Chapter 7.7, Problem 4CYU

To determine

To construct an exponential function to model the given graph.

Answer to Problem 4CYU

  $y=0.25{\left(6\right)}^x$

Explanation of Solution

Given:

Graph passes through the points (1, 1.5) and (3, 54).

Calculation:

Given curve passes through the points (1, 1.5) and (3, 54).

Let the required exponential function be

  $y=a\cdot b^x$

Now put (1, 1.5) in $y=a\cdot b^x$ , we get

  $\begin{array}{l}1.5=a\cdot b^1\\ \Rightarrow a\cdot b=1.5\to \left(1\right)\end{array}$

Now put (3, 54) in $y=a\cdot b^x$ , we get

  $\begin{array}{l}54=a\cdot b^3\\ \Rightarrow a\cdot b^3=54\to \left(2\right)\end{array}$

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

  $\begin{array}{l}\frac{a\cdot b^3}{a\cdot b}=\frac{54}{1.5}\\ \Rightarrow b^2=36\\ \Rightarrow b=6\end{array}$

Now put $b=6$ in equation (1), we get

  $\begin{array}{l}a\cdot 6=1.5\\ \Rightarrow a=\frac{1.5}{6}\\ \Rightarrow a=0.25\end{array}$

Hence the required exponential function is,

  $y=0.25{\left(6\right)}^x$