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

Learning Outcomes

For the exponential function $f(x)=r^x$ ,

Perform vertical and horizontal shifts
Determine the equation of a transformed function
Determine the transformations of the exponential function $f(x)=r^{(x-h)}+k$

Vertical Shifts

If we shift the graph of the exponential function $f(x)=2^x$ up 5 units, all of the points on the graph increase their $y$ -coordinates by 5, but their $x$ -coordinates remain the same. Therefore, the equation of the function $f(x)=2^x$ after it has been shifted up 5 units transforms to $f(x)=2^x+5$ . Table 1 shows the changes to specific values of this function, which are graphed in figure 1.

$x$	$2^x$	$2^x+5$	Figure 1. Shifting the graph of $f(x)=2^x$ up 5 units.
-3	$\dfrac{1}{8}$	$\dfrac{41}{8}$
-2	$\dfrac{1}{4}$	$\dfrac{21}{4}$
-1	$\dfrac{1}{2}$	$\dfrac{11}{2}$
0	1	6
1	2	7
2	4	9
3	8	13
Table 1. $f(x)=2^x$ is transformed to $f(x)=2^x+5$

If we shift the graph of the function $f(x)=2^x$ down 3 units, all of the points on the graph decrease their $y$ -coordinates by 3, but their $x$ -coordinates remain the same. Therefore, the equation of the function $f(x)=2^x$ after it has been shifted down 3 units transforms to $f(x)=2^x-3$ . Table 2 shows the changes to specific values of this function, which are graphed in figure 2.

$x$	$2^x$	$2^x-3$	Figure 2. Shifting the graph of $f(x)=2^x$ down 3 units.
-3	$\dfrac{1}{8}$	$-\dfrac{23}{8}$
-2	$\dfrac{1}{4}$	$-\dfrac{11}{4}$
-1	$\dfrac{1}{2}$	$-\dfrac{5}{2}$
0	1	-2
1	2	-1
2	4	1
3	8	5
Table 2. $f(x)=2^x$ is transformed to $f(x)=2^x-3$

These vertical shifts can be applied to any exponential function. Change the values of $r$ and $k$ in figure 3 by moving the green and red dots and note the changes that happen.

Figure 3. Manipulation of $f(x)=r^x+k$

Vertical shifts

A vertical shift of the graph of $f(x)=r^x$ with $r>0,\;r\neq1$ , adds a constant, $k$ , resulting in the transformed function:

$f(x)=r^x + k$

If $k>0$ , the graph shifts upwards by $k$ units and if $k<0$ the graph shifts downwards by $k$ units.

TRY IT 1

Manipulate the values of $r$ and $k$ in figure 3 to answer the following questions:

1. What happens to the point (0, 1) on the graph of $f(x)=2^x$ when $k=5$ ?

2. What happens to the horizontal asymptote $y=0$ of the function $f(x)=2^x$ when $k=5$ ?

3. What happens to the point (0, 1) on the graph of $f(x)=3^x$ when $k=-2$ ?

4. What happens to the horizontal asymptote $y=0$ of the function $f(x)=3^x$ when $k=-2$ ?

5. What happens to every point on the graph of $y=r^x$ for $r>0,\;r\neq1$ when $k=4$ ?

6. What happens to the horizontal asymptote of the graph of $y=r^x$ for $r>0,\;r\neq1$ when $k=-3$ ?

Show Answer

(0, 1) moves up 5 units to (0, 6).
The asymptote $y=0$ moves up 5 units to the asymptote $y=5$ .
(0, 1) moves down 2 units to (0, –1).
The asymptote $y=0$ moves down 2 units to the asymptote $y=-2$ .
The $y$ -value of every point increases by 4 while the $x$ -value stays the same.
The asymptote $y=0$ moves down 3 units to the asymptote $y=-3$ .

Try It 2

Determine the exponential function that comes from transforming the parent function $f(x)=2^x$ :

1. 3 units up

2. 7 units down

Show Answer

$f(x)=2^x+3$
$f(x)=2^x-7$

Horizontal Shifts

If we shift the graph of the function $f(x)=2^x$ right 4 units, all of the points on the graph increase their $x$ -coordinates by 4, but their $y$ -coordinates remain the same. The $y$ -intercept (0, 1) in the original graph moves to (4, 1) (figure 4). Any point $(x, y)$ on the original graph is moved to $(x+4, y)$ .

But what happens to the original function $f(x)=2^x$ ? An automatic assumption may be that since $x$ moves to $x+4$ that the function will become $f(x)=2^{x+4}$ . But that is NOT the case. Remember that the $y$ -intercept is moved to (4, 1) and if we substitute $x=4$ into the function $f(x)=2^{x+4}$ we get $f(4)=2^(4+4)=256 \neq 1$ !! The way to get a function value of 1 is for the transformed function to be $f(x)=2^{x-4}$ . Then $f(4)=2^{4-4}=2^0=1$ . So the function $f(x)=2^x$ transforms to $f(x)=2^{x-4}$ after being shifted 4 units to the right. The reason is that when we move the function 4 units to the right, the $x$ -value increases by 4 and to keep the corresponding $y$ -coordinate the same in the transformed function, the $x$ -coordinate of the transformed function needs to subtract 4 to get back to the original $x$ that is associated with the original $y$ -value. Table 3 shows the changes to specific values of this function, and the graph is shown in figure 4.

$x$	$x-4$	$2^{x-4}$	Figure 4. Shift the graph right 4 units.
1	-3	$\dfrac{1}{8}$
2	-2	$\dfrac{1}{4}$
3	-1	$\dfrac{1}{2}$
4	0	1
5	1	2
6	2	4
7	3	8
Table 3. Shifting the graph right by 4 units transforms $f(x)=2^x$ into $f(x)=2^{x-4}$

On the other hand, if we shift the graph of the function $f(x)=2^x$ left by 7 units, all of the points on the graph decrease their $x$ -coordinates by 7, but their $y$ -coordinates remain the same. So any point $(x, y)$ on the original graph moves to $(x-7, y)$ . Consequently, to keep the same $y$ -values we need to increase the $x$ -value by 7 in the transformed function. The equation of the function after being shifted left 7 units is $f(x)=2^{x+7}$ . Table 4 shows the changes to specific values of this function, and the graph is shown in figure 5.

$x$	$x+7$	$2^{x+7}$	Figure 5. Shifting the graph left 7 units.
-10	-3	$\dfrac{1}{8}$
-9	-2	$\dfrac{1}{4}$
-8	-1	$\dfrac{1}{2}$
-7	0	1
-6	1	2
-5	2	4
-4	3	8
Table 4. Shifting the graph left by 7 units transforms $f(x)=2^x$ into $f(x)=2^{x+7}$

horizontal shifts

A horizontal shift of the graph of $f(x)=r^x$ subtracts a constant, $h$ , from the variable $x$ resulting in the transformed function

$f(x)=r^{x-h}$

If $h>0$ the graph shifts to the right and if $h<0$ the graph shifts to the left.

Figure 6. Manipulation of $f(x)=r^{x-h}$

TRY IT 3

Manipulate the values of $r$ and $h$ in figure 6 to answer the following questions:

1. What happens to the point (0, 1) on the graph of $f(x)=2^x$ when $h=5$ ?

2. What happens to the horizontal asymptote $y=0$ of the function $f(x)=2^x$ when $h=5$ ?

3. What happens to the point (0, 1) on the graph of $f(x)=3^x$ when $h=-2$ ?

4. What happens to the horizontal asymptote $y=0$ of the function $f(x)=3^x$ when $h=-2$ ?

5. What happens to every point on the graph of $y=r^x$ for $r>0,\;r\neq1$ when $h=4$ ?

6. What happens to every point on the graph of $y=r^x$ for $r>0,\;r\neq1$ when $h=-5$ ?

Show Answer

(0, 1) moves right 5 units to (5, 1).
The asymptote $y=0$ does not change.
(0, 1) moves left 2 units to (–2, 1).
The asymptote $y=0$ does not change.
The $x$ -value of every point increases by 4 while the $y$ -value stays the same.
The $x$ -value of every point decreases by 5 while the $y$ -value stays the same.

We can now combine vertical and horizontal transformations.

Horizontal and vertical shifts

The parent function $f(x)=r^x$ , $r>0, r\neq1$ , transforms to $f(x)=r^{x-h}+k$ when it is moved $h$ units horizontally and $k$ units vertically.

In figure 7, we can manipulate the values of $r$ , $h$ , and $k$ to determine what happens to the graph of $f(x)=r^x$ .

Figure 7. Animation of $f(x)=r^{x-h}+k$

Try It 4

Use the animation in figure 7 to determine the following:

1. What happens to the point (0, 1) on the function $f(x)=2^x$ when $h=4$ and $k=5$ ?

2. What happens to the point (1, 2) on the function $f(x)=2^x$ when $h=4$ and $k=5$ ?

3. What happens to any point $(x, y)$ on the function $f(x)=2^x$ when $h=4$ and $k=5$ ?

4. What happens to the point (0, 1) on the function $f(x)=3^x$ when $h=-3$ and $k=-2$ ?

5. What happens to the point (1, 2) on the function $f(x)=2^x$ when $h=-3$ and $k=-2$ ?

6. What happens to any point $(x, y)$ on the function $f(x)=2^x$ when $h=-3$ and $k=-2$ ?

7. What happens to any point $(x, y)$ on the function $f(x)=2^x$ when it is shifted $h$ units horizontally and $k$ units vertically?

Show Answer

(0, 1) moves to (4, 6)
(1, 2) moves to (5, 7)
$(x, y)$ moves to $(x+4, y+5)$
(0, 1) moves to (–3, –1)
(1, 2) moves to (–2, 0)
$(x, y)$ moves to $(x-3, y-2)$
$(x, y)$ moves to $(x+h, y+k)$

Example 1

Explain the transformations that need to happen to the function $f(x)=3^x$ to get the function $f(x)=3^{x+5}-7$ .

Solution

First we identify $h$ and $k$ in the function $f(x)=3^{x-h}+k$ : $h=-5$ since $x+5=x-(-5)$ . $k=-7$ .

The function $f(x)=3^x$ 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 $f(x)=4^x$ to get the function $f(x)=4^{x-7}-2$ .

Show Answer

$f(x)=4^x$ is shifted right 7 units and down 2 units.

Example 2

Determine the transformed function after $g(x)=2^x$ is shifted right 4 units and up 3 units.

Solution

Shifting right 4 units means $h=4$ . Shifting up 3 units means $k=3$ .

So the transformed function is $g(x)=2^{x-4}+3$ .

Try It 6

Determine the transformed function after $g(x)=7^x$ is shifted left 6 units and down 2 units.

Show Answer

$g(x)=7^{x+6}-2$

Chapter 5: Exponential Functions