Chapter 8.2, Problem 18PPS

To determine

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

Answer to Problem 18PPS

The exponential function that passes through the given points is $y=128{\left(4.926\right)}^x$ .

Explanation of Solution

Given information:

The given points are $(0,128)$ and $\left(5,371,293\right)$ .

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

Substitute 0 for $x$ and $128$ for $y$ in the general form of exponential function $y=a{b}^x$ to find the value of $a$ .

  $\begin{array}{c}y=a{b}^x\\ 128=a{b}^0\\ a=128\end{array}$

Substitute $5$ for $x$ , 128 for $a$ , and 371,293 for $y$ in the general form of exponential function $y=a{b}^x$ to find the value of $b$ .

  $\begin{array}{c}371,293=128\times b^5\\ b^5=\frac{371,293}{128}\\ b=\sqrt[5]{2900.7265625}\\ =4.926\end{array}$

Substitute $128$ for $a$ and $4.926$ for $b$ in the general form of exponential function $y=a{b}^x$ to find the value of $y$ .

  $y=128{\left(4.926\right)}^x$

Therefore, the exponential function that passes through the given points is $y=128{\left(4.926\right)}^x$ .