3.4 Graphs of Exponential Functions

Let’s start with a basic exponential function such that

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

where $a > 1$ $a > 1$ . There are three properties of a graph of an exponential function for any $a$ $a$ greater than 1. First, $f (x = 0) = a^{0} = 1$ $f (x = 0) = a^{0} = 1$ ; 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 $x$ $x$ . This means that the higher the value of $x$ $x$ is, the greater the outcome of the function. Finally, $f (x) > 0$ $f (x) > 0$ for all $x$ $x$ . Based on these properties, the exponential function graph looks like this:

The graph of f(x) = a^x, a > 1. A concave up, increasing curve is drawn with a y-intercept of (0, 1).

Figure 3.1: Graph of

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

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

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

The graph of f(x) = a^x, o < a < 1. A dashed blue curve, concave up, increasing, is drawn with a y-intercept at (0, 1). A solid orange line is drawn, concave up, decreasing, with a y-intercept at (0, 1).

Figure 3.2: Graph of

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

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

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

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

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 $1 < a$ $1 < a$ , and the orange line represents when $0 < a < 1$ $0 < a < 1$ .)

The graph of f(x) = -a^x, 0 < a < 1. A dotted orange line, concave up, decreasing, is drawn with a y-intercept at (0, 1). A solid orange line, concave down, increasing, is drawn with a y-intercept at (0, -1). The graph of f(x) = -a^x, 1 < a. A dotted blue line, concave up, increasing, is drawn with a y-intercept at (0, 1). A solid blue line, concave down, decreasing, is drawn with a y-intercept at (0, -1).

Figure 3.3: Graph of

f (x) = - a^{x}

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

Something to note is that the most common choice for $a$ $a$ in an exponential function is $e$ $e$ , where $e \approx 2.718281828 . . .$ $e \approx 2.718281828 . . .$ This number $e$ $e$ 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 $b$ $b$ is added to the exponent, it will shift the graph to the left by $b$ $b$ . If a positive real number $b$ $b$ is subtracted from the exponent, it will shift the graph to the right by $b$ $b$ . If a positive real number $c$ $c$ is added to the entire function, it will shift the graph upward by $c$ $c$ . If a positive real number $c$ $c$ is subtracted from the entire function, it will shift the graph downward by $c$ $c$ .

Image 1: A graph of f(x) = a^(x + b), a > 1, b > 0. A curve, concave up, increasing, is drawn through (-(1-b), a). A graph of f(x) = a^(x + b), a > 1, b < 0. A curve, concave up, increasing, is drawn through ((1-b), a). Additionally, a dashed line is drawn as a curve, concave up, increasing, through (0, 1). Image 2: A graph of f(x) = a^x + c, a > 1, c > 0. A curve, concave up, increasing, is drawn through (0, (a+c)). A graph of f(x) = a^x + c, a > 1, c < 0. A curve, concave up, increasing, is drawn through (0, -(a+c)). Additionally, a dashed line is drawn as a curve, concave up, increasing, through (0, 1).

Figure 3.4: Graphs of

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

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

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

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

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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