MathematicsPreCalculus Mathematics at Nebraska

SubsectionGraphs of Exponential Functions

The graphs of exponential functions have two characteristic shapes, depending on whether the base, $b\text{,}$ is greater than $1$ or less than $1\text{.}$ As typical examples, consider the graphs of $f (x) = 2^x$ and $g(x) =\left(\dfrac{1}{2}\right)^x$ shown in Figure190. Some values for $f$ and $g$ are recorded in Tables188 and 189.

$x$	$f(x)$
$-3$	$\frac{1}{8}$
$-2$	$\frac{1}{4}$
$-1$	$\frac{1}{2}$
$0$	$1$
$1$	$2$
$2$	$4$
$3$	$8$

$x$	$g(x)$
$-3$	$8$
$-2$	$4$
$-1$	$2$
$0$	$1$
$1$	$\frac{1}{2}$
$2$	$\frac{1}{4}$
$3$	$\frac{1}{8}$

Notice that $f (x) = 2^x$ is an increasing function and $g(x) = \left(\dfrac{1}{2}\right)^x$ is a decreasing function. In general, exponential functions have the following properties.

In Table188 you can see that as the $x$-values decrease toward negative infinity, the corresponding $y$-values decrease toward zero. As a result, the graph of $f$ decreases toward the $x$-axis as we move to the left. Thus, the $x$-axis, $y=0 \text{,}$ is a horizontal asymptote for exponential functions no matter what value $b\text{,}$ the growth factor, happens to be. We can see this illustrated in Figure190, since the $y$-values of each exponential get closer to zero. (For more info on horizontal asymptotes, see AppendixB.)

For exponential functions with $0 \lt b \lt 1\text{,}$ this takes place close to the positive $x$-axis, as illustrated in Figure190b. For exponential functions with $1 \lt b\text{,}$ this takes place close to the negative $x$-axis, as illustrated in Figure190a.

In Example191, we compare two increasing exponential functions. The larger the value of the base, $b\text{,}$ the faster the function grows. In this example, both functions have $a = 1\text{.}$

Example191

Compare the graphs of $f (x) = 3^x$ and $g(x) = 4^x\text{.}$

Solution

We evaluate each function for several convenient values, as shown in Table192.

Plot the points for each function and connect them with smooth curves. For positive \(x\)-values, \(g(x)\) is always larger than \(f (x)\text{,}\) and is increasing more rapidly. In Figure193, \(g(x) = 4^x\) climbs more rapidly than \(f (x) = 3^x\text{.}\) Both graphs cross the \(y\)-axis at (0, 1).

\(x\)	\(f(x)\)	\(g(x)\)
\(-2\)	\(\dfrac{1}{9}\)	\(\dfrac{1}{16}\)
\(-1\)	\(\dfrac{1}{3}\)	\(\dfrac{1}{4}\)
\(0\)	\(1\)	\(1\)
\(1\)	\(3\)	\(4\)
\(2\)	\(9\)	\(16\)

For decreasing exponential functions, those with bases between $0$ and $1\text{,}$ the smaller the base, the more steeply the graph decreases. For example, compare the graphs of $p(x) = 0.8^x$ and $q(x) = 0.5^x$ shown in Figure194.