Chapter 7.7, Problem 12PPS

To determine

To construct an exponential function to model the given graph.

Answer to Problem 12PPS

  $y=-2{\left(2\right)}^x$

Explanation of Solution

Given:

  

Calculation:

Given curve passes through the points $\left(-1,-1\right)\&\left(2,-8\right)$ .

Let the required exponential function be

  $y=a\cdot b^x$

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

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

Now put $\left(2,-8\right)$ in $y=a\cdot b^x$ , we get

  $\begin{array}{l}-8=a\cdot b^2\\ \Rightarrow a\cdot b^2=-8\to \left(2\right)\end{array}$

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

  $\begin{array}{l}\frac{a\cdot b^2}{\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$b$}\right.}=\frac{-8}{-1}\\ \Rightarrow b^3=8\\ \Rightarrow b=2\end{array}$

Now put $b=2$ in equation (1), we get

  $\begin{array}{l}\frac{a}{2}=-1\\ \Rightarrow a=-2\end{array}$

Hence the required exponential function is,

  $y=-2{\left(2\right)}^x$