Chapter 9.9, Problem 36STP

To determine

An equation

Answer to Problem 36STP

The given data represents an exponential model

The equation is $y=3{\left(2\right)}^x$

Explanation of Solution

Given:

  

Concept Used:

If first difference of y-values is equal, then model be linear

If second difference be equal, then the model be quadratic

If first and second differences are not equal then the model will be exponential

Calculation:

In order to check whether the given data is linear, quadratic or exponential, find the first difference as shown below:

  $\begin{array}{c}\text{First differences:}\\ d=y_2-y_1\\ =6-3\\ =3\\ d=y_3-y_2\\ =12-6\\ =6\\ d=y_4-y_3\\ =24-12\\ d=12\\ \text{Second differences:}\\ d=6-3\\ d=3\\ d=12-6\\ d=6\\ \text{Ratio of y-values:}\\ r_1=\frac{a_2}{a_1}\\ =\frac{6}{3}=2\\ r_2=\frac{a_3}{a_2}\\ =\frac{12}{6}=2\end{array}$

Thus, the first difference of y-values is not equal but the second difference is also no equal , so the given data represents a exponential model

General equation of exponential function is given by $y=a{b}^x$ the ratio is 2 so $b=2$

To find the value of a we use a ordered pair from the table

  $\begin{array}{c}\left(0,3\right)\\ y=a{b}^x\\ 3=a{\left(4\right)}^0\\ 3=a\times 1\\ 3=a\\ \Rightarrow a=3\end{array}$

Thus, the equation is $y=3{\left(2\right)}^x$