Graphing Exponential Functions Lesson

An exponential function is of the form:
f(x) = a^x
a > 0
a ≠ 1
x is any real number
When we think about the graph of f(x) = a^x:

• (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)
-2	1/9	(-2, 1/9)
-1	1/3	(-1, 1/3)
0	1	(0, 1)
1	3	(1, 3)
2	9	(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.