Transformations of Graphs of Exponential Functions

Translations of Exponential Functions

A translation of an exponential function has the form

$f(x)=ab^{x+c}+d$

Where the parent function, $y=b^x, b > 1$, is

• shifted horizontally $c$ units to the left.
• stretched vertically by a factor of $|a|$ if $|a| > 0$.
• compressed vertically by a factor of $|a|$ if $0 < |a| < 1$.
• shifted vertically $d$ units.
• reflected about the $x$-axis when $a < 0$.

Note the order of the shifts, transformations, and reflections follow the order of operations.

Example 6

Writing a Function from a Description

Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.

• $f(x)=e^x$ is vertically stretched by a factor of $2$, reflected across the $y$-axis, and then shifted up $4$ units.

Solution

We want to find an equation of the general form $f(x)=ab^{x+c}+d$. We use the description provided to find $a, b, c$, and $d$.

• We are given the parent function $f(x)=e^x$, so $b=e$.
• The function is stretched by a factor of $2$ , so $a=2$.
• The function is reflected about the $y$-axis. We replace $x$ with $−x$ to get: $e^{−x}$.
• The graph is shifted vertically 4 units, so $d=4$.

Substituting in the general form we get,

$\begin{aligned} f(x) &=a b^{x+c}+d \\ &=2 e^{-x+0}+4 \\ &=2 e^{-x}+4 \end{aligned}$

The domain is $(− \infty, \infty)$; the range is $(4, \infty)$; the horizontal asymptote is $y=4$.

Try It #6

Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.

$f(x)=e^x$ is compressed vertically by a factor of $\frac{1}{3}$, reflected across the $x$-axis and then shifted down $2$ units.