Chapter 7.7, Problem 11PPS

To determine

To construct an exponential function to model the given graph.

Answer to Problem 11PPS

  $y=10{\left(0.5\right)}^x$

Explanation of Solution

Given:

  

Calculation:

Given curve passes through the points $\left(-3,80\right)\&\left(-1,20\right)$ .

Let the required exponential function be

  $y=a\cdot b^x$

Now put $\left(-3,80\right)$ in $y=a\cdot b^x$ , we get

  $\begin{array}{l}80=a\cdot b^{-3}\\ \Rightarrow \frac{a}{b^3}=80\to \left(1\right)\end{array}$

Now put $\left(-1,20\right)$ in $y=a\cdot b^x$ , we get

  $\begin{array}{l}20=a\cdot b^{-1}\\ \Rightarrow \frac{a}{b}=20\to \left(2\right)\end{array}$

Now dividing equation (1) by equation (2), we get

  $\begin{array}{l}\frac{\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$b^3$}\right.}{\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$b$}\right.}=\frac{80}{20}\\ \Rightarrow \frac{1}{b^2}=4\\ \Rightarrow b^2=\frac{1}{4}\\ \Rightarrow b=\frac{1}{2}\\ \Rightarrow b=0.5\end{array}$

Now put $b=0.5$ in equation (2), we get

  $\begin{array}{l}\frac{a}{0.5}=20\\ \Rightarrow a=10\end{array}$

Hence the required exponential function is,

  $y=10{\left(0.5\right)}^x$