Chapter 3.5, Problem 33E

To determine

Tofind:

The equation of the exponential function represented by the graph.

Answer to Problem 33E

The required equation of the exponential model is $f\left(x\right)=4e^{-0.28x}$ .

Explanation of Solution

Given:

A table of values for an exponential function:

  $\begin{array}{ccc}x& 0& 5\\ y& 4& 1\end{array}$

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 $\left(0,4\right)$ in the exponential model, we get:

  $\begin{array}{l}f\left(0\right)=a{e}^{b\left(0\right)}\\ \Rightarrow 4=a{e}^0\\ \Rightarrow a=4\end{array}$

The equation now becomes $f\left(x\right)=4e^{bx}$ . Upon substituting the point $\left(5,1\right)$ in this exponential model, we get:

  $\begin{array}{l}f\left(5\right)=4e^{b\left(5\right)}\\ \Rightarrow 1=4e^{5b}\\ \Rightarrow \ln \left(\frac{1}{4}\right)=5b\\ \Rightarrow b=-\frac{\ln 4}{5}=-0.28\end{array}$

Therefore, the required exponential model becomes:

  $f\left(x\right)=4e^{-0.28x}$