Introduction

Welcome back to AP Precalculus! Today’s topic will be similar to the one from the previous topic, and will go into finding the best model for a function. Without further ado, let’s get started.

Residuals

Let’s go back to the graph from the previous topic, which shows both data-points and the exponential regression for those data points.

You might notice that none of the datapoints are on the graph of the function. As discussed in the previous topic, a regression is not a perfect mapping of the data to a function, but rather a function meant to approximate the data. This is especially noticeable at the lower $x$-values, where the regression function undershoots the datapoints by a considerable degree.

In order to further examine the error in the function, we can find the residuals of the regression. The residuals of a regression is the difference between the data point and the function modeling it. The residual plot is obtained by plotting the residuals on a graph. Below is the residual plot of the graph above.

Looking at this plot, we can see that the residuals follow a somewhat smooth curve at the beginning, but then go all over the place near the end. The points at the beginning are above the line, which reflects our earlier statement that the data points at the beginning are above the graph of the function. When a residual is greater than $0$, we say that the function underestimates the function at that point. When a residual is less than $0$, we say that the function overestimates the function at that point.

If you look at this graph, while the points initially show a slight pattern, the points are generally scattered about near the end, so we would say that the residuals of this graph do not have a pattern. However, take a look at this residual plot.

This is the residual plot of the linear regression of the function $x^2$. Unlike the plot we saw before, the residual plot follows a clear pattern.

Whether the residual plot follows a clear pattern determines whether the model (or regression type) is appropriate. In the first example, the residual plot didn’t show a clear pattern, and so the model is appropriate for the function. On the other hand, in the second example, the residual plot shows a clear pattern, meaning the model is not appropriate for the data.

As a quick side note, the AP exam will often be very obvious about what is a clear pattern or not when they give you a residual graph, since all CollegeBoard is testing you on is if you know how to figure out if a model is appropriate for the data.

Finding an appropriate model

We previously discussed some basic descriptions on if a model is appropriate in the previous topic, and CollegeBoard expects you to know how to figure out whether a linear, quadratic, or exponential model works best. A couple of tips on this section

Questions