Chapter 3.5, Problem 34E

To determine

Tofind:

The equation of the exponential function represented by the graph.

Answer to Problem 34E

The required equation of the exponential model is $f\left(x\right)=5e^{-0.77x}$ .

Explanation of Solution

Given:

A table of values for an exponential function:

  $\begin{array}{ccc}x& 0& 3\\ y& 5& \frac{1}{2}\end{array}$

Formula used:

An exponential model is given by $f\left(x\right)=a{e}^{bx}$

Calculation:

We will substitute the points given on the graph one by one into the equation of an exponential model, and this way we will figure out the values of a and b.

Upon substituting the point $\left(0,5\right)$ in the exponential model, we get:

  $\begin{array}{l}f\left(0\right)=a{e}^{b\left(0\right)}\\ \Rightarrow 5=a{e}^0\\ \Rightarrow a=5\end{array}$

The equation now becomes $f\left(x\right)=5e^{bx}$ . Upon substituting the point $\left(3,\frac{1}{2}\right)$ in this exponential model, we get:

  $\begin{array}{l}f\left(3\right)=5e^{b\left(3\right)}\\ \Rightarrow \frac{1}{2}=5e^{3b}\\ \Rightarrow \ln \left(\frac{1}{10}\right)=3b\\ \Rightarrow b=-\frac{\ln 10}{3}=-0.77\end{array}$

Therefore, the required exponential model becomes:

  $f\left(x\right)=5e^{-0.77x}$