A scatter plot of the data set can help us to visualize the data behavior.
See solution.
Practice makes perfect
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-values corresponding to equally-spaced x-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.
y = ax^3+bx^2+cx+d
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.
