Family Overview

In Plain Language

Before building a formula, students choose a function type that matches the situation, then say out loud the assumptions and limits of that choice.

• Linear fits roughly constant rates of change.
• Quadratic fits roughly linear rates of change, a unique max/min, symmetry, or many area/2D stories.
• Cubic often fits volume/3D stories; higher polynomials fit multiple zeros or multiple turns.
• Finite differences: roughly constant nonzero $n$th differences suggest degree $n$.
• $n+1$ points with distinct inputs can be modeled by a polynomial of degree at most $n$.
• Piecewise models are for scenarios that behave differently on different intervals.

Assumptions cover what stays consistent, how quantities change together, and needed domain/range restrictions (context, math, rounding, extremes).

Why It Matters

AP Precalculus modeling FRQs punish jumping to a regression without justifying the type or naming limits. Selection + articulation is the bridge into Topic 1.14 construction.

How Families Can Help

• Ask: “What clue made you pick this family?”
• Ask: “Where does this model stop being believable?”
• Practice: match three stories to linear / quadratic / piecewise.

What Not to Do

• Do not default to linear for every table.
• Do not ignore piecewise when the story clearly switches rules.
• Do not skip domain/range limits (time ≥ 0, round money, etc.).
• Do not treat “the calculator suggested quadratic” as a substitute for CED clues.