5.2.1: Transformations of the Exponential Function––Vertical and Horizontal Shifts

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]
y=2^x+5

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]
y=2^x-3

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]  

shifting right

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]  

shifting left

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.