Topic 1.13 Function Model Selection and Assumption Articulation: select an appropriate type of function to model a data set or context, and articulate the assumptions and limitations of the chosen model.

What this topic is asking

The College Board (Topic 1.13) wants you to select an appropriate function model for a data set or context, and to articulate the assumptions behind that choice. Selection rests on the rate-of-change behavior: constant first differences suggest linear, constant second differences suggest quadratic, a turning behavior suggests a higher polynomial, and an asymptotic approach suggests rational. You must also state what the model assumes and where it breaks down.

Matching behavior to a family

The rate-of-change tests from Topics 1.2 and 1.3 do most of the work, and the end-behavior ideas from 1.6 and 1.7 distinguish polynomial from rational when the data trail off.

Articulating assumptions

Choosing a model is a claim about the world, and that claim rests on assumptions. Common ones include: the relationship is smooth and continuous; the observed pattern continues outside the data (extrapolation); the spacing or measurement is reliable; and no unmodelled factor (a policy change, a saturation effect) disrupts the trend. Stating these explicitly is required, because a model that fits the data can still mislead if its assumptions fail.

Why articulating limitations matters

A model is a tool with a domain of validity, not a universal truth. A linear model of growth assumes the rate never changes, which fails once resources run short. A quadratic model assumes a single, symmetric turning point. A rational model assumes the asymptotic limit is real and that the process never actually reaches it. The exam rewards naming the specific assumption that the situation depends on, and the specific way the model would mislead if that assumption broke, rather than a vague "the model might be wrong".

A useful way to think about selection is to separate the question "what shape does the data have" from the question "what shape does the context demand". Sometimes the numbers look quadratic but the context (say, a quantity that cannot go negative) rules out part of the parabola, so the model must be restricted in domain. Holding both the data pattern and the contextual constraints in view, and letting the more restrictive of the two govern the model and its stated limitations, is the judgement this topic is really testing, and it sets up the explicit construction work of Topic 1.14.

Try this

Q1. Data approaches the line $y = 50$ as time grows but never reaches it. Which model fits? [1 point]

• Cue. A rational model, because of the horizontal asymptote at $y = 50$ that polynomials cannot produce.

Q2. A data set has constant first differences. Which model is appropriate, and what does it assume? [1 point]

• Cue. Linear; it assumes the rate of change stays constant across the whole range modelled.