Chapter 3.1
Q. 3.1.2
Q. 3.1.2
Graphing an Exponential Function
Graph: f(x)=2^x.
Step-by-Step
Verified Solution
We begin by setting up a table of coordinates.
| x | f(x)=2^x | |
| We selected integers from -3 to 3 , inclusive, to include three negative numbers, 0 , and three positive numbers. We also wanted to keep the resulting computations for y relatively simple. | -3 | f(-3)=2^{-3}=\frac{1}{2^3}=\frac{1}{8} |
| -2 | f(-2)=2^{-2}=\frac{1}{2^2}=\frac{1}{4} | |
| -1 | f(-1)=2^{-1}=\frac{1}{2^1}=\frac{1}{2} | |
| 0 | f(0)=2^0=1 | |
| 1 | f(1)=2^1=2 | |
| 2 | f(2)=2^2=4 | |
| 3 | f(2)=2^3=8 |
We plot these points, connecting them with a continuous curve. Figure 3.2 shows the graph of f(x)=2^x. Observe that the graph approaches, but never touches, the negative portion of the x-axis. Thus, the x-axis, or y = 0, is a horizontal asymptote. The range is the set of all positive real numbers. Although we used integers for x in our table of coordinates, you can use a calculator to find additional points. For example, f(0.3)=2^{0.3} \approx 1.231 and f(0.95)=2^{0.95} \approx 1.932. The points (0.3, 1.231) and (0.95, 1.932) approximately fit the graph.
