Chapter 9.9, Problem 24PPS

To determine

Which kinds of model best describe the data and then write an equation for the function

Answer to Problem 24PPS

The given data represents a quadratic model

The equation is $y=4.2x^2$

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:

  $\begin{array}{c}\text{First differences:}\\ d=y_2-y_1\\ =4.2-0\\ =4.2\\ d=y_3-y_2\\ =16.8-4.2\\ =12.6\\ d=y_4-y_3\\ =37.8-16.8\\ d=21\\ \text{Second differences:}\\ d=12.6-4.2\\ d=8.4\\ d=21-12.6\\ d=8.4\end{array}$

Thus, the first difference of y-values is not equal but the second difference is equal, so the given data represents a quadratic model

General equation of quadratic function is given by $y=a{x}^2$

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

  $\begin{array}{c}\left(1,4.2\right)\\ y=a{x}^2\\ 4.2=a{\left(1\right)}^2\\ 4.2=a\\ 4.2=a\\ \Rightarrow a=4.2\end{array}$

Thus, the equation is $y=4.2x^2$