Learning Objectives
For the logarithmic function ,
- Perform vertical and horizontal shifts
- Perform vertical compressions and stretches
- Perform reflections across the -axis
- Perform reflections across the -axis
- Determine the transformations of the logarithmic function
- 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 up 5 units, all of the points on the graph increase their -coordinates by 5, but their -coordinates remain the same. Therefore, the equation of the function after it has been shifted up 5 units transforms to . The vertical asymptote at remains the same. Table 1 shows the changes to specific values of this function, which are replicated on the graph in figure 1.
Figure 1. Shifting the graph of up 5 units. |
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Table 1. is transformed to . |
If we shift the graph of the function down 6 units, all of the points on the graph decrease their -coordinates by 6, but their -coordinates remain the same. Therefore, the equation of the function after it has been shifted down 6 units transforms to .The vertical asymptote at remains the same. Table 2 shows the changes to specific values of this function, which are replicated on the graph in figure 2.
Figure 2. Shifting the graph of down 6 units. |
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Table 2. is transformed to . |
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 and . Pay attention to what happens to the graph and the relationship between the value of and the transformed function.
Manipulation 1. Vertical shifts
Vertical shifts
We can represent a vertical shift of the graph of by adding or subtracting a constant, , to the function:
If , the graph shifts upwards and if the graph shifts downwards.
Example 1
- If is shifted vertically up by 7 units, what is the equation of the transformed function?
- If is shifted vertically down by 4 units, what is the equation of the transformed function?
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
Solution
With vertical shifts, the parent function is transformed to .
- so the transformed function is
- so the transformed function is
- so the transformation was a vertical shift up by 9 units.
- so the transformation was a vertical shift down by 3 units.
Try It 1
- If is shifted vertically up by 2 units, what is the equation of the transformed function?
- If is shifted vertically down by 9 units, what is the equation of the transformed function?
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
Horizontal Shifts
If we shift the graph of the function right 8 units, all of the points on the graph increase their -coordinates by 8, but their -coordinates remain the same. The -intercept (1, 0) in the original graph is moved to (9, 0) (figure 3). The vertical asymptote at shifts right by 8 units to . Any point on the original graph is moved to .
But what happens to the original function ? An automatic assumption may be that since moves to that the function will become . But that is NOT the case. Remember that the -intercept is moved to (9, 0) and if we substitute into the function we get !! The way to get a function value of 0 is for the transformed function to be . Then . So the function transforms to after being shifted 8 units to the right. The reason is that when we move the function 8 units to the right, the -value increases by 8 and to keep the corresponding -coordinate the same in the transformed function, the -coordinate of the transformed function needs to subtract 8 to get back to the original that is associated with the original -value. Table 3 shows the changes to specific values of this function, and the graph is shown in figure 3.
Figure 3. Shifting the graph right 8 units. |
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Table 3. Shifting the graph right by 8 units transforms into . |
Notice that the vertical asymptote also shifts from to .
On the other hand, if we shift the graph of the function left by 11 units, all of the points on the graph decrease their -coordinates by 11, but their -coordinates remain the same. So any point on the original graph moves to . Consequently, to keep the same -values we need to increase the -value by 11 in the transformed function. The equation of the function after being shifted left 11 units is . Table 4 shows the changes to specific values of this function, and the graph is shown in figure 4.
Figure 4. Shifting the graph left 11 units. |
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–3 | |||
–2 | |||
–1 | |||
–10 | 1 | 0 | |
–9 | 2 | 1 | |
–7 | 4 | 2 | |
–3 | 8 | 3 | |
Table 4. Shifting the graph left by 11 units transforms into . |
Notice that the vertical asymptote also shifts from to .
Move the red dots in manipulation 2 to change the values of and . Pay attention to what happens to the graph and the relationship between the value of and the transformed function.
Manipulation 2. Horizontal shifts
horizontal shifts
We can represent a horizontal shift of the graph of by adding or subtracting a constant, , to the variable .
If the graph shifts toward the right and if the graph shifts to the left. The vertical asymptote shifts to .
Example 2
- If is shifted right by 2 units, what is the equation of the transformed function?
- If is shifted left by 9 units, what is the equation of the transformed function?
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
Solution
With horizontal shifts, the parent function is transformed to .
- so the transformed function is
- so the transformed function is
- so the transformation was a horizontal shift left by 3 units.
- so the transformation was a horizontal shift right by 8 units
Try It 2
- If is shifted right by 7 units, what is the equation of the transformed function?
- If is shifted left by 4 units, what is the equation of the transformed function?
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
We can combine vertical and horizontal shifts by transforming .
Move the red dots in manipulation 3 to change the values of and . Pay attention to what happens to the graph and the relationship between the values of and and the transformed function.
Manipulation 3. Vertical and horizontal shifts
Example 3
- If is shifted vertically down by 3 units and right by 2 units, what is the equation of the transformed function?
- If is shifted left by 9 units and up by 6 units, what is the equation of the transformed function?
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
Solution
With horizontal and vertical shifts, the parent function is transformed to .
- so the transformed function is
- so the transformed function is
- so the transformation was a horizontal shift right by 2 units and up by 7 units.
- so the transformation was a horizontal shift left by 5 units and down by 4 units.
Try It 3
- If is shifted down by 7 units and right by 5 units, what is the equation of the transformed function?
- If is shifted left by 4 units and up by 3 units, what is the equation of the transformed function?
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
Vertical Stretching and Compressing
If we vertically stretch the graph of the function by a factor of 2, all of the-coordinates of the points on the graph are multiplied by 2, but their -coordinates remain the same. The equation of the function after the graph is stretched up by a factor of 2 is . The reason for multiplying by 2 is that each -coordinate is doubled, and since , is doubled. Table 5 shows this change and the graph is shown in figure 5.
Figure 5. Stretching the graph vertically. |
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-3 | -6 | ||
-2 | -4 | ||
-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 into . |
On the other hand, if we vertically compress the graph of the function to half of its original height, we multiply the function by the factor . All of the -coordinates of the points on the graph are halved, but their -coordinates remain the same. This means the -coordinates are divided by 2, or multiplied by . The equation of the function after being compressed is . The reason for multiplying by is that each -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.
Figure 6. Compressing the graph vertically. |
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-3 | |||
-2 | -1 | ||
-1 | |||
1 | 0 | 0 | |
2 | 1 | ||
4 | 2 | 1 | |
8 | 3 | ||
Table 6. Compressing the graph vertically by a factor of transforms into . |
Notice that vertical stretching and compressing do not change the vertical asymptote.
Move the red dots in manipulation 4 to change the values of and . Pay attention to what happens to the graph and the relationship between the value of and the transformed function.
Manipulation 4. Vertical stretching and compressing
vertical stretching and compressing
A stretch or compression of the graph of can be represented by multiplying the function by a constant, .
The magnitude of indicates the stretch/compression of the graph. If , the graph is stretched up by a factor of If [latex]0
Example 4
- If is stretched by a factor of 7, what is the equation of the transformed function?
- If is compressed to one-third its height, what is the equation of the transformed function?
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
Solution
With stretching and compression, the parent function is transformed to .
- so the transformed function is
- so the transformed function is
- so the transformation was a compression to one-sixth its height
- so the transformation was a vertical stretch by a factor of 5
Try It 4
- If is stretched by a factor of 2, what is the equation of the transformed function?
- If is compressed to one-eighth its height, what is the equation of the transformed function?
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
Now we can combine vertical stretches and compressions with horizontal and vertical shifts.
Example 5
- If is stretched by a factor of 7, and moved down by 5 units what is the equation of the transformed function?
- If 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?
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
Solution
With stretching and compression, combined with shifting the parent function is transformed to .
- so the transformed function is
- so the transformed function is
- so the transformation was a compression to one-fifth its height, a shift right by 9 units, and a shift down by 6 units
- 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
- If is stretched by a factor of 5, and moved down by 2 units what is the equation of the transformed function?
- If 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?
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
Reflections
Across the -axis
When the graph of the function is reflected across the -axis, the -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 -coordinates remain the same. The equation of the function after is reflected across the -axis is . 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.
Figure 7. Reflecting the graph of across the -axis. |
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-3 | 3 | ||
-2 | 2 | ||
-1 | 1 | ||
1 | 0 | 0 | |
2 | 1 | -1 | |
4 | 2 | -2 | |
8 | 3 | -3 | |
Table 7. Reflecting the graph of across the -axis transforms into . |
Notice that reflecting across the -axis does not change the vertical asymptote.
Across the -axis
When the graph of the function is reflected across the -axis, the -coordinates of all of the points on the graph change their signs, from positive to negative values, while the -coordinates remain the same. The equation of the function after is reflected across the -axis is . 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.
Figure 8. Reflecting the graph of across the -axis. |
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–3 | |||
–2 | |||
–1 | |||
–1 | 1 | 0 | |
–2 | 2 | 1 | |
–4 | 4 | 2 | |
–8 | 8 | 3 | |
Table 8. Reflecting the graph of across the -axis transforms into |
Example 6
- If is reflected across the -axis what is the equation of the transformed function?
- If is reflected across the -axis what is the equation of the transformed function?
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
Solution
- The -values change sign so the transformed function is
- The -values change sign so the transformed function is
- so the transformation was a reflection across the -axis and a shift down by 6 units
- and is so the transformation was a vertical stretch by a factor of 3, a reflection across the , and a reflection across the -axis
Try It 6
- If is reflected across the -axis what is the equation of the transformed function?
- If is reflected across the -axis what is the equation of the transformed function?
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
Combining Transformations
After learning all the transformations for the function , we should be able to write a transformed function given specific transformations, and also determine what transformations have been performed on the function , given an arbitrary transformed function .
Example 7
What transformations have been done to the parent function to get the transformed function ?
Solution
We need to identify and whether or not has a negative sign in front of it. To do this we line up the transformed function with the standard function :
means it has been stretched by a factor of 3 and reflected across the -axis.
means it has been shifted left by 3 units.
means it has been shifted down by 6 units.
Try It 7
- What transformation was made to the parent function if the transformed function is ?
- What transformation was made to the parent function if the transformed function is ?
Example 8
- If is reflected across the -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?
- If is reflected across the -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
- so the transformed function is . The vertical asymptote is shifted left by 2 units from to .
- and has a negative coefficient so the transformed function is . Since there are no horizontal shifts, nothing happens to the vertical asymptote.
Try It 8
- If 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?
- If is reflected across the -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 and or to reflect the graph across the -axis. Pay attention to what happens to the graph and the relationship between the values of and the transformed function.
Manipulation 5. Transformations on
Candela Citations
- Transformations of the logarithmic function f(x)=log_b{x}. Authored by: Leo Chang and Hazel McKenna. Provided by: Utah Valley University. License: CC BY: Attribution
- All graphs created using desmos graphing calculator. Authored by: Leo Chang. Provided by: Utah Valley University. Located at: http://www.desmos.com/calculator. License: CC BY: Attribution
- All Examples and Try Its: hjm624; hjm832; hjm302; 442; hjm356; hjm098; hjm529; hjm831. Authored by: Hazel McKenna. Provided by: Utah Valley University. License: CC BY: Attribution
- All manipulations created using Desmos. Authored by: Hazel McKenna. Provided by: Utah Valley University. Located at: http://desmos.com. License: CC BY: Attribution