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Chapter 3.1

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.

3.2