Lesson Objectives
• Learn the basic definition of an exponential function
• Learn how to sketch the graph of an exponential function
• Learn how to graph function transformations of exponential functions

How to Sketch the Graph of an Exponential Function

Before we get into how to graph an exponential function, let's begin with a basic definition of the exponential function and the required restrictions to use the set of real numbers as its domain.

Exponential Function

$$\text{For}\hspace{.2em}a > 0, a ≠ 1, x ∈ \mathbb{R}$$ $$f(x)=a^x$$ Let's think a bit on our restrictions. First, our base, which is a (the base), is a positive real number. If a (the base) is allowed to be zero or negative, then we will run into issues. We should know at this point that something like: $$(-4)^{\frac{1}{2}}=\sqrt{-4}=2i$$ In this section, we are not involving non-real complex numbers or those numbers that involve the imaginary unit i. Let's suppose we let a (the base) be equal to -4 and x (the exponent) be equal to 1/2. $$(-4)^{\frac{1}{2}}\rightarrow \text{Not Real}$$ As another example of what could go wrong, suppose we let a (the base) be equal to 0 and x (the exponent) be equal to -1. $$0^{-1}=\frac{1}{0}\rightarrow \text{Undefined}$$ The last restriction of a ≠ 1 is needed since 1 raised to any power is equal to 1, which would give us a linear function f(x) = 1.

Graphs of Exponential Functions

$$f(x)=a^x, a > 1$$
• This graph is continuous and increasing over its entire domain
• The x-axis or y = 0 is a horizontal asymptote as x → -∞
$$f(x)=a^x, 0 < a < 1$$
• This graph is continuous and decreasing over its entire domain
• The x-axis or y = 0 is a horizontal asymptote as x → +∞
In general, when we think about the graph of f(x) = ax:
• (0,1) is on the graph
• Since a can't be 0, and any non-zero number raised to the power of 0 is 1
• The graph approaches the x-axis, but will never touch it. It forms an asymptote.
• The domain consists of all real numbers or the interval: (-∞, ∞)
• The range consists of all positive real numbers, or the interval: (0, ∞)
• When a > 1, the graph rises from left to right
• When 0 < a < 1, the graph falls from left to right
• The points (-1, 1/a), (0, 1), and (1, a) are on the graph
We graph an exponential function in the usual way. We find and plot enough ordered pairs and then connect the points with a smooth curve. Let's look at an example.
Example 1: Sketch the graph of each. $$f(x) = 3^x$$ Let's create a table with some ordered pairs:
x y (x, y)
-21/9(-2, 1/9)
-11/3(-1, 1/3)
01(0, 1)
13(1, 3)
29(2, 9)
Now we can plot the points on the coordinate plane and connect the points using a smooth curve. As the graph moves from right to left, it approaches the x-axis but does not touch it.

Function Transformations with Exponential Functions

Previously, we learned about reflecting across an axis.
• The graph of y = -f(x) is the same as the graph of y = f(x) reflected across the x-axis
• The graph of y = f(-x) is the same as the graph of y = f(x) reflected across the y-axis
Example 2: Sketch the graph of each. $$g(x)=\left(\frac{1}{3}\right)^x$$ Instead of creating points, let's graph this function by reflecting our previous graph across the y-axis. $$f(x)=3^x$$ $$g(x)=f(-x)$$ $$f(-x)=3^{-x}=\frac{1}{3^x}=\left(\frac{1}{3}\right)^{x}$$ Example 3: Sketch the graph of each. $$g(x)=-3^x$$ Instead of creating points, let's graph this function by reflecting our previous graph across the x-axis. $$f(x)=3^x$$ $$g(x)=-f(x)$$ $$-f(x)=-3^x$$

Skills Check:

Example #1

Find the Range. $$f(x)=-2^{x}$$

A
$$\{y | y < 0\}$$
B
All Real Numbers
C
$$\{y | y > 0\}$$
D
$$\{y | y ≥ 0\}$$
E
$$\{y | y ≤ 0\}$$

Example #2

Find the transformation from f(x) to g(x). $$f(x)=5^x$$ $$g(x)=-5^{x}$$

A
Shifted down by 5 units
B
Shifted left by 5 units
C
Reflected across the line y = x
D
Reflected across the x-axis
E
Reflected across the y-axis

Example #3

Fill in the blanks. $$f(x)=10^x$$ $$g(x)=\left(\frac{1}{10}\right)^x$$ The given function f is a(n) "___" function, while g is a(n) "___" function.

A
increasing, constant
B
increasing, increasing
C
decreasing, increasing
D
increasing, decreasing
E
decreasing, decreasing