5.2.2: Transformations of the Exponential Function––Stretches, Compressions and Reflections

Learning Outcomes

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

Perform vertical compressions and stretches
Perform reflections across the $x$ -axis
Perform reflections across the $y$ -axis
Determine the function given specific transformations
Determine the transformations of the exponential function $f(x)=ar^{(x-h)}+k$

Stretching Up and Compressing Down

If we vertically stretch the graph of the function $f(x)=2^x$ by a factor of two, all of the $y$ -coordinates of the points on the graph are multiplied by 2, but their $x$ -coordinates remain the same. The equation of the function after the graph is stretched by a factor of 2 is $f(x)=2\left(2^x\right)$ . The reason for multiplying $2^x$ by 2 is that each $y$ -coordinate is doubled, and since $y=2^x$ , $2^x$ is doubled. Table 1 shows this change and the graph is shown in figure 1. Any point $(x, y)$ on the graph of $f(x)=2^x$ moves to $(x, 2y)$ when the graph is stretched by a factor of 2.

$x$	$2^x$	$2\left(2^x\right)$	Figure 1. Stretching the graph vertically by a factor of 2
$-3$	$\dfrac{1}{8}$	$\dfrac{1}{4}$
$-2$	$\dfrac{1}{4}$	$\dfrac{1}{2}$
$-1$	$\dfrac{1}{2}$	$1$
$0$	$1$	$2$
$1$	$2$	$4$
$2$	$4$	$8$
$3$	$8$	$16$
Table 1. Stretching the graph vertically by a factor of 2 transforms $f(x)=2^x$ into $f(x)=2\left(2^x\right)$

On the other hand, if we vertically compress the graph of the function $f(x)=2^x$ by half, all of the $y$ -coordinates of the points on the graph are halved, but their $x$ -coordinates remain the same. This means the $y$ -coordinates are divided by 2, or multiplied by $\frac{1}{2}$ . The equation of the function after being compressed by half is $f(x)=\dfrac{1}{2}\left(2^x\right)$ . The reason for multiplying $2^x$ by $\frac{1}{2}$ is that each $y$ -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 $(x, y)$ on the graph of $f(x)=2^x$ moves to $\left(x, \frac{1}{2}y\right)$ when the graph is compressed.

$x$	$2^x$	$\dfrac{1}{2}\left(2^x\right)$	Figure 2. Compressing the graph.
$-3$	$\dfrac{1}{8}$	$\dfrac{1}{16}$
$-2$	$\dfrac{1}{4}$	$\dfrac{1}{8}$
$-1$	$\dfrac{1}{2}$	$\dfrac{1}{4}$
$0$	$1$	$\dfrac{1}{2}$
$1$	$2$	$1$
$2$	$4$	$2$
$3$	$8$	$4$
Table 2. Compressing the graph vertically by half transforms $f(x)=2^x$ into $f(x)=\dfrac{1}{2}\left(2^x\right)$

Stretching and compression can, of course, be applied to any exponential function $f(x)=r^x$ , with $r>0$ and $r\neq1$ .

vertical stretching and compressing

A stretch or compression of the graph of $f(x)=r^x$ can be represented by multiplying the function by a constant $a>0$ .

$f(x)=a\cdot r^x$

The magnitude of $a$ indicates the stretch or compression of the graph. If $a>1$ , the graph is stretched upwards by a factor of $a.$ If [latex]0initial value of the function is transformed from $1$ to $a$ .

We can change the value of $a$ in figure 3 by moving the purple dot to see what happens to the graph of $f(x)=r^x$ . Moving the red dot changes the value of $r$ .

Figure 3. Manipulation of $f(x)=ar^x$

Notice that the value of $a$ becomes the initial value of $f(x)$ . That is $f(0)=a$ .

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 $f(x)=2^x$ when it is transformed to $f(x)=4\left(2^x\right)$ ?

2. What happens to the point (0, 1) on the graph of $f(x)=3^x$ when it is transformed to $f(x)=4\left(3^x\right)$ ?

3. What happens to the point (0, 1) on the graph of $f(x)=\left(\dfrac{1}{2}\right)^x$ when it is transformed to $f(x)=4\left(\dfrac{1}{2}\right)^x$ ?

4. What happens to the point (0, 1) on the graph of $f(x)=r^x$ when it is transformed to $f(x)=4r^x$ ?

5. What happens to the point (0, 1) on the graph of $f(x)=r^x$ when it is transformed to $f(x)=ar^x$ ?

Solution

1. With $r=2$ and $a=4$ , the point (0, 1) moves to (0, 4).

2. With $r=3$ and $a=4$ , the point (0, 1) moves to (0, 4).

3. With $r=\dfrac{1}{2}$ and $a=4$ , the point (0, 1) moves to (0, 4).

4. With $a=4$ , the point (0, 1) moves to $(0, 4)$ .

5. The point (0, 1) moves to $(0, a)$ .

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 $f(x)=2^x$ when it is transformed to $f(x)=4\left(2^x\right)$ ?

2. What happens to the point (1, 3) on the graph of $f(x)=3^x$ when it is transformed to $f(x)=4\left(3^x\right)$ ?

3. What happens to the point $\left(1, \dfrac{1}{2}\right)$ on the graph of $f(x)=\left(\dfrac{1}{2}\right)^x$ when it is transformed to $f(x)=4\left(\dfrac{1}{2}\right)^x$ ?

4. What happens to the point $(1, r)$ on the graph of $f(x)=r^x$ when it is transformed to $f(x)=4\left(r^x\right)$ ?

5. What happens to the point $(1, r)$ on the graph of $f(x)=r^x$ when it is transformed to $f(x)=a\left(r^x\right)$ ?

6. What happens to the asymptote $y=0$ when $f(x)$ is stretched or compressed?

Show Answer

1. With $r=2$ and $a=4$ , the point (1, 2) moves to (1, 8).

2. With $r=3$ and $a=4$ , the point (1, 3) moves to (1, 12).

3. With $r=\dfrac{1}{2}$ and $a=4$ , the point $\left(1, \dfrac{1}{2}\right)$ moves to (1, 2).

4. With $a=4$ , the point $(1, r)$ moves to $(1, 4r)$ .

5. The point $(1, r)$ moves to $(1, ar)$ .

6. The asymptote stays the same.

Example 2

Determine what happens to the point (2, 9) when the function $f(x)=3^x$ is:

1. stretched by a factor of 5

2. compressed to $\dfrac{1}{3}$ its height

Solution

1. When a graph is stretched the $x$ -coordinate stays the same, while the $y$ -coordinate is multiplied by the stretch factor, 5. Hence, (2, 9) becomes (2, 45).

2. When a graph is compressed the $x$ -coordinate stays the same, while the $y$ -coordinate is multiplied by $\dfrac{1}{3}$ . Hence, (2, 9) becomes (2, 3).

Try It 2

Determine what happens to the point (–2, 9) when the function $f(x)=\left(\dfrac{1}{3}\right)^x$ is:

1. stretched by a factor of 4

2. compressed to $\dfrac{1}{6}$ its height

Show Answer

1. (–2, 9) becomes (–2, 36).

2. (–2, 9) becomes $\left(–2, \dfrac{3}{2}\right)$ .

Example 3

The function $f(x)=4^x$ is transformed. Determine the equation of the transformed function when:

1. $f(x)$ is stretched by a factor of 2.

2. $f(x)$ is stretched by a factor of 5.

3. $f(x)$ is compressed to $\dfrac{1}{3}$ its height.

4. $f(x)$ is compressed to $\dfrac{1}{4}$ its height.

Solution

1. Since $a=2$ the transformed function is $f(x)=2\left(4^x\right)$ .

2. Since $a=5$ the transformed function is $f(x)=5\left(4^x\right)$ .

3. Since $a=\dfrac{1}{3}$ the transformed function is $f(x)=\dfrac{1}{3}\left(4^x\right)$ .

4. Since $a=\dfrac{1}{4}$ the transformed function is $f(x)=\dfrac{1}{4}\left(4^x\right)$ .

Try It 3

Determine the transformation made to the parent function $g(x)=5^x$ .

1. $g(x)=4\left(5^x\right)$

2. $g(x)=\dfrac{1}{4}\left(5^x\right)$

3. $g(x)=7\left(5^x\right)$

4. $g(x)=\dfrac{1}{3}\left(5^x\right)$

Show Answer

$g(x)=5^x$ is stretched by a factor of 4.
$g(x)=5^x$ is compressed to $\dfrac{1}{4}$ its height.
$g(x)=5^x$ is stretched by a factor of 7.
$g(x)=5^x$ is compressed to $\dfrac{1}{3}$ its height.

Reflection across the $x$ -axis

When the graph of the function $f(x)=2^x$ is reflected across the $x$ -axis, the $y$ -coordinates of all of the points on the graph change their signs, from positive to negative values, while the $x$ -coordinates remain the same. The equation of the function after $f(x)=2^x$ is reflected across the $x$ -axis is $f(x)=-2^x$ . 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 $x$ -coordinate stays the same, the $y$ -coordinate becomes $-y$ .

$x$	$y=2^x$	$-y=2^x$ or $y=-2^x$	Figure 4. Reflecting the graph of $f(x)=2^x$ across the $x$ -axis.
$-3$	$\dfrac{1}{8}$	$-\dfrac{1}{8}$
$-2$	$\dfrac{1}{4}$	$-\dfrac{1}{4}$
$-1$	$\dfrac{1}{2}$	$-\dfrac{1}{2}$
$0$	$1$	$-1$
$1$	$2$	$-2$
$2$	$4$	$-4$
$3$	$8$	$-8$
Table 3. Reflecting the graph of $f(x)=2^x$ across the $x$ -axis transforms $f(x)=2^x$ into $f(x)=-2^x$

Example 4

The function $f(x)=4^x$ is reflected across the $x$ -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 $x$ -axis, the $x$ -coordinate stays the same while the $y$ -coordinate changes sign. So (2, 16) is transformed to (2, –16).

Try It 4

The function $f(x)=\left(\dfrac{2}{3}\right)^x$ is reflected across the $x$ -axis. What happens to the point $\left(-1, \dfrac{3}{2}\right)$ that lies on the parent function after the transformation?

Show Answer

$\left(-1, \dfrac{3}{2}\right)$ is transformed to $\left(-1, -\dfrac{3}{2}\right)$

Reflection across the $y$ -axis

When the graph of the function $f(x)=2^x$ is reflected across the $y$ -axis, the $x$ -coordinates of all of the points on the graph change their signs while the $y$ -coordinates remain the same. The equation of the function after $f(x)=2^x$ is reflected across the $y$ -axis is $f(x)=2^{-x}$ . 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 $y$ -coordinate stays the same, the $x$ -coordinate becomes $-x$ .

$x$	$2^x$	$2^{-x}$	Figure 5. Reflecting the graph of $f(x)=2^x$ across the $y$ -axis.
$-3$	$\dfrac{1}{8}$	$8$
$-2$	$\dfrac{1}{4}$	$4$
$-1$	$\dfrac{1}{2}$	$2$
$0$	$1$	$1$
$1$	$2$	$\dfrac{1}{2}$
$2$	$4$	$\dfrac{1}{4}$
$3$	$8$	$\dfrac{1}{8}$
Table 4. Reflecting the graph of $f(x)=2^x$ across the $y$ -axis transforms $f(x)=2^x$ into $f(x)=2^{-x}$

Example 5

The function $f(x)=4^x$ is reflected across the $y$ -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 $y$ -axis, the $y$ -coordinate stays the same while the $x$ -coordinate changes sign. So (2, 16) is transformed to (–2, 16).

Try It 5

The function $f(x)=7^x$ is reflected across the $y$ -axis. What happens to the point $\left(-1, \dfrac{1}{7}\right)$ that lies on the parent function, after the transformation?

Show Answer

$\left(-1, \dfrac{1}{7}\right)$ is transformed to $\left(1, \dfrac{1}{7}\right)$

Determine the transformations of the exponential function $f(x)=ar^{x-h}+k$

All of the transformations we have applied to the parent function $f(x)=r^x$ can be combined. The result is a general exponential function $f(x)=ar^{x-h}+k$ . Given any function in the form $f(x)=ar^{x-h}+k$ , we can determine from the values of $a, h,$ and $k$ the transformations that were performed on the parent function $f(x)=r^x$ . 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 $f(x)=2^x$ to get the function $f(x)=-3\left(2^{x+4}\right)-6$ ?

Solution

First we identify $a, h,$ and $k$ :

$a=-3,\;h=-4,\;k=-6$

A negative value of $a$ means the function $f(x)=2^x$ has been reflected across the $x$ -axis. $a=-3$ means it has also been stretched by a factor of 3.

$h=-4$ means the parent function has been shifted horizontally 4 units left, while $k=-6$ means it has been shifted vertically down by 6 units.

Try It 6

What transformations were performed on the parent function $f(x)=2^x$ to get the function $f(x)=\dfrac{1}{4}\left(2^{x-6}\right)+3$ ?

Show Answer

The parent function has been compressed to $\dfrac{1}{4}$ its height, shifted horizontally 6 units right, and shifted vertically 3 units up.

Example 7

If the parent function $g(x)=3^x$ is stretched by a factor of 7, reflected across the $y$ -axis, and shifted vertically 3 units down, what is the equation of the transformed function?

Solution

Stretched by a factor of 7 means $a=7$ .

Reflected across the $y$ -axis means $x$ becomes $-x$ .

Shifted vertically down by 3 units means $k=-3$ .

So, the transformed function is $f(x)=7\left(3^{-x}\right)-3$ .

Try It 7

If the parent function $g(x)=5^x$ is compressed to $\dfrac{1}{4}$ its height, reflected across the $x$ -axis, shifted horizontally left by 9 units, and shifted vertically up by 4 units, what is the equation of the transformed function?

Show Answer

$f(x)=-\dfrac{1}{4}\left(5^{x+9}\right)+4$

Move the dots in figure 6, to see what happens to the graph of the parent function $f(x)=r^x$ when it is transformed by changing the values of the constants $a, h, k$ and by making $x$ negative to reflect across the $y-axis$ .

Figure 6. Transformation of $f(x)=r^x$

Chapter 5: Exponential Functions