Transformations of Graphs of Exponential Functions

Shifts of the Parent Function $F(X) = B^X$

For any constants $c$ and $d$, the function $f(x)=b^{x+c}+d$ shifts the parent function $f(x)=b^x$

• vertically $d$ units, in the same direction of the sign of $d$.
• horizontally $c$ units, in the opposite direction of the sign of $c$.
• The $y$-intercept becomes $(0,b^c+d)$.
• The horizontal asymptote becomes $y=d$.
• The range becomes $(d, \infty)$.
• The domain, $(−\infty, \infty)$, remains unchanged.

HOW TO

Given an exponential function with the form $f(x)=b^{x+c}+d$, graph the translation.

1. Draw the horizontal asymptote $y=d$.

2. Identify the shift as $(−c,d)$. Shift the graph of $f(x)=b^x$ left $c$ units if $c$ is positive, and right $c$ units if $c$ is negative.

3. Shift the graph of $f(x)=b^x$ up $d$ units if $d$ is positive, and down $d$ units if $d$ is negative.

4. State the domain, $(− \infty, \infty)$, the range, $(d, \infty)$, and the horizontal asymptote $y=d$.

Example 2

Graphing a Shift of an Exponential Function

Graph $f(x)=2^{x+1}−3$. State the domain, range, and asymptote.

Solution

We have an exponential equation of the form $f(x)=b^{x+c}+d$, with $b=2$, $c=1$, and $d=−3$.

Draw the horizontal asymptote $y=d$, so draw $y=−3$.

Identify the shift as $(−c,d)$, so the shift is $(−1,−3)$.

Shift the graph of $f(x)=b^x$ left 1 units and down 3 units.

Figure 7

The domain is $(− \infty, \infty)$; the range is $(−3, \infty)$; the horizontal asymptote is $y=−3$.

Try It #2

Graph $f(x)=2^{x−1}+3$. State domain, range, and asymptote.

Try It #3

Solve $4=7.85(1.15)^x−2.27$ graphically. Round to the nearest thousandth.

Stretches and Compressions of the Parent Function $f(x)=b^x$

For any factor $a > 0$, the function $f(x)=a(b)^x$

• is stretched vertically by a factor of $a$ if $|a| > 1$.
• is compressed vertically by a factor of $a$ if $|a| < 1$.
• has a $y$-intercept of $(0,a)$.
• has a horizontal asymptote at $y=0$, a range of $(0, \infty)$, and a domain of $(−\infty, \infty)$, which are unchanged from the parent function.

Example 4

Graphing the Stretch of an Exponential Function

Sketch a graph of $f(x)=4(\frac{1}{2})^x$. State the domain, range, and asymptote.

Solution

Before graphing, identify the behavior and key points on the graph.

• Since $b=12$ is between zero and one, the left tail of the graph will increase without bound as $x$ decreases, and the right tail will approach the $x$-axis as $x$ increases.
• Since $a=4$, the graph of $f(x)=(\frac{1}{2})^x$ will be stretched by a factor of $4$.

• Create a table of points as shown in Table 4.

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$f(x)=4(\frac{1}{2})^x$	$32$	$16$	$8$	$4$	$2$	$1$	$0.5$

Table 4

• Plot the $y$-intercept, $(0,4)$, along with two other points. We can use $(−1,8)$ and $(1,2)$.

Draw a smooth curve connecting the points, as shown in Figure 9.

Figure 9

The domain is $(− \infty, \infty)$; the range is $(0, \infty)$; the horizontal asymptote is $y=0$.

Try It #4

Sketch the graph of $f(x)=\frac{1}{2} (4)^x$. State the domain, range, and asymptote.

Reflections of the Parent Function $f(x)=b^x$

The function $f(x)=−b^x$

• reflects the parent function $f(x)=b^x$ about the $x$-axis.
• has a $y$-intercept of $(0,−1)$.
• has a range of $(− \infty,0)$.
• has a horizontal asymptote at $y=0$ and domain of $(−\infty, \infty)$, which are unchanged from the parent function.

The function $f(x)=b{−x}$

• reflects the parent function $f(x)=b^x$ about the $y$-axis.
• has a $y$-intercept of $(0,1)$, a horizontal asymptote at $y=0$, a range of $(0, \infty)$, and a domain of $(−\infty, \infty)$, which are unchanged from the parent function.

Example 5

Writing and Graphing the Reflection of an Exponential Function

Find and graph the equation for a function, $g(x)$, that reflects $f(x)=(\frac{1}{4})^x$ about the $x$-axis. State its domain, range, and asymptote.

Solution

Since we want to reflect the parent function $f(x)=(\frac{1}{4})^x$ about the $x$-axis, we multiply $f(x)$ by $−1$ to get, $g(x)=−(\frac{1}{4})^x$. Next we create a table of points as in Table 5.

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$g(x)=−(\frac{1}{4})^x$	$-64$	$-16$	$-4$	$-1$	$-0.25$	$-0.0625$	$-0.0156$

Table 5

Plot the $y$-intercept, $(0,−1)$, along with two other points. We can use $(−1,−4)$ and $(1,−0.25)$.

Draw a smooth curve connecting the points:

Figure 11

The domain is $(− \infty, \infty)$; the range is $(− \infty, 0)$; the horizontal asymptote is $y=0$.

Try It #5

Find and graph the equation for a function, $g(x)$, that reflects $f(x)=1.25^x$ about the $y$-axis. State its domain, range, and asymptote.