3.4 Graphs of Exponential Functions

Let’s start with a basic exponential function such that

$f (x) = a^{x}$

where $a > 1$ . There are three properties of a graph of an exponential function for any $a$ greater than 1. First, $f (x = 0) = a^{0} = 1$ ; as we learned in the previous section, a real number raised to the power of 0 is 1. Next, the exponential function is increasing with $x$ . This means that the higher the value of $x$ is, the greater the outcome of the function. Finally, $f (x) > 0$ for all $x$ . Based on these properties, the exponential function graph looks like this:

Figure 3.1: Graph of

f (x) = a^{x}, a > 1

When $0 < a < 1$ , the graph of $f (x)$ still goes through y-intercept (0, 1) and is always positive for all $x$ , but the function is decreasing.

Figure 3.2: Graph of

f (x) = a^{x}, 0 < a < 1

When there is a negative sign in front of $a$ , such that

$f (x) = - a^{x}$

then the graph of exponential function is rotated at the y-axis, as shown in Figure 3.3. (In this graph, the blue line represents when $1 < a$ , and the orange line represents when $0 < a < 1$ .)

Figure 3.3: Graph of

f (x) = - a^{x}

Something to note is that the most common choice for $a$ in an exponential function is $e$ , where $e \approx 2.718281828 . . .$ This number $e$ is called Euler’s number and is frequently used to describe growth and decay. In finance, it is used to calculate compound interest.

In addition, if a positive real number $b$ is added to the exponent, it will shift the graph to the left by $b$ . If a positive real number $b$ is subtracted from the exponent, it will shift the graph to the right by $b$ . If a positive real number $c$ is added to the entire function, it will shift the graph upward by $c$ . If a positive real number $c$ is subtracted from the entire function, it will shift the graph downward by $c$ .

Figure 3.4: Graphs of

f (x) = a^{x + b}

and

f (x) = a^{x} + c

Let’s look at a couple of examples. Can you graph the following exponential functions?

$\begin{aligned} f (x) = e^{x + 3} - 2 \\ g (x) = e^{x - 1} + 1 \end{aligned}$

Let’s start with $f (x)$ . Since 3 is added to the exponent, and 2 is subtracted from the entire function, the graph is moved left by 3 and down by 2. How about $g (x)$ ? Since 1 is subtracted from the exponent, and 1 is added to the entire function, the graph is moved right by 1 and up by 1.

Figure 3.5: Graph of

f (x) = e^{x + 3} - 2

and

g (x) = e^{x - 1} + 1

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