AP PRECALCULUS — UNIT 1B · POLYNOMIAL & RATIONAL FUNCTIONS
1.13 — Function Model Selection and Assumption Articulation
Notes — Matching Data Shape to a Function Family
💡 Learning Objectives By the end of this lesson you will be able to (AP CED 1.13.A, 1.13.B):
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1. The Model-Selection Question
When faced with real-world data or a described scenario, the modeler must choose WHICH kind of function best captures the behavior. Each family of functions has a characteristic shape and a characteristic rate of change. The choice of model is a judgment call — different families may fit the data roughly equally well — and it must be justified, not merely stated.
2. Shape Signatures of Common Function Families
Each family has a distinctive signature. Use these to match data to model:
- Linear (degree 1): constant rate of change. First differences of evenly-spaced data are constant.
- Quadratic (degree 2): roughly symmetric around a single max or min. Second differences are constant.
- Cubic (degree 3) and higher polynomials: multiple turning points, alternating concavity.
- Rational: presence of vertical or horizontal asymptotes, long-run steady state, or singular behavior near a specific input.
- Piecewise: distinct regimes of behavior — the data splits naturally into regions with different shapes.
Matching data shape to function family. The same four data sets could be described verbally ('rises steadily', 'peaks then falls', …) — those verbal cues should match your model choice.
💡 Quick Check Which function family would you use for each scenario?
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3. Geometric Reasoning
Geometric contexts often suggest a model directly. Lengths give linear. Areas give quadratic (two dimensions). Volumes give cubic (three dimensions). If a problem says 'the cross-section has side x,' think about whether the quantity scales linearly, quadratically, or cubically with x.
4. Constant nth Differences → Polynomial of Degree n
For evenly-spaced inputs, take first differences (consecutive y-values subtracted). If those are constant, the data is linear. If not, take second differences — if constant, quadratic. Continue until the nth differences are constant; the underlying model has degree n.
📘 Example Example 1 — Reading degree from a table The table (x, y) has values (0, 1), (1, 3), (2, 9), (3, 21), (4, 41). Which polynomial degree fits? First differences: 2, 6, 12, 20. Not constant. Second differences: 4, 6, 8. Not constant. Third differences: 2, 2. CONSTANT. So the model is degree 3 (cubic). |
5. Articulating Assumptions
No model is correct without its assumptions. A model implicitly assumes certain things stay constant, that quantities change together in a specific way, and that the data lies within a valid range. Stating these assumptions is part of communicating a model on the AP Exam.
💡 Definitions Assumption categories to articulate:
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📘 Example Example 2 — Articulating assumptions A company models its profit by P(x) = 40x − 2000 dollars, where x is the number of units sold. Linear model assumes: (a) each unit brings a constant profit of $40, (b) fixed costs are $2000 regardless of x, (c) x is a nonnegative integer (can't sell negative units). Potential failure points: large-volume discounts would break the linear assumption; capacity limits would truncate the model's domain. |
🎯 AP Tip On an AP modeling FRQ, writing 'this is linear because the rate is constant' is half the answer. The other half is stating at least one assumption and one restriction that the model relies on. Name BOTH to earn full credit. |
📘 Try This Decide the most appropriate model family for each, and state one assumption.
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6. Summary
- Match shape to family: linear, quadratic, polynomial (higher degree), rational, piecewise.
- Constant nth differences of evenly-spaced data ⇒ polynomial of degree n.
- Geometric contexts often dictate degree: lengths → 1, areas → 2, volumes → 3.
- Every model carries assumptions and restrictions — state them explicitly.