Chapter 3.5, Problem 31E

To determine

Tofind:

The equation of the exponential function represented by the graph.

Answer to Problem 31E

The required equation of the exponential model is $f\left(x\right)=1e^{0.77x}$ .

Explanation of Solution

Given:

The graph of an exponential function:

  

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 $\left(0,1\right)$ in the exponential model, we get:

  $\begin{array}{l}f\left(0\right)=a{e}^{b\left(0\right)}\\ \Rightarrow 1=a{e}^0\\ \Rightarrow a=1\end{array}$

The equation now becomes $f\left(x\right)=1e^{bx}$ . Upon substituting the point $\left(3,10\right)$ in this exponential model, we get:

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

Therefore, the required exponential model becomes:

  $f\left(x\right)=1e^{0.77x}$