Chapter 7.2, Problem 19PPS

To determine

To find: the exponential function that passes through the given points.

Answer to Problem 19PPS

The exponential function that passes through the given points is $y=144{\left(3.5\right)}^x$ .

Explanation of Solution

Given information:

The given points are $(0,144)$ and $\left(4,21,609\right)$ .

The general form of the exponential function can be written as $y=a{b}^x$ .

Substitute 0 for $x$ and $144$ for $y$ in the general form of exponential function $y=a{b}^x$ to find the value of $a$ .

  $\begin{array}{c}y=a{b}^x\\ 144=a{b}^0\\ a=144\end{array}$

Substitute $4$ for $x$ , $144$ for $a$ , and 21,609 for $y$ in the general form of exponential function $y=a{b}^x$ to find the value of $b$ .

  $\begin{array}{c}21,609=144\times b^5\\ b^4=\frac{21,609}{144}\\ b=\sqrt[4]{150.0625}\\ \approx 3.5\end{array}$

Substitute $144$ for $a$ and $3.5$ for $b$ in the general form of exponential function $y=a{b}^x$ to find the value of $y$ .

  $y=144{\left(3.5\right)}^x$

Therefore, the exponential function that passes through the given points is $y=144{\left(3.5\right)}^x$ .