2.6 Competing Function Model Validation

Comparing model types

Different function types grow in fundamentally different ways: linear functions grow by equal differences over equal intervals, quadratic/polynomial functions grow by predictable rates of change patterns, and exponential functions grow by equal factors over equal intervals. To decide which model best fits data, examine successive differences and ratios of output values for equally spaced inputs.

Using regression to validate models

Given a data set, you can compute regression equations (linear, quadratic, exponential, etc.) and compare how well each fits using residuals or correlation values. The model with residuals showing no clear pattern (scattered randomly) and the smallest overall error is the better fit. Context matters too — real-world constraints can rule out a model even if it fits numerically well.

Justifying the better model

To validate a model, check whether the rate of change behavior in the data matches the model's defining property: constant first differences (linear), constant second differences (quadratic), or constant ratios (exponential). Compare residual plots — a good model shows randomly scattered residuals, while a poor model shows a clear pattern (like a curve), indicating systematic error.

Key terms

How do you identify a linear model from data?

Constant first differences between consecutive y-values over equal x-intervals.

How do you identify a quadratic model from data?

Constant second differences between consecutive y-values over equal x-intervals.

How do you identify an exponential model from data?

Constant ratios between consecutive y-values over equal x-intervals.

What does a random residual plot indicate?

The chosen model is a good fit — it has captured the systematic pattern, leaving only random error.

Besides fit statistics, what else validates a model?

Contextual reasonableness — the model's predictions should make sense within real-world constraints.

Try an AP-style question

A linear model's residual plot shows a clear U-shape. What does this suggest?

1. The linear model fits the data very well
2. The data have no pattern, so any model works
3. A linear model is not appropriate; the data are curved
4. The residuals are all zero, so predictions are exact