Chapter 3.1, Problem 16E

To determine

To match: The given exponential function with the correct graph.

Answer to Problem 16E

The equation $f\left(x\right)=2^{x-2}$ is matched to graph $\left(\text{b}\right)$ .

Explanation of Solution

Given information:

The exponential function $f\left(x\right)=2^{x-2}$ . The given graphs are:

  

Calculation:

Consider the provided function $f\left(x\right)=2^{x-2}$ . Construct a table with different values of x .

Substitute $x=0$ in the above function.

  $\begin{array}{c}f\left(0\right)=2^{0-2}\\ =2^{-2}\\ =\frac{1}{4}\end{array}$

Substitute $x=1$ in the above function.

  $\begin{array}{c}f\left(1\right)=2^{1-2}\\ =2^{-1}\\ =\frac{1}{2}\end{array}$

Substitute $x=-1$ in the above function.

  $\begin{array}{c}f\left(-1\right)=2^{-1-2}\\ =2^{-3}\\ =\frac{1}{2^3}\\ =\frac{1}{8}\end{array}$

Substitute $x=2$ in the above function.

  $\begin{array}{c}f\left(2\right)=2^{2-2}\\ =2^0\\ =1\end{array}$

The above obtained values are tabulated as,

  $\begin{array}{ccccc}x& -1& 0& 1& 2\\ f\left(x\right)& \frac{1}{8}& \frac{1}{4}& \frac{1}{2}& 1\end{array}$

It is observed that the graph of the function passes through the point $\left(0,\frac{1}{4}\right)$ . The graph of the equation is an increasing function of x since as values of x increases the corresponding value of the function $f\left(x\right)=2^{x-2}$ increases.

Thus, from all the graphs, the equation is matched to graph $\left(\text{b}\right)$ .