Lesson Objectives
• Learn how to sketch the graph of an exponential function

## How to Sketch the Graph of an Exponential Function

An exponential function is of the form:
f(x) = ax
a > 0
a ≠ 1
x is any real number
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
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.