Chapter 7.7, Problem 2GP

To determine

To construct an exponential function to model the given graph.

Answer to Problem 2GP

  $y=2{\left(4\right)}^x$

Explanation of Solution

Given:

Graph passes through the points (1, 8) and (4, 512).

Calculation:

Given curve passes through the points (1, 8) and (4, 512).

Let the required exponential function be

  $y=a\cdot b^x$

Now put (1, 8) in $y=a\cdot b^x$ , we get

  $\begin{array}{l}8=a\cdot b^1\\ \Rightarrow a\cdot b=8\to \left(1\right)\end{array}$

Now put (4, 512) in $y=a\cdot b^x$ , we get

  $\begin{array}{l}512=a\cdot b^4\\ \Rightarrow a\cdot b^4=512\to \left(2\right)\end{array}$

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

  $\begin{array}{l}\frac{a\cdot b^4}{a\cdot b}=\frac{512}{8}\\ \Rightarrow b^3=64\\ \Rightarrow b=4\end{array}$

Now put $b=4$ in equation (1), we get

  $\begin{array}{l}a\cdot 4=8\\ \Rightarrow a=\frac{8}{4}\\ \Rightarrow a=2\end{array}$

Hence the required exponential function is,

  $y=2{\left(4\right)}^x$