Chapter 7, Problem 62SGR

To determine

The exponential function from the given graph.

Answer to Problem 62SGR

The exponential function is $y=\frac{1}{2}{(2)}^x$ .

Explanation of Solution

Given information:

A graph given with three points $(0,\frac{1}{2})$ , $(2,2)$ and $(3,4)$ .

Calculation:

Let, the exponential function be $y=a{b}^x$ .

It contains the given three points, $(0,\frac{1}{2})$ , $(2,2)$ and $(3,4)$ .

By using the point $(0,\frac{1}{2})$ , we plug in $x=0$ and $y=\frac{1}{2}$ .

  $\begin{array}{l}y=a{b}^x\\ \Rightarrow \frac{1}{2}=a{b}^0\\ \Rightarrow \frac{1}{2}=a(1)\\ \Rightarrow a=\frac{1}{2}\end{array}$

So, the exponential equation becomes $y=\frac{1}{2}{b}^x$ .

Now by using the point $(3,4)$ , we plug in $x=3$ and $y=4$ .

  $\begin{array}{l}y=\frac{1}{2}{b}^x\\ \Rightarrow 4=\frac{1}{2}{b}^3\\ \Rightarrow 4\times 2=b^3\\ \Rightarrow 8=b^3\\ \Rightarrow b=\sqrt[3]{8}\\ \Rightarrow b=2\end{array}$

So, the exponential function is $y=\frac{1}{2}{(2)}^x$ .