Exercise 2 Page 219

There are different approaches we can take to decide which polynomial model would be the best choice given real-life data. We will discuss how to do this by using finite differences and by using the regression feature of a graphic calculator.

Using Finite Differences

We can determine if the real-life data fits a polynomial of specific degree by calculating the differences of the $y\text{-values}$ corresponding to equally-spaced $x\text{-values}.$ These are called finite differences. We can see an example below.

If the first differences are constant the data set fits a linear model. If the second differences are constant, then the data set fits a quadratic model, and so on. As we can see from the example data above, in this case a cubic model is appropriate. To find the specific cubic model we can use the standard form of a cubic function. By substituting $4$ different points, we can set up a system of equations. We solve to find the parameters $a,$ $b,$ $c$ and $d.$

Using The Regression Feature of the Graphic Calculator

Another way to find an appropriate polynomial model is by first drawing a scatter plot. With it we can visualize the behavior of the data set and determine a reasonable polynomial model. Consider the data plot shown below.

We can see that the behavior is clearly not linear. In this case, the data suggests a quadratic model, so we can use the regression feature of the calculator to find the specific quadratic model that best fits the data.