Exponential functions grow exponentially—that is, very, very quickly. Two squared is 4; 2 cubed is 8, but by the time you get to 2 ^{7}, you have, in four small steps from 8, already reached 128, and it only grows faster from there. Four more steps, for example, bring the value to 2,048.
Any function defined by y = b ^{x}, where b > 0, b ≠ 1, and x is a real number, is called an exponential function.
Example 1
Graph y = 2 ^{x}.
First find a sufficient number of ordered pairs to see the shape of the graph.
x
2 ^{x}= y
( x, y)
| –3 | |
| –2 | |
| –1 | |
| 0 | 2^{0} = 1 | (0, 1)
| 1 | 2^{1} = 2 | (1, 2)
| 2 | 2^{2} = 4 | (2, 4)
| 3 | 2^{3} = 8 | (3, 8)
Find these points and connect them to form a smooth curve. No value for x makes y become zero. The more negative x becomes, the smaller y becomes. The negative x‐axis becomes an asymptote for this function. The graph is shown in Figure 1.
Figure 1. Exponential growth of 2.
Example 2
Graph .
Make a chart and graph the ordered pairs.
x
[ x, f ( x)]
| –3 | | (–3, 8)
| –2 | | (–2, 4)
| –1 | | (–1, 2)
| 0 | | (0, 1)
| 1 | |
| 2 | |
| 3 | |
The graph is shown in Figure 2.
All exponential functions, f ( x) = b ^{x}, b > 0, b ≠ 1, will contain the ordered pair (0, 1), since b ^{0} = 1 for all b ≠ 0. Exponential functions with b > 1 will have a basic shape like that in the graph shown in Figure 1, and exponential functions with b < 1 will have a basic shape like that of Figure 2.
The graph of x = b ^{y}is called the inverse of the graph of y = b ^{x}because the x and y variables are interchanged. Remember, the graphs of inverses are symmetrical around the line y = x. That is, if the graph of y = b ^{x}is “folded over” the line y = x and then retraced, it creates the graph of x = b ^{y}. Whatever ordered pairs satisfy y = b ^{x}, the reversed ordered pairs would satisfy x = b ^{y}.
Figure 2. Exponential growth of .
Example 3
Graph y = 3 ^{x}and x = 3 ^{y}on the same set of axes.
x
3 ^{x}= y
( x, y)
| –3 | |
| –2 | |
| –1 | |
| 0 | (3)^{0} = 1 | (0, 1)
| 1 | (3)^{1} = 3 | (1, 3)
| 2 | (3)^{2} = 9 | (2, 9)
| 3 | (3)^{3} = 27 | (3, 27)
x
3 ^{y}= x
( x, y)
| | |
| | |
| | |
| 1 | (3)^{0} = 1 | (1, 0)
| 3 | (3)^{1} = 3 | (3, 1)
| 9 | (3)^{2} = 9 | (9, 2)
| 27 | (3)^{3} = 27 | (27, 3)
The graphs of y = 3 ^{x}and x = 3 ^{y}are shown in Figure 3.
Figure 3. In this graph, y = 3 ^{x}and x = 3 ^{y}.