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

Please choose the best answer.

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). $$f(x)=5^x$$ $$g(x)=-5^{x}$$

Please choose the best answer.

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.

Please choose the best answer.

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

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