Learning Outcomes
For the exponential function [latex]f(x)=r^x[/latex],
- Perform vertical compressions and stretches
- Perform reflections across the [latex]x[/latex]-axis
- Perform reflections across the [latex]y[/latex]-axis
- Determine the function given specific transformations
- Determine the transformations of the exponential function [latex]f(x)=ar^{(x-h)}+k[/latex]
Stretching Up and Compressing Down
If we vertically stretch the graph of the function [latex]f(x)=2^x[/latex] by a factor of two, 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 by a factor of 2 is [latex]f(x)=2\left(2^x\right)[/latex]. The reason for multiplying [latex]2^x[/latex] by 2 is that each [latex]y[/latex]-coordinate is doubled, and since [latex]y=2^x[/latex], [latex]2^x[/latex] is doubled. Table 1 shows this change and the graph is shown in figure 1. Any point [latex](x, y)[/latex] on the graph of [latex]f(x)=2^x[/latex] moves to [latex](x, 2y)[/latex] when the graph is stretched by a factor of 2.
| [latex]x[/latex] | [latex]2^x[/latex] | [latex]2\left(2^x\right)[/latex] |
Figure 1. Stretching the graph vertically by a factor of 2 |
| [latex]-3[/latex] | [latex]\dfrac{1}{8}[/latex] | [latex]\dfrac{1}{4}[/latex] | |
| [latex]-2[/latex] | [latex]\dfrac{1}{4}[/latex] | [latex]\dfrac{1}{2}[/latex] | |
| [latex]-1[/latex] | [latex]\dfrac{1}{2}[/latex] | [latex]1[/latex] | |
| [latex]0[/latex] | [latex]1[/latex] | [latex]2[/latex] | |
| [latex]1[/latex] | [latex]2[/latex] | [latex]4[/latex] | |
| [latex]2[/latex] | [latex]4[/latex] | [latex]8[/latex] | |
| [latex]3[/latex] | [latex]8[/latex] | [latex]16[/latex] | |
| Table 1. Stretching the graph vertically by a factor of 2 transforms [latex]f(x)=2^x[/latex] into [latex]f(x)=2\left(2^x\right)[/latex] | |||
On the other hand, if we vertically compress the graph of the function [latex]f(x)=2^x[/latex] by half, 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]\frac{1}{2}[/latex]. The equation of the function after being compressed by half is [latex]f(x)=\dfrac{1}{2}\left(2^x\right)[/latex]. The reason for multiplying [latex]2^x[/latex] by [latex]\frac{1}{2}[/latex] is that each [latex]y[/latex]-coordinate becomes half of the original value when it is divided by 2. Table 2 shows this change and the graph is shown in figure 2. Any point [latex](x, y)[/latex] on the graph of [latex]f(x)=2^x[/latex] moves to [latex]\left(x, \frac{1}{2}y\right)[/latex] when the graph is compressed.
| [latex]x[/latex] | [latex]2^x[/latex] | [latex]\dfrac{1}{2}\left(2^x\right)[/latex] |
Figure 2. Compressing the graph. |
| [latex]-3[/latex] | [latex]\dfrac{1}{8}[/latex] | [latex]\dfrac{1}{16}[/latex] | |
| [latex]-2[/latex] | [latex]\dfrac{1}{4}[/latex] | [latex]\dfrac{1}{8}[/latex] | |
| [latex]-1[/latex] | [latex]\dfrac{1}{2}[/latex] | [latex]\dfrac{1}{4}[/latex] | |
| [latex]0[/latex] | [latex]1[/latex] | [latex]\dfrac{1}{2}[/latex] | |
| [latex]1[/latex] | [latex]2[/latex] | [latex]1[/latex] | |
| [latex]2[/latex] | [latex]4[/latex] | [latex]2[/latex] | |
| [latex]3[/latex] | [latex]8[/latex] | [latex]4[/latex] | |
| Table 2. Compressing the graph vertically by half transforms [latex]f(x)=2^x[/latex] into [latex]f(x)=\dfrac{1}{2}\left(2^x\right)[/latex] | |||
Stretching and compression can, of course, be applied to any exponential function [latex]f(x)=r^x[/latex], with [latex]r>0[/latex] and [latex]r\neq1[/latex].
vertical stretching and compressing
A stretch or compression of the graph of [latex]f(x)=r^x[/latex] can be represented by multiplying the function by a constant [latex]a>0[/latex].
[latex]f(x)=a\cdot r^x[/latex]
The magnitude of [latex]a[/latex] indicates the stretch or compression of the graph. If [latex]a>1[/latex], the graph is stretched upwards by a factor of [latex]a.[/latex] If [latex]0<a<1[/latex], the graph is compressed down to [latex]a[/latex] times its original height.
In addition, [latex]f(0)=a[/latex], so the initial value of the function is transformed from [latex]1[/latex] to [latex]a[/latex].
We can change the value of [latex]a[/latex] in figure 3 by moving the purple dot to see what happens to the graph of [latex]f(x)=r^x[/latex]. Moving the red dot changes the value of [latex]r[/latex].
Figure 3. Manipulation of [latex]f(x)=ar^x[/latex]
Notice that the value of [latex]a[/latex] becomes the initial value of [latex]f(x)[/latex]. That is [latex]f(0)=a[/latex].
Example 1
Manipulate the animation 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 it is transformed to [latex]f(x)=4\left(2^x\right)[/latex]?
2. What happens to the point (0, 1) on the graph of [latex]f(x)=3^x[/latex] when it is transformed to [latex]f(x)=4\left(3^x\right)[/latex]?
3. What happens to the point (0, 1) on the graph of [latex]f(x)=\left(\dfrac{1}{2}\right)^x[/latex] when it is transformed to [latex]f(x)=4\left(\dfrac{1}{2}\right)^x[/latex]?
4. What happens to the point (0, 1) on the graph of [latex]f(x)=r^x[/latex] when it is transformed to [latex]f(x)=4r^x[/latex]?
5. What happens to the point (0, 1) on the graph of [latex]f(x)=r^x[/latex] when it is transformed to [latex]f(x)=ar^x[/latex]?
Solution
1. With [latex]r=2[/latex] and [latex]a=4[/latex], the point (0, 1) moves to (0, 4).
2. With [latex]r=3[/latex] and [latex]a=4[/latex], the point (0, 1) moves to (0, 4).
3. With [latex]r=\dfrac{1}{2}[/latex] and [latex]a=4[/latex], the point (0, 1) moves to (0, 4).
4. With [latex]a=4[/latex], the point (0, 1) moves to [latex](0, 4)[/latex].
5. The point (0, 1) moves to [latex](0, a)[/latex].
Try It 1
Manipulate the animation in figure 3 to answer the following questions:
1. What happens to the point (1, 2) on the graph of [latex]f(x)=2^x[/latex] when it is transformed to [latex]f(x)=4\left(2^x\right)[/latex]?
2. What happens to the point (1, 3) on the graph of [latex]f(x)=3^x[/latex] when it is transformed to [latex]f(x)=4\left(3^x\right)[/latex]?
3. What happens to the point [latex]\left(1, \dfrac{1}{2}\right)[/latex] on the graph of [latex]f(x)=\left(\dfrac{1}{2}\right)^x[/latex] when it is transformed to [latex]f(x)=4\left(\dfrac{1}{2}\right)^x[/latex]?
4. What happens to the point [latex](1, r)[/latex] on the graph of [latex]f(x)=r^x[/latex] when it is transformed to [latex]f(x)=4\left(r^x\right)[/latex]?
5. What happens to the point [latex](1, r)[/latex] on the graph of [latex]f(x)=r^x[/latex] when it is transformed to [latex]f(x)=a\left(r^x\right)[/latex]?
6. What happens to the asymptote [latex]y=0[/latex] when [latex]f(x)[/latex] is stretched or compressed?
Example 2
Determine what happens to the point (2, 9) when the function [latex]f(x)=3^x[/latex] is:
1. stretched by a factor of 5
2. compressed to [latex]\dfrac{1}{3}[/latex] its height
Solution
1. When a graph is stretched the [latex]x[/latex]-coordinate stays the same, while the [latex]y[/latex]-coordinate is multiplied by the stretch factor, 5. Hence, (2, 9) becomes (2, 45).
2. When a graph is compressed the [latex]x[/latex]-coordinate stays the same, while the [latex]y[/latex]-coordinate is multiplied by[latex]\dfrac{1}{3}[/latex]. Hence, (2, 9) becomes (2, 3).
Try It 2
Determine what happens to the point (–2, 9) when the function [latex]f(x)=\left(\dfrac{1}{3}\right)^x[/latex] is:
1. stretched by a factor of 4
2. compressed to [latex]\dfrac{1}{6}[/latex] its height
Example 3
The function [latex]f(x)=4^x[/latex] is transformed. Determine the equation of the transformed function when:
1. [latex]f(x)[/latex] is stretched by a factor of 2.
2. [latex]f(x)[/latex] is stretched by a factor of 5.
3. [latex]f(x)[/latex] is compressed to [latex]\dfrac{1}{3}[/latex] its height.
4. [latex]f(x)[/latex] is compressed to [latex]\dfrac{1}{4}[/latex] its height.
Solution
1. Since [latex]a=2[/latex] the transformed function is [latex]f(x)=2\left(4^x\right)[/latex].
2. Since [latex]a=5[/latex] the transformed function is [latex]f(x)=5\left(4^x\right)[/latex].
3. Since [latex]a=\dfrac{1}{3}[/latex] the transformed function is [latex]f(x)=\dfrac{1}{3}\left(4^x\right)[/latex].
4. Since [latex]a=\dfrac{1}{4}[/latex] the transformed function is [latex]f(x)=\dfrac{1}{4}\left(4^x\right)[/latex].
Try It 3
Determine the transformation made to the parent function [latex]g(x)=5^x[/latex].
1. [latex]g(x)=4\left(5^x\right)[/latex]
2. [latex]g(x)=\dfrac{1}{4}\left(5^x\right)[/latex]
3. [latex]g(x)=7\left(5^x\right)[/latex]
4. [latex]g(x)=\dfrac{1}{3}\left(5^x\right)[/latex]
Reflection across the [latex]x[/latex]-axis
When the graph of the function [latex]f(x)=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, while the [latex]x[/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=2^x[/latex] is reflected across the [latex]x[/latex]-axis is [latex]f(x)=-2^x[/latex]. The graph changes from increasing upwards to decreasing downwards. Table 3 shows the effect of such a reflection on the function values and the graph is shown in figure 4. While the [latex]x[/latex]-coordinate stays the same, the [latex]y[/latex]-coordinate becomes [latex]-y[/latex].
| [latex]x[/latex] | [latex]y=2^x[/latex] | [latex]-y=2^x[/latex] or [latex]y=-2^x[/latex] |
Figure 4. Reflecting the graph of [latex]f(x)=2^x[/latex] across the [latex]x[/latex]-axis. |
| [latex]-3[/latex] | [latex]\dfrac{1}{8}[/latex] | [latex]-\dfrac{1}{8}[/latex] | |
| [latex]-2[/latex] | [latex]\dfrac{1}{4}[/latex] | [latex]-\dfrac{1}{4}[/latex] | |
| [latex]-1[/latex] | [latex]\dfrac{1}{2}[/latex] | [latex]-\dfrac{1}{2}[/latex] | |
| [latex]0[/latex] | [latex]1[/latex] | [latex]-1[/latex] | |
| [latex]1[/latex] | [latex]2[/latex] | [latex]-2[/latex] | |
| [latex]2[/latex] | [latex]4[/latex] | [latex]-4[/latex] | |
| [latex]3[/latex] | [latex]8[/latex] | [latex]-8[/latex] | |
| Table 3. Reflecting the graph of [latex]f(x)=2^x[/latex] across the [latex]x[/latex]-axis transforms [latex]f(x)=2^x[/latex] into [latex]f(x)=-2^x[/latex] | |||
Example 4
The function [latex]f(x)=4^x[/latex] is reflected across the [latex]x[/latex]-axis. What happens to the point (2, 16) that lies on the parent function after the transformation?
Solution
When a function is reflected across the [latex]x[/latex]-axis, the [latex]x[/latex]-coordinate stays the same while the [latex]y[/latex]-coordinate changes sign. So (2, 16) is transformed to (2, –16).
Try It 4
The function [latex]f(x)=\left(\dfrac{2}{3}\right)^x[/latex] is reflected across the [latex]x[/latex]-axis. What happens to the point [latex]\left(-1, \dfrac{3}{2}\right)[/latex] that lies on the parent function after the transformation?
Reflection across the [latex]y[/latex]-axis
When the graph of the function [latex]f(x)=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 while the [latex]y[/latex]-coordinates remain the same. The equation of the function after [latex]f(x)=2^x[/latex] is reflected across the [latex]y[/latex]-axis is [latex]f(x)=2^{-x}[/latex]. The graph changes from increasing from the left to decreasing from the left. Table 4 shows the effect of such a reflection on the functions values and the graph is shown in figure 5. While the [latex]y[/latex]-coordinate stays the same, the [latex]x[/latex]-coordinate becomes [latex]-x[/latex].
| [latex]x[/latex] | [latex]2^x[/latex] | [latex]2^{-x}[/latex] |
Figure 5. Reflecting the graph of [latex]f(x)=2^x[/latex] across the [latex]y[/latex]-axis. |
| [latex]-3[/latex] | [latex]\dfrac{1}{8}[/latex] | [latex]8[/latex] | |
| [latex]-2[/latex] | [latex]\dfrac{1}{4}[/latex] | [latex]4[/latex] | |
| [latex]-1[/latex] | [latex]\dfrac{1}{2}[/latex] | [latex]2[/latex] | |
| [latex]0[/latex] | [latex]1[/latex] | [latex]1[/latex] | |
| [latex]1[/latex] | [latex]2[/latex] | [latex]\dfrac{1}{2}[/latex] | |
| [latex]2[/latex] | [latex]4[/latex] | [latex]\dfrac{1}{4}[/latex] | |
| [latex]3[/latex] | [latex]8[/latex] | [latex]\dfrac{1}{8}[/latex] | |
| Table 4. Reflecting the graph of [latex]f(x)=2^x[/latex] across the [latex]y[/latex]-axis transforms [latex]f(x)=2^x[/latex] into [latex]f(x)=2^{-x}[/latex] | |||
Example 5
The function [latex]f(x)=4^x[/latex] is reflected across the [latex]y[/latex]-axis. What happens to the point (2, 16) that lies on the parent function, after the transformation?
Solution
When a function is reflected across the [latex]y[/latex]-axis, the [latex]y[/latex]-coordinate stays the same while the [latex]x[/latex]-coordinate changes sign. So (2, 16) is transformed to (–2, 16).
Try It 5
The function [latex]f(x)=7^x[/latex] is reflected across the [latex]y[/latex]-axis. What happens to the point [latex]\left(-1, \dfrac{1}{7}\right)[/latex] that lies on the parent function, after the transformation?
Determine the transformations of the exponential function [latex]f(x)=ar^{x-h}+k[/latex]
All of the transformations we have applied to the parent function [latex]f(x)=r^x[/latex] can be combined. The result is a general exponential function [latex]f(x)=ar^{x-h}+k[/latex]. Given any function in the form [latex]f(x)=ar^{x-h}+k[/latex], we can determine from the values of [latex]a, h,[/latex] and [latex]k[/latex] the transformations that were performed on the parent function [latex]f(x)=r^x[/latex]. Likewise, if we know the transformations, we can write the equation of the transformed function.
Example 6
What transformations were performed on the parent function [latex]f(x)=2^x[/latex] to get the function [latex]f(x)=-3\left(2^{x+4}\right)-6[/latex]?
Solution
First we identify [latex]a, h,[/latex] and [latex]k[/latex]:
[latex]a=-3,\;h=-4,\;k=-6[/latex]
A negative value of [latex]a[/latex] means the function [latex]f(x)=2^x[/latex] has been reflected across the [latex]x[/latex]-axis. [latex]a=-3[/latex] means it has also been stretched by a factor of 3.
[latex]h=-4[/latex] means the parent function has been shifted horizontally 4 units left, while [latex]k=-6[/latex] means it has been shifted vertically down by 6 units.
Try It 6
What transformations were performed on the parent function [latex]f(x)=2^x[/latex] to get the function [latex]f(x)=\dfrac{1}{4}\left(2^{x-6}\right)+3[/latex]?
Example 7
If the parent function [latex]g(x)=3^x[/latex] is stretched by a factor of 7, reflected across the [latex]y[/latex]-axis, and shifted vertically 3 units down, what is the equation of the transformed function?
Solution
Stretched by a factor of 7 means [latex]a=7[/latex].
Reflected across the [latex]y[/latex]-axis means [latex]x[/latex] becomes [latex]-x[/latex].
Shifted vertically down by 3 units means [latex]k=-3[/latex].
So, the transformed function is [latex]f(x)=7\left(3^{-x}\right)-3[/latex].
Try It 7
If the parent function [latex]g(x)=5^x[/latex] is compressed to [latex]\dfrac{1}{4}[/latex] its height, reflected across the [latex]x[/latex]-axis, shifted horizontally left by 9 units, and shifted vertically up by 4 units, what is the equation of the transformed function?
Move the dots in figure 6, to see what happens to the graph of the parent function [latex]f(x)=r^x[/latex] when it is transformed by changing the values of the constants [latex]a, h, k[/latex] and by making [latex]x[/latex] negative to reflect across the [latex]y-axis[/latex].
Figure 6. Transformation of [latex]f(x)=r^x[/latex]
