Chapter 7.7, Problem 3CYU

To determine

To construct an exponential function to model the given graph.

Answer to Problem 3CYU

  $y=4{\left(3\right)}^x$

Explanation of Solution

Given:

Graph passes through the points (1, 12) and (3, 108).

Calculation:

Given curve passes through the points (1, 12) and (3, 108).

Let the required exponential function be

  $y=a\cdot b^x$

Now put (1, 12) in $y=a\cdot b^x$ , we get

  $\begin{array}{l}12=a\cdot b^1\\ \Rightarrow a\cdot b=12\to \left(1\right)\end{array}$

Now put (3, 108) in $y=a\cdot b^x$ , we get

  $\begin{array}{l}108=a\cdot b^3\\ \Rightarrow a\cdot b^3=108\to \left(2\right)\end{array}$

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

  $\begin{array}{l}\frac{a\cdot b^3}{a\cdot b}=\frac{108}{12}\\ \Rightarrow b^2=9\\ \Rightarrow b=3\end{array}$

Now put $b=3$ in equation (1), we get

  $\begin{array}{l}a\cdot 3=12\\ \Rightarrow a=\frac{12}{3}\\ \Rightarrow a=4\end{array}$

Hence the required exponential function is,

  $y=4{\left(3\right)}^x$