Topic 2.6 Competing Function Model Validation: compare competing function models for a data set by analyzing residuals, and validate or reject a model based on the pattern and size of its residuals.

What this topic is asking

The College Board (Topic 2.6) wants you to validate a function model and choose between competing models using residuals. A residual is the difference between an observed value and the model's prediction. A good model leaves residuals that scatter randomly around zero; a pattern in the residuals means the model is missing structure and a different one is needed.

What a residual is

A single residual says little; the pattern of all the residuals is what validates or rejects a model.

Reading a residual plot

The key insight is that a curve in the residuals is a sign the model is the wrong family, even if the fitted curve looks close to the points. The residual plot magnifies the leftover structure that the eye misses.

Why pattern beats closeness

Students often pick the model whose curve appears to pass nearest the points, but that can mislead. A linear model can run close to gently curving data while still bending away systematically, which shows up as a curved residual plot. The residual plot removes the overall trend and exposes only the leftover error, so a pattern there is decisive evidence that the model type is wrong. This is why the exam asks for residual reasoning rather than a visual impression: random residuals validate a model, and patterned residuals reject it, regardless of how good the original fit looked.

A useful companion idea is that residuals also flag where a model fails locally. If the residuals are random except for one large outlier, the model may be fine apart from an anomalous data point. If the residuals grow steadily larger toward one end, the model fits the start but drifts at the extreme, which often signals that exponential structure is being forced onto data that is really polynomial, or vice versa. Reading both the overall pattern and its local features turns the residual plot into a diagnostic, not just a yes-or-no validator.

Try this

Q1. A residual is positive at a data point. Did the model overpredict or underpredict there? [1 point]

• Cue. Underpredicted: a positive residual means actual exceeds predicted, so the model's value was too low.

Q2. Two models have random residual plots, but one has residuals about half the size of the other. Which is better? [1 point]

• Cue. The one with the smaller residuals; both are patternless, so the smaller errors win.