Chapter 6.1, Problem 63E

To determine

To write: an exponential function f so that the slope from the point $\left(0,f\left(0\right)\right)$ to the point $\left(2,f\left(2\right)\right)$ is equal to $12$ .

Answer to Problem 63E

  $f(x)=5^x.$

Explanation of Solution

Given information: The slope from the point $\left(0,f\left(0\right)\right)$ to the point $\left(2,f\left(2\right)\right)$ is equal to $12$ .

Calculation:

Let the exponential function be:

  $\begin{array}{l}f(x)=b^x\\ \text{so}\text{that:}\\ f(0)=b^0=1\\ f(2)=b^2\end{array}$

The slope is:

  $\begin{array}{l}m=\frac{f(2)-f(0)}{2-0}=\frac{b^2-1}{2}\\ \text{Equate}\text{the}\text{slope}\text{to}\text{12:}\\ \frac{b^2-1}{2}=12\\ b^2-1=24\\ b^2=25\\ b=5\\ \text{The}\text{exponential}\text{function}\text{is:}\\ f(x)=5^x.\end{array}$