Learning Outcomes
For the exponential function [latex]f(x)=r^x[/latex],
- Perform vertical and horizontal shifts
- Determine the equation of a transformed function
- Determine the transformations of the exponential function [latex]f(x)=r^{(x-h)}+k[/latex]
Vertical Shifts
If we shift the graph of the exponential function [latex]f(x)=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)=2^x[/latex] after it has been shifted up 5 units transforms to [latex]f(x)=2^x+5[/latex]. Table 1 shows the changes to specific values of this function, which are graphed in figure 1.
| [latex]x[/latex] | [latex]2^x[/latex] | [latex]2^x+5[/latex] |
Figure 1. Shifting the graph of [latex]f(x)=2^x[/latex] up 5 units. |
|---|---|---|---|
| -3 | [latex]\dfrac{1}{8}[/latex] | [latex]\dfrac{41}{8}[/latex] | |
| -2 | [latex]\dfrac{1}{4}[/latex] | [latex]\dfrac{21}{4}[/latex] | |
| -1 | [latex]\dfrac{1}{2}[/latex] | [latex]\dfrac{11}{2}[/latex] | |
| 0 | 1 | 6 | |
| 1 | 2 | 7 | |
| 2 | 4 | 9 | |
| 3 | 8 | 13 | |
| Table 1. [latex]f(x)=2^x[/latex] is transformed to [latex]f(x)=2^x+5[/latex] | |||
If we shift the graph of the function [latex]f(x)=2^x[/latex] down 3 units, all of the points on the graph decrease their [latex]y[/latex]-coordinates by 3, but their [latex]x[/latex]-coordinates remain the same. Therefore, the equation of the function [latex]f(x)=2^x[/latex] after it has been shifted down 3 units transforms to [latex]f(x)=2^x-3[/latex]. Table 2 shows the changes to specific values of this function, which are graphed in figure 2.
| [latex]x[/latex] | [latex]2^x[/latex] | [latex]2^x-3[/latex] |
Figure 2. Shifting the graph of [latex]f(x)=2^x[/latex] down 3 units. |
| -3 | [latex]\dfrac{1}{8}[/latex] | [latex]-\dfrac{23}{8}[/latex] | |
| -2 | [latex]\dfrac{1}{4}[/latex] | [latex]-\dfrac{11}{4}[/latex] | |
| -1 | [latex]\dfrac{1}{2}[/latex] | [latex]-\dfrac{5}{2}[/latex] | |
| 0 | 1 | -2 | |
| 1 | 2 | -1 | |
| 2 | 4 | 1 | |
| 3 | 8 | 5 | |
| Table 2. [latex]f(x)=2^x[/latex] is transformed to [latex]f(x)=2^x-3[/latex] | |||
These vertical shifts can be applied to any exponential function. Change the values of [latex]r[/latex] and [latex]k[/latex] in figure 3 by moving the green and red dots and note the changes that happen.
Figure 3. Manipulation of [latex]f(x)=r^x+k[/latex]
Vertical shifts
A vertical shift of the graph of [latex]f(x)=r^x[/latex] with [latex]r>0,\;r\neq1[/latex], adds a constant, [latex]k[/latex], resulting in the transformed function:
[latex]f(x)=r^x + k[/latex]
If [latex]k>0[/latex], the graph shifts upwards by [latex]k[/latex] units and if [latex]k<0[/latex] the graph shifts downwards by [latex]k[/latex] units.
TRY IT 1
Manipulate the values of [latex]r[/latex] and [latex]k[/latex] in figure 3 to answer the following questions:
1. What happens to the point (0, 1) on the graph of [latex]f(x)=2^x[/latex] when [latex]k=5[/latex]?
2. What happens to the horizontal asymptote [latex]y=0[/latex] of the function [latex]f(x)=2^x[/latex] when [latex]k=5[/latex]?
3. What happens to the point (0, 1) on the graph of [latex]f(x)=3^x[/latex] when [latex]k=-2[/latex]?
4. What happens to the horizontal asymptote [latex]y=0[/latex] of the function [latex]f(x)=3^x[/latex] when [latex]k=-2[/latex]?
5. What happens to every point on the graph of [latex]y=r^x[/latex] for [latex]r>0,\;r\neq1[/latex] when [latex]k=4[/latex]?
6. What happens to the horizontal asymptote of the graph of [latex]y=r^x[/latex] for [latex]r>0,\;r\neq1[/latex] when [latex]k=-3[/latex]?
Try It 2
Determine the exponential function that comes from transforming the parent function [latex]f(x)=2^x[/latex]:
1. 3 units up
2. 7 units down
Horizontal Shifts
If we shift the graph of the function [latex]f(x)=2^x[/latex] right 4 units, all of the points on the graph increase their [latex]x[/latex]-coordinates by 4, but their [latex]y[/latex]-coordinates remain the same. The [latex]y[/latex]-intercept (0, 1) in the original graph moves to (4, 1) (figure 4). Any point [latex](x, y)[/latex] on the original graph is moved to [latex](x+4, y)[/latex].
But what happens to the original function [latex]f(x)=2^x[/latex]? An automatic assumption may be that since [latex]x[/latex] moves to [latex]x+4[/latex] that the function will become [latex]f(x)=2^{x+4}[/latex]. But that is NOT the case. Remember that the [latex]y[/latex]-intercept is moved to (4, 1) and if we substitute [latex]x=4[/latex] into the function [latex]f(x)=2^{x+4}[/latex] we get [latex]f(4)=2^(4+4)=256 \neq 1[/latex]!! The way to get a function value of 1 is for the transformed function to be [latex]f(x)=2^{x-4}[/latex]. Then [latex]f(4)=2^{4-4}=2^0=1[/latex]. So the function [latex]f(x)=2^x[/latex] transforms to [latex]f(x)=2^{x-4}[/latex] after being shifted 4 units to the right. The reason is that when we move the function 4 units to the right, the [latex]x[/latex]-value increases by 4 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 4 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 4.
| [latex]x[/latex] | [latex]x-4[/latex] | [latex]2^{x-4}[/latex] |
Figure 4. Shift the graph right 4 units. |
| 1 | -3 | [latex]\dfrac{1}{8}[/latex] | |
| 2 | -2 | [latex]\dfrac{1}{4}[/latex] | |
| 3 | -1 | [latex]\dfrac{1}{2}[/latex] | |
| 4 | 0 | 1 | |
| 5 | 1 | 2 | |
| 6 | 2 | 4 | |
| 7 | 3 | 8 | |
| Table 3. Shifting the graph right by 4 units transforms [latex]f(x)=2^x[/latex] into [latex]f(x)=2^{x-4}[/latex] | |||
On the other hand, if we shift the graph of the function [latex]f(x)=2^x[/latex] left by 7 units, all of the points on the graph decrease their [latex]x[/latex]-coordinates by 7, but their [latex]y[/latex]-coordinates remain the same. So any point [latex](x, y)[/latex] on the original graph moves to [latex](x-7, y)[/latex]. Consequently, to keep the same [latex]y[/latex]-values we need to increase the [latex]x[/latex]-value by 7 in the transformed function. The equation of the function after being shifted left 7 units is [latex]f(x)=2^{x+7}[/latex]. Table 4 shows the changes to specific values of this function, and the graph is shown in figure 5.
| [latex]x[/latex] | [latex]x+7[/latex] | [latex]2^{x+7}[/latex] |
Figure 5. Shifting the graph left 7 units. |
| -10 | -3 | [latex]\dfrac{1}{8}[/latex] | |
| -9 | -2 | [latex]\dfrac{1}{4}[/latex] | |
| -8 | -1 | [latex]\dfrac{1}{2}[/latex] | |
| -7 | 0 | 1 | |
| -6 | 1 | 2 | |
| -5 | 2 | 4 | |
| -4 | 3 | 8 | |
| Table 4. Shifting the graph left by 7 units transforms [latex]f(x)=2^x[/latex] into [latex]f(x)=2^{x+7}[/latex] | |||
horizontal shifts
A horizontal shift of the graph of [latex]f(x)=r^x[/latex] subtracts a constant, [latex]h[/latex], from the variable [latex]x[/latex] resulting in the transformed function
[latex]f(x)=r^{x-h} [/latex]
If [latex]h>0[/latex] the graph shifts to the right and if [latex]h<0[/latex] the graph shifts to the left.
Figure 6. Manipulation of [latex]f(x)=r^{x-h}[/latex]
TRY IT 3
Manipulate the values of [latex]r[/latex] and [latex]h[/latex] in figure 6 to answer the following questions:
1. What happens to the point (0, 1) on the graph of [latex]f(x)=2^x[/latex] when [latex]h=5[/latex]?
2. What happens to the horizontal asymptote [latex]y=0[/latex] of the function [latex]f(x)=2^x[/latex] when [latex]h=5[/latex]?
3. What happens to the point (0, 1) on the graph of [latex]f(x)=3^x[/latex] when [latex]h=-2[/latex]?
4. What happens to the horizontal asymptote [latex]y=0[/latex] of the function [latex]f(x)=3^x[/latex] when [latex]h=-2[/latex]?
5. What happens to every point on the graph of [latex]y=r^x[/latex] for [latex]r>0,\;r\neq1[/latex] when [latex]h=4[/latex]?
6. What happens to every point on the graph of [latex]y=r^x[/latex] for [latex]r>0,\;r\neq1[/latex] when [latex]h=-5[/latex]?
We can now combine vertical and horizontal transformations.
Horizontal and vertical shifts
The parent function [latex]f(x)=r^x[/latex], [latex]r>0, r\neq1[/latex], transforms to [latex]f(x)=r^{x-h}+k[/latex] when it is moved [latex]h[/latex] units horizontally and [latex]k[/latex] units vertically.
In figure 7, we can manipulate the values of [latex]r[/latex], [latex]h[/latex], and [latex]k[/latex] to determine what happens to the graph of [latex]f(x)=r^x[/latex].
Figure 7. Animation of [latex]f(x)=r^{x-h}+k[/latex]
Try It 4
Use the animation in figure 7 to determine the following:
1. What happens to the point (0, 1) on the function [latex]f(x)=2^x[/latex] when [latex]h=4[/latex] and [latex]k=5[/latex]?
2. What happens to the point (1, 2) on the function [latex]f(x)=2^x[/latex] when [latex]h=4[/latex] and [latex]k=5[/latex]?
3. What happens to any point [latex](x, y)[/latex] on the function [latex]f(x)=2^x[/latex] when [latex]h=4[/latex] and [latex]k=5[/latex]?
4. What happens to the point (0, 1) on the function [latex]f(x)=3^x[/latex] when [latex]h=-3[/latex] and [latex]k=-2[/latex]?
5. What happens to the point (1, 2) on the function [latex]f(x)=2^x[/latex] when [latex]h=-3[/latex] and [latex]k=-2[/latex]?
6. What happens to any point [latex](x, y)[/latex] on the function [latex]f(x)=2^x[/latex] when [latex]h=-3[/latex] and [latex]k=-2[/latex]?
7. What happens to any point [latex](x, y)[/latex] on the function [latex]f(x)=2^x[/latex] when it is shifted [latex]h[/latex] units horizontally and [latex]k[/latex] units vertically?
Example 1
Explain the transformations that need to happen to the function [latex]f(x)=3^x[/latex] to get the function [latex]f(x)=3^{x+5}-7[/latex].
Solution
First we identify [latex]h[/latex] and [latex]k[/latex] in the function [latex]f(x)=3^{x-h}+k[/latex]: [latex]h=-5[/latex] since [latex]x+5=x-(-5)[/latex]. [latex]k=-7[/latex].
The function [latex]f(x)=3^x[/latex] is shifted horizontal left by 5 units and vertically down by 7 units.
Try It 5
Explain the transformations that need to happen to the function [latex]f(x)=4^x[/latex] to get the function [latex]f(x)=4^{x-7}-2[/latex].
Example 2
Determine the transformed function after [latex]g(x)=2^x[/latex] is shifted right 4 units and up 3 units.
Solution
Shifting right 4 units means [latex]h=4[/latex]. Shifting up 3 units means [latex]k=3[/latex].
So the transformed function is [latex]g(x)=2^{x-4}+3[/latex].
Try It 6
Determine the transformed function after [latex]g(x)=7^x[/latex] is shifted left 6 units and down 2 units.
