Function Model Selection and Assumption Articulation

What AP Precalc asks here

Given a dataset (table) or a contextual description, choose the appropriate function family. Use the finite-difference test on equally spaced inputs: constant first differences → linear; constant second differences → quadratic; constant ratios of successive outputs → exponential. For non-equally-spaced data, work with AROCs instead of raw differences. AP free-response items also require you to articulate the assumptions of validity (domain, units, range of meaningful inputs).

Compute first differences. If they are constant, the data is linear. If not, compute second differences. If those are constant, the data is quadratic.

Compute the ratio of successive outputs. A constant ratio means exponential growth/decay; the ratio is the growth factor per unit input.

Match the verbal cue to a function family: constant rate → linear; constant acceleration → quadratic; percent change per unit time → exponential.