We now define exponential functions. We assume that
We require the base to be positive in order to avoid the imaginary numbers that would occur by taking even roots of negative numbers—an example is
The following are examples of exponential functions:
Note that, in contrast to functions like
Let’s now consider graphs of exponential functions.
Graph the exponential function
We compute some function values and list the results in a table.
x | y | (x, y) |
---|---|---|
0 | 1 | (0, 1) |
1 | 2 | (1, 2) |
2 | 4 | (2, 4) |
3 | 8 | (3, 8) |
−1 | ||
−2 | ||
−3 |
Next, we plot these points and connect them with a smooth curve. Be sure to plot enough points to determine how steeply the curve rises.
Note that as x increases, the function values increase without bound. As x decreases, the function values decrease, getting close to 0. That is, as
Now Try Exercise 11.
Graph the exponential function
Before we plot points and draw the curve, note that
Points of |
Points of |
---|---|
(0, 1) | (0, 1) |
(1, 2) | (−1, 2) |
(2, 4) | (−2, 4) |
(3, 8) | (−3, 8) |
This tells us that this graph is a reflection of the graph of
Next, we plot these points and connect them with a smooth curve.
Note that as x increases, the function values decrease, getting close to 0. The x-axis,
Now Try Exercise 15.
To graph other types of exponential functions, keep in mind the ideas of translation, stretching, and reflection. All these concepts allow us to visualize the graph before drawing it.
Graph each of the following. Before doing so, describe how each graph can be obtained from the graph of
The graph of
x | |
---|---|
−1 | |
0 | |
1 | |
2 | 1 |
3 | 2 |
4 | 4 |
5 | 8 |
The graph of
x | |
---|---|
−2 | |
−1 | |
0 | −3 |
1 | −2 |
2 | 0 |
3 | 4 |
The graph of
x | |
---|---|
−3 | −3 |
−2 | 1 |
−1 | 3 |
0 | 4 |
1 | |
2 |
Now Try Exercises 27 and 33.