1.13 Function Model Selection and Assumption Articulation

Choosing the right model

To model a data set or context, compare patterns of change. Linear functions have constant first differences (constant rate of change). Quadratic functions have constant second differences. Exponential functions have roughly constant ratios between consecutive outputs. Examining differences/ratios in a table helps you decide which family — linear, quadratic, polynomial, or exponential — best fits the data before fitting a curve.

Assumptions and restrictions

Every model relies on assumptions about the underlying process (e.g., constant growth rate, symmetry, no extreme events) and has a domain of validity — the model may fail outside observed data (extrapolation risk). Precalculus modeling requires stating assumptions explicitly, such as treating discrete data as continuous or assuming trends persist, and recognizing when a model is inappropriate.

Evaluating model fit

After selecting a candidate function, check its reasonableness: does it pass near the data points, do residuals show no clear pattern, and do end behaviors make sense in context? A good model balances simplicity with accuracy, and its limitations (like unrealistic long-term behavior) should be acknowledged rather than ignored.

Key terms

Constant first differences

Indicates data is best modeled by a linear function.

Constant second differences

Indicates data is best modeled by a quadratic function.

Constant ratio between outputs

Indicates data is best modeled by an exponential function.

Extrapolation

Using a model to predict values outside the range of the observed data; can be unreliable.

Model assumption

A condition assumed to hold true for the model to be valid, such as constant growth rate or no outliers.

Try an AP-style question

Which model fits best?

1. Linear
2. Quadratic
3. Exponential
4. Cubic