AP PRECALCULUS — UNIT 1B · POLYNOMIAL & RATIONAL FUNCTIONS
1.14 — Function Model Construction and Application
Notes — Building and Using Polynomial / Rational Models
💡 Learning Objectives
By the end of this lesson you will be able to (AP CED 1.14.A):
- Construct a polynomial or rational function model from a verbal description or data.
- Apply the model to answer questions: compute outputs, solve equations, describe end behavior.
- Interpret the model's predictions in the context of the problem.
- Identify any limits of the model and reasonable domain restrictions.
1. The Four-Step Modeling Process
Building a usable function model from scratch follows four steps:
- IDENTIFY the variables. Decide which quantity is the independent input and which is the dependent output. Assign clear letter names and units.
- RELATE the variables using the geometry or physics of the situation. Write an equation expressing the output in terms of the input.
- RESTRICT the domain based on context. Negative lengths, nonsense values, or unreachable inputs should be excluded.
- APPLY the model to answer the question. Compute specific values, find extrema, describe long-run behavior, or validate by plugging in known points.
2. A Classic Optimization Example
📘 Example
Example 1 — Open-Top Box Problem
A rectangular piece of cardboard measures 12 in by 8 in. Squares of side x are cut from each corner, and the sides are folded up to form an open-topped box. Express the volume V as a function of x, state the domain, and find (approximately) the x that maximizes V.
Identify: input = x (side length of cut square, inches); output = V (volume, cubic inches).
Relate: after cutting, the base is (12 − 2x) by (8 − 2x), and the height is x.
V(x) = x · (12 − 2x) · (8 − 2x).
Restrict: we need x > 0 (positive cut) and 8 − 2x > 0 (shorter side must have positive length), so 0 < x < 4.
Apply: expand V(x) = x(96 − 40x + 4x²) = 96x − 40x² + 4x³. A graphing utility shows the maximum near x ≈ 1.57 in, giving V ≈ 67.6 in³.
Left: the cardboard with square cuts. Right: the resulting volume function V(x) = x(12 − 2x)(8 − 2x) on the domain 0 < x < 4.
⚠️ Common Mistake
Forgetting the domain restriction is a top-cited error on AP modeling FRQs. Always ask: what values of x make physical sense? The domain is part of the model's definition — not an afterthought.
3. Rational Function Models
Rational models appear when a fixed cost is being spread over a variable quantity (average cost), when a quantity saturates toward an equilibrium (concentration), or when a denominator represents a limiting resource.
📘 Example
Example 2 — Average cost model
A company has fixed daily costs of $2000 and a variable cost of $15 per unit produced. Express the average cost per unit as a function of x, the number of units produced.
Total cost: TC(x) = 2000 + 15x.
Average cost: AC(x) = TC(x) / x = (2000 + 15x) / x = 2000/x + 15, valid for x > 0.
Apply: as x → ∞, AC(x) → 15. The horizontal asymptote y = 15 is the long-run steady-state average cost per unit.
💡 Quick Check
Quick model: a pollutant's concentration drops by half every 4 hours. What function family captures this? What if the pollutant were diluted by an ever-growing volume of clean water at a constant rate?
- Half-every-4-hours = exponential decay (Unit 2).
- Diluted by growing clean-water volume = rational function of time (denominator grows).
4. Using Models to Answer Questions
Once the model is built, use it to answer the actual question. Common question types on AP FRQs:
- Evaluate: plug in a value and interpret the output with units.
- Solve: find the input that gives a specific output (may need factoring or technology).
- Extrema: find the input that maximizes or minimizes the output (technology allowed).
- Long-run behavior: describe limits at infinity and interpret the meaning.
- Rate of change: describe how outputs change as inputs change (average rate over an interval).
🎯 AP Tip
Modeling FRQs frequently pair with context-heavy interpretation. Every numerical answer must be followed by a SENTENCE of interpretation with UNITS. 'V(2) = 48 cubic inches' is correct math; the AP response also needs 'so a cut of 2 inches produces a box of volume 48 cubic inches.'
📘 Try This
Build a quick model: a rectangular garden with 40 ft of fencing, one side against a house (not fenced).
- Let the width (perpendicular to house) be w; length = 40 − 2w.
- Area: A(w) = w(40 − 2w) = 40w − 2w².
- Domain: 0 < w < 20.
- Maximum area: vertex of parabola at w = 10, giving A(10) = 200 sq ft.
5. Summary
- Four steps: IDENTIFY, RELATE, RESTRICT, APPLY.
- Polynomial models often arise from geometric (area, volume) setups.
- Rational models often arise from average-cost, concentration, or saturation settings.
- Always state the domain, always interpret answers with units, always double-check the model against one known data point.