Mathfolis

Function Model Construction and Application

Unit 1 · Polynomial and Rational Functions

What AP Precalc asks here

After selecting a family (Topic 1.13), construct the specific model: pick a parameterization, use the data to set up equations for the unknown coefficients, and solve. Then use the model to predict at a new input, invert it for a target output, or interpret its parameters in context (with units). The College Board's FRQ rubrics deduct for missing interpretation and missing domain-of-validity statements.

Parameter counts (constraints needed)

Linear
y=mx+b(2 parameters, 2 constraints)y = mx + b \quad\text{(2 parameters, 2 constraints)}
Quadratic
y=ax2+bx+c(3 parameters, 3 constraints)y = ax^2 + bx + c \quad\text{(3 parameters, 3 constraints)}
Quadratic (vertex form)
y=a(x−h)2+k(3 parameters)y = a(x - h)^2 + k \quad\text{(3 parameters)}
AP Tip: Always state the units and a domain of validity for a model. A linear heart-rate model that predicts 550 bpm at 60 minutes is mathematically valid but physiologically meaningless.
Type 1

Build a linear model from data

Use two points (or one point + slope) to set up y=mx+by = mx + b. Solve for mm and bb, then apply.

Example 1
A taxi charges $3.00 base fare plus a per-mile rate. After 4 miles the total is $13.00. What is the per-mile rate? (A) $2.00 (B) $2.25 (C) $2.50 (D) $3.25

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Type 2

Build a quadratic from three constraints

Use vertex form when you know the vertex (plus one other point), or solve a 3x3 system when you have three general points.

Example 2
Build a quadratic with vertex at (3,−1)(3, -1) and passing through (5,7)(5, 7). (A) f(x)=(x−3)2−1f(x) = (x - 3)^2 - 1 (B) f(x)=2(x−3)2−1f(x) = 2(x - 3)^2 - 1 (C) f(x)=2(x+3)2−1f(x) = 2(x + 3)^2 - 1 (D) f(x)=(x−3)2+7f(x) = (x - 3)^2 + 7

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Type 3

Build a rational model from features

Place the zero(s) in the numerator and the vertical asymptote(s) in the denominator. Match degrees and leading coefficients to hit the requested horizontal asymptote.

Example 3
Build a rational function with vertical asymptote (VA) at x=3x = 3, horizontal asymptote (HA) y=2y = 2, and zero at x=−1x = -1. (A) r(x)=x+1x−3r(x) = \dfrac{x + 1}{x - 3} (B) r(x)=2(x+1)x−3r(x) = \dfrac{2(x + 1)}{x - 3} (C) r(x)=2(x−1)x+3r(x) = \dfrac{2(x - 1)}{x + 3} (D) r(x)=(x+1)2(x−3)r(x) = \dfrac{(x + 1)}{2(x - 3)}

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