift of the graph of [latex]f(x)=\log_2{x}[/latex] by adding or subtracting a constant, [latex]k[/latex], to the function:

[latex]f(x)=\log_2{x}+k[/latex]

 If [latex]k>0[/latex], the graph shifts upwards and if [latex]k<0[/latex] the graph shifts downwards.

Example 1

  1. If [latex]f(x)=\log_3x[/latex] is shifted vertically up by 7 units, what is the equation of the transformed function?
  2. If [latex]f(x)=\log_7x[/latex] is shifted vertically down by 4 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=\log_5x+9[/latex]?
  4. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=\log_2x-3[/latex]?

Solution

With vertical shifts, the parent function [latex]f(x)=\log_bx[/latex] is transformed to [latex]f(x)=\log_bx+k[/latex].

  1. [latex]k=7[/latex] so the transformed function is [latex]f(x)=\log_3x+7[/latex]
  2. [latex]k=-4[/latex] so the transformed function is [latex]f(x)=\log_7x-4[/latex]
  3. [latex]k=9[/latex] so the transformation was a vertical shift up by 9 units.
  4. [latex]k=-3[/latex] so the transformation was a vertical shift down by 3 units.

Try It 1

  1. If [latex]f(x)=\log_3x[/latex] is shifted vertically up by 2 units, what is the equation of the transformed function?
  2. If [latex]f(x)=\log_7x[/latex] is shifted vertically down by 9 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=\log_5x+3[/latex]?
  4. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=\log_2x-8[/latex]?

Horizontal Shifts

If we shift the graph of the function [latex]f(x)=\log_2{x}[/latex] right 8 units, all of the points on the graph increase their [latex]x[/latex]-coordinates by 8, but their [latex]y[/latex]-coordinates remain the same. The [latex]x[/latex]-intercept (1, 0) in the original graph is moved to (9, 0) (figure 3). The vertical asymptote at [latex]x=0[/latex] shifts right by 8 units to [latex]x=8[/latex]. Any point [latex](x, y)[/latex] on the original graph is moved to [latex](x+8, y)[/latex].

But what happens to the original function [latex]f(x)=\log_2{x}[/latex]? An automatic assumption may be that since [latex]x[/latex] moves to [latex]x+8[/latex] that the function will become [latex]f(x)=\log_2{(x+8)}[/latex]. But that is NOT the case. Remember that the [latex]x[/latex]-intercept is moved to (9, 0) and if we substitute [latex]x=9[/latex] into the function [latex]f(x)=\log_2{(x+8)}[/latex] we get [latex]f(9)=\log_2{(9+8)}=4.0875 \neq 1[/latex]!! The way to get a function value of 0 is for the transformed function to be [latex]f(x)=\log_2{(x-8)}[/latex]. Then [latex]f(9)=\log_2{(9-8)}=0[/latex]. So the function [latex]f(x)=\log_2{x}[/latex] transforms to [latex]f(x)=\log_2{(x-8)}[/latex] after being shifted 8 units to the right. The reason is that when we move the function 8 units to the right, the [latex]x[/latex]-value increases by 8 and to keep the corresponding [latex]y[/latex]-coordinate the same in the transformed function, the [latex]x[/latex]-coordinate of the transformed function needs to subtract 8 to get back to the original [latex]x[/latex] that is associated with the original [latex]y[/latex]-value. Table 3 shows the changes to specific values of this function, and the graph is shown in figure 3.

[latex]x[/latex] [latex]x-8[/latex] [latex]\log_2{(x-8)}[/latex]

Figure 3. Shifting the graph right 8 units.
[latex]\dfrac{65}{8}[/latex] [latex]\dfrac{1}{8}[/latex] [latex]-3[/latex]
[latex]\dfrac{33}{4}[/latex] [latex]\dfrac{1}{4}[/latex] [latex]-2[/latex]
[latex]\dfrac{17}{2}[/latex] [latex]\dfrac{1}{2}[/latex] [latex]-1[/latex]
[latex]9[/latex] [latex]1[/latex] [latex]0[/latex]
[latex]10[/latex] [latex]2[/latex] [latex]1[/latex]
[latex]12[/latex] [latex]4[/latex] [latex]2[/latex]
[latex]16[/latex] [latex]8[/latex] [latex]3[/latex]
Table 3. Shifting the graph right by 8 units transforms [latex]f(x)=\log_2{x}[/latex] into [latex]f(x)=\log_2{(x-8)}[/latex].

Notice that the vertical asymptote also shifts from [latex]x=0[/latex] to [latex]x=8[/latex].

On the other hand, if we shift the graph of the function [latex]f(x)=\log_2{x}[/latex] left by 11 units, all of the points on the graph decrease their [latex]x[/latex]-coordinates by 11, but their [latex]y[/latex]-coordinates remain the same. So any point [latex](x, y)[/latex] on the original graph moves to [latex](x-11, y)[/latex]. Consequently, to keep the same [latex]y[/latex]-values we need to increase the [latex]x[/latex]-value by 11 in the transformed function. The equation of the function after being shifted left 11 units is [latex]f(x)=\log_2{(x+11)}[/latex]. Table 4 shows the changes to specific values of this function, and the graph is shown in figure 4.

[latex]x[/latex] [latex]x+11[/latex] [latex]f(x)=\log_2{(x+11)}[/latex]

Figure 4. Shifting the graph left 11 units.
[latex]-\dfrac{87}{8}[/latex] [latex]\dfrac{1}{8}[/latex] –3
[latex]-\dfrac{43}{4}[/latex] [latex]\dfrac{1}{4}[/latex] –2
[latex]-\dfrac{21}{2}[/latex] [latex]\dfrac{1}{2}[/latex] –1
–10 1 0
–9 2 1
–7 4 2
–3 8 3
Table 4. Shifting the graph left by 11 units transforms [latex]f(x)=\log_2{x}[/latex] into [latex]f(x)=\log_2{(x+11)}[/latex].

Notice that the vertical asymptote also shifts from [latex]x=0[/latex] to [latex]x=-11[/latex].

Move the red dots in manipulation 2 to change the values of [latex]b[/latex] and [latex]h[/latex]. Pay attention to what happens to the graph and the relationship between the value of [latex]h[/latex] and the transformed function.


Manipulation 2. Horizontal shifts

horizontal shifts

We can represent a horizontal shift of the graph of [latex]f(x)=\log_2{x}[/latex] by adding or subtracting a constant, [latex]h[/latex], to the variable [latex]x[/latex].

[latex]f(x)=\log_2{(x}-h)[/latex]

If [latex]h>0[/latex] the graph shifts toward the right and if [latex]h<0[/latex] the graph shifts to the left.

The vertical asymptote [latex]x=0[/latex] shifts to [latex]x=h[/latex].

Example 2

  1. If [latex]f(x)=\log_3x[/latex] is shifted right by 2 units, what is the equation of the transformed function?
  2. If [latex]f(x)=\log_7x[/latex] is shifted left by 9 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=\log_5{(x+3)}[/latex]?
  4. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=\log_2{(x-8)}[/latex]?

Solution

With horizontal shifts, the parent function [latex]f(x)=\log_bx[/latex] is transformed to [latex]f(x)=\log_b{(x-h)}[/latex].

  1. [latex]h=2[/latex] so the transformed function is [latex]f(x)=\log_3{(x-2)}[/latex]
  2. [latex]h=-9[/latex] so the transformed function is [latex]f(x)=\log_7{(x+9)}[/latex]
  3. [latex]h=-3[/latex] so the transformation was a horizontal shift left by 3 units.
  4. [latex]h=8[/latex] so the transformation was a horizontal shift right by 8 units

Try It 2

  1. If [latex]f(x)=\log_3x[/latex] is shifted right by 7 units, what is the equation of the transformed function?
  2. If [latex]f(x)=\log_7x[/latex] is shifted left by 4 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=\log_5{(x-5)}[/latex]?
  4. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=\log_2{(x+4)}[/latex]?

We can combine vertical and horizontal shifts by transforming [latex]f(x)=\log_b{(x-h)}+k[/latex].

Move the red dots in manipulation 3 to change the values of [latex]b, \;h[/latex] and [latex]k[/latex]. Pay attention to what happens to the graph and the relationship between the values of [latex]h[/latex] and[latex]h[/latex] and the transformed function.


Manipulation 3. Vertical and horizontal shifts

Example 3

  1. If [latex]f(x)=\log_3x[/latex] is shifted vertically down by 3 units and right by 2 units, what is the equation of the transformed function?
  2. If [latex]f(x)=\log_7x[/latex] is shifted left by 9 units and up by 6 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=\log_5{(x-2)}+7[/latex]?
  4. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=\log_2{(x+5)}-4[/latex]?

Solution

With horizontal and vertical shifts, the parent function [latex]f(x)=\log_bx[/latex] is transformed to [latex]f(x)=\log_b{(x-h)}+k[/latex].

  1. [latex]h=2,\;k=-3[/latex] so the transformed function is [latex]f(x)=\log_3{(x-2)}-3[/latex]
  2. [latex]h=-9,\;k=6[/latex] so the transformed function is [latex]f(x)=\log_7{(x+9)}+6[/latex]
  3. [latex]h=2,\;k=7[/latex] so the transformation was a horizontal shift right by 2 units and up by 7 units.
  4. [latex]h=-5,\;k=-4[/latex] so the transformation was a horizontal shift left by 5 units and down by 4 units.

Try It 3

  1. If [latex]f(x)=\log_3x[/latex] is shifted down by 7 units and right by 5 units, what is the equation of the transformed function?
  2. If [latex]f(x)=\log_7x[/latex] is shifted left by 4 units and up by 3 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=\log_5{(x-2)}-1[/latex]?
  4. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=\log_2{(x+4)}+7[/latex]?

Vertical Stretching and Compressing

If we vertically stretch the graph of the function [latex]f(x)=\log_2{x}[/latex] by a factor of 2, all of the[latex]y[/latex]-coordinates of the points on the graph are multiplied by 2, but their [latex]x[/latex]-coordinates remain the same. The equation of the function after the graph is stretched up by a factor of 2 is [latex]f(x)=2\log_2{x}[/latex]. The reason for multiplying  [latex]\log_2{x}[/latex] by 2 is that each [latex]y[/latex]-coordinate is doubled, and since [latex]y=\log_2{x}[/latex], [latex]\log_2{x}[/latex] is doubled. Table 5 shows this change and the graph is shown in figure 5.

[latex]x[/latex] [latex]\log_2{x}[/latex] [latex]f(x)=2\log_2{x}[/latex]

Figure 5. Stretching the graph vertically.
[latex]\dfrac{1}{8}[/latex] -3 -6
[latex]\dfrac{1}{4}[/latex] -2 -4
[latex]\dfrac{1}{2}[/latex] -1 -2
1 0 0
2 1 2
4 2 4
8 3 6
Table 5. Stretching the graph vertically by a factor of 2 transforms [latex]f(x)=\log_2{x}[/latex] into [latex]f(x)=2\log_2{x}[/latex].

On the other hand, if we vertically compress the graph of the function [latex]f(x)=\log_2{x}[/latex] to half of its original height, we multiply the function by the factor [latex]\dfrac{1}{2}[/latex]. All of the [latex]y[/latex]-coordinates of the points on the graph are halved, but their [latex]x[/latex]-coordinates remain the same. This means the [latex]y[/latex]-coordinates are divided by 2, or multiplied by [latex]\dfrac{1}{2}[/latex]. The equation of the function after being compressed is [latex]f(x)=\dfrac{1}{2}\times\log_2{x}[/latex]. The reason for multiplying [latex]\log_2{x}[/latex] by [latex]\dfrac{1}{2}[/latex] is that each [latex]y[/latex]-coordinate becomes half of the original value when it is divided by 2. Table 6 shows this change and the graph is shown in figure 6.

[latex]x[/latex] [latex]\log_2{x}[/latex] [latex]f(x)=\dfrac{1}{2}\log_2{x}[/latex]

Figure 6. Compressing the graph vertically.
[latex]\dfrac{1}{8}[/latex] -3 [latex]-\dfrac{3}{2}[/latex]
[latex]\dfrac{1}{4}[/latex] -2 -1
[latex]\dfrac{1}{2}[/latex] -1 [latex]-\dfrac{1}{2}[/latex]
1 0 0
2 1 [latex]\dfrac{1}{2}[/latex]
4 2 1
8 3 [latex]\dfrac{3}{2}[/latex]
Table 6. Compressing the graph vertically by a factor of [latex]\dfrac{1}{2}[/latex] transforms [latex]f(x)=\log_2{x}[/latex] into [latex]f(x)=\dfrac{1}{2}\log_2{x}[/latex].

Notice that vertical stretching and compressing do not change the vertical asymptote.

Move the red dots in manipulation 4 to change the values of [latex]b[/latex] and [latex]a[/latex]. Pay attention to what happens to the graph and the relationship between the value of [latex]a[/latex] and the transformed function.


Manipulation 4. Vertical stretching and compressing

vertical stretching and compressing

A stretch or compression of the graph of [latex]f(x)=\log_2{x}[/latex] can be represented by multiplying the function by a constant, [latex]a>0[/latex].

[latex]f(x)=a\log_2{x}[/latex]

The magnitude of [latex]a[/latex] indicates the stretch/compression of the graph. If [latex]a>1[/latex], the graph is stretched up by a factor of [latex]a.[/latex] If [latex]0<a<1[/latex], the graph is compressed down by a factor of [latex]a[/latex].

Example 4

  1. If [latex]f(x)=\log_3x[/latex] is stretched by a factor of 7, what is the equation of the transformed function?
  2. If [latex]f(x)=\log_7x[/latex] is compressed to one-third its height, what is the equation of the transformed function?
  3. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=\frac{1}{6}\log_5x[/latex]?
  4. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=5\log_2x[/latex]?

Solution

With stretching and compression, the parent function [latex]f(x)=\log_bx[/latex] is transformed to [latex]f(x)=a\log_bx[/latex].

  1. [latex]a=7[/latex] so the transformed function is [latex]f(x)=7\log_3x[/latex]
  2. [latex]a=\frac{1}{3}[/latex] so the transformed function is [latex]f(x)=\frac{1}{3}\log_7x[/latex]
  3. [latex]a=\frac{1}{6}[/latex] so the transformation was a compression to one-sixth its height
  4. [latex]a=5[/latex] so the transformation was a vertical stretch by a factor of 5

Try It 4

  1. If [latex]f(x)=\log_3x[/latex] is stretched by a factor of 2, what is the equation of the transformed function?
  2. If [latex]f(x)=\log_7x[/latex] is compressed to one-eighth its height, what is the equation of the transformed function?
  3. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=\frac{1}{4}\log_5x[/latex]?
  4. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=7\log_2x[/latex]?

Now we can combine vertical stretches and compressions with horizontal and vertical shifts.

Example 5

  1. If [latex]f(x)=\log_3x[/latex] is stretched by a factor of 7, and moved down by 5 units what is the equation of the transformed function?
  2. If [latex]f(x)=\log_7x[/latex] is compressed to one-third its height, moved left by 4 units and moved up by 2 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=\frac{1}{5}\log_5{(x-9)}-6[/latex]?
  4. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=3\log_2{(x+5)}+7[/latex]?

Solution

With stretching and compression, combined with shifting the parent function [latex]f(x)=\log_bx[/latex] is transformed to [latex]f(x)=a\log_b{(x-h)}+k[/latex].

  1. [latex]a=7,\; k=-5[/latex] so the transformed function is [latex]f(x)=7\log_3x-5[/latex]
  2. [latex]a=\frac{1}{3},\;h=-4,\;k=2[/latex] so the transformed function is [latex]f(x)=\frac{1}{3}\log_7{(x+4)}+2[/latex]
  3. [latex]a=\frac{1}{5},\;h=9,\;k=-6[/latex] so the transformation was a compression to one-fifth its height, a shift right by 9 units, and a shift down by 6 units
  4. [latex]a=3,\;-5,\;k=7[/latex] so the transformation was a vertical stretch by a factor of 3, a shift left by 5 units, and a shift up by 7 units

Try It 5

  1. If [latex]f(x)=\log_3x[/latex] is stretched by a factor of 5, and moved down by 2 units what is the equation of the transformed function?
  2. If [latex]f(x)=\log_7x[/latex] is compressed to one-half its height, moved left by 7 units and moved up by 3 units, what is the equation of the transformed function?
  3. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=\frac{1}{3}\log_5{(x-2)}-3[/latex]?
  4. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=9\log_2{(x+1)}+5[/latex]?

Reflections

Across the [latex]x[/latex]-axis

When the graph of the function [latex]f(x)=\log_2{x}[/latex] is reflected across the [latex]x[/latex]-axis, the [latex]y[/latex]-coordinates of all of the points on the graph change their signs, from positive to negative values or from negative to positive values, while the [latex]x[/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=\log_2{x}[/latex] is reflected across the [latex]x[/latex]-axis is [latex]f(x)=-\log_2{x}[/latex]. The graph changes from increasing upwards to decreasing downwards. Table 7 shows the effect of such a reflection on the functions values and the graph is shown in figure 7.

[latex]x[/latex] [latex]\log_2{x}[/latex] [latex]-\log_2{x}[/latex]

Figure 7. Reflecting the graph of [latex]f(x)=\log_2{x}[/latex] across the [latex]x[/latex]-axis.
[latex]\dfrac{1}{8}[/latex] -3 3
[latex]\dfrac{1}{4}[/latex] -2 2
[latex]\dfrac{1}{2}[/latex] -1 1
1 0 0
2 1 -1
4 2 -2
8 3 -3
Table 7. Reflecting the graph of [latex]f(x)=\log_2{x}[/latex] across the [latex]x[/latex]-axis transforms [latex]f(x)=\log_2{x}[/latex] into [latex]f(x)=-\log_2{x}[/latex].

Notice that reflecting across the [latex]x[/latex]-axis does not change the vertical asymptote.

Across the [latex]y[/latex]-axis

When the graph of the function [latex]f(x)=\log_2{x}[/latex] is reflected across the [latex]y[/latex]-axis, the [latex]x[/latex]-coordinates of all of the points on the graph change their signs, from positive to negative values, while the [latex]y[/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=\log_2{x}[/latex] is reflected across the [latex]y[/latex]-axis is [latex]f(x)=\log_2{(-x)}[/latex]. The graph changes from increasing from the left to decreasing from the left. Table 8 shows the effect of such a reflection on the functions values and the graph is shown in figure 8.

[latex]x[/latex] [latex]-x[/latex] [latex]\log_2{(-x)}[/latex]

Figure 8. Reflecting the graph of [latex]f(x)=\log_2{x}[/latex] across the [latex]y[/latex]-axis.
[latex]-\dfrac{1}{8}[/latex] [latex]\dfrac{1}{8}[/latex] –3
[latex]-\dfrac{1}{4}[/latex] [latex]\dfrac{1}{4}[/latex] –2
[latex]-\dfrac{1}{2}[/latex] [latex]\dfrac{1}{2}[/latex] –1
–1 1 0
–2 2 1
–4 4 2
–8 8 3
Table 8. Reflecting the graph of [latex]f(x)=\log_2{x}[/latex] across the [latex]y[/latex]-axis transforms [latex]f(x)=\log_2{x}[/latex] into [latex]f(x)=\log_2{(-x)}[/latex]
Notice that reflecting across the [latex]x[/latex]-axis does not change the vertical asymptote.

Example 6

  1. If [latex]f(x)=\log_3x[/latex] is reflected across the [latex]x[/latex]-axis what is the equation of the transformed function?
  2. If [latex]f(x)=\log_7x[/latex] is reflected across the [latex]y[/latex]-axis what is the equation of the transformed function?
  3. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=-\log_5{x}-6[/latex]?
  4. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=-3\log_2{-x}[/latex]?

Solution

  1. The [latex]y[/latex]-values change sign so the transformed function is [latex]f(x)=-\log_3x[/latex]
  2.  The [latex]x[/latex]-values change sign so the transformed function is [latex]f(x)=\log_7{(-x)}[/latex]
  3. [latex]a=-1,\;k=-6[/latex] so the transformation was a reflection across the [latex]x[/latex]-axis and a shift down by 6 units
  4. [latex]a=-3[/latex] and [latex]x[/latex] is [latex]-x[/latex] so the transformation was a vertical stretch by a factor of 3, a reflection across the [latex]x[/latex], and a reflection across the [latex]y[/latex]-axis

Try It 6

  1. If [latex]f(x)=\log_9x[/latex] is reflected across the [latex]x[/latex]-axis what is the equation of the transformed function?
  2. If [latex]f(x)=\log_4x[/latex] is reflected across the [latex]y[/latex]-axis what is the equation of the transformed function?
  3. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=\log_5{(-x)}+4[/latex]?
  4. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=2\log_2{(-x)}[/latex]?

Combining Transformations

After learning all the transformations for the function [latex]f(x)=\log_b{x}[/latex], we should be able to write a transformed function given specific transformations, and also determine what transformations have been performed on the function [latex]f(x)=\log_b{x}[/latex], given an arbitrary transformed function [latex]f(x)=a\log_b{(x-h)}+k[/latex].

Example 7

What transformations have been done to the parent function [latex]f(x)=\log_2{x}[/latex] to get the transformed function [latex]f(x)=-3\log_2{(x+3)}-6[/latex]?

Solution

We need to identify [latex]a,\;h,\;k[/latex] and whether or not [latex]x[/latex] has a negative sign in front of it. To do this we line up the transformed function [latex]f(x)=-3\log_2{(x+3)}-6[/latex] with the standard function [latex]f(x)=a\log_2{(x-h)}+k[/latex]:

[latex]a=-3[/latex] means it has been stretched by a factor of 3 and reflected across the [latex]x[/latex]-axis.

[latex]h=-3[/latex] means it has been shifted left by 3 units.

[latex]k=-6[/latex] means it has been shifted down by 6 units.

Try It 7

  1. What transformation was made to the parent function [latex]f(x)=\log_5x[/latex] if the transformed function is [latex]f(x)=\log_5{-x}+7[/latex]?
  2. What transformation was made to the parent function [latex]f(x)=\log_2x[/latex] if the transformed function is [latex]f(x)=-2\log_2{(x-5)}-4[/latex]?

Example 8

  1. If [latex]f(x)=\log_3x[/latex] is reflected across the [latex]x[/latex]-axis, stretched by a factor of 3, and shifted left by 2 units, what is the equation of the transformed function? What happens to the vertical asymptote?
  2. If [latex]f(x)=\log_7x[/latex] is reflected across the [latex]y[/latex]-axis, compressed to half its height, and shifted up by 7 units, what is the equation of the transformed function? What happens to the vertical asymptote?

Solution

  1. [latex]a=-3,\;h=-2[/latex] so the transformed function is [latex]-3\log_3{(x+2)}[/latex]. The vertical asymptote is shifted left by 2 units from [latex]x=0[/latex] to [latex]x=-2[/latex].
  2. [latex]a=\frac{1}{2},\;k=7[/latex] and [latex]x[/latex] has a negative coefficient so the transformed function is [latex]f(x)=\frac{1}{2}\log_7{(-x)}+7[/latex]. Since there are no horizontal shifts, nothing happens to the vertical asymptote.

Try It 8

  1. If [latex]f(x)=\log_3x[/latex] is stretched by a factor of 7, shifted right by 4 units and shifted down by 4 units, what is the equation of the transformed function? What happens to the vertical asymptote?
  2. If [latex]f(x)=\log_7x[/latex] is reflected across the [latex]x[/latex]-axis, compressed to one-sixth its height, shifted left by 1 unit, and shifted up by 7 units, what is the equation of the transformed function? What happens to the vertical asymptote?

Move the red dots in manipulation 5 to change the values of [latex]a, h, k[/latex] and [latex]b[/latex] or to reflect the graph across the [latex]y[/latex]-axis. Pay attention to what happens to the graph and the relationship between the values of [latex]a,\;h,\;k[/latex] and the transformed function.


Manipulation 5. Transformations on [latex]f(x)=\log_b{x}[/latex]

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