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## Q. 3.1.2

Graphing an Exponential Function

Graph:    $f(x)=2^x$.

## 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. 