Modeling with Functions
Build mathematical models for real-world relationships by identifying which function family fits the data or context. Use regression and optimization to interpret models and draw conclusions.
A lesson of Pre-Calculus, in Functions and Their Graphs, about 10 minutes.
By the end of this lesson, you will be able to build mathematical models from verbal descriptions and geometric constraints, optimize using quadratic and other functions, and interpret model outputs in real-world context.
The Big Idea
Mathematical modeling is the art of translating a real-world scenario into an equation, solving the math, and interpreting the result. The hardest part is usually the setup — not the algebra. Pre-calculus modeling problems are templates for the optimization problems that dominate calculus: maximize area, minimize cost, find the optimal configuration.
The Modeling Framework
- Read carefully — what are you asked to find?
- Define variables — what are the independent and dependent quantities?
- Write the objective function — express what you're maximizing/minimizing
- Write constraint equations — express the fixed conditions
- Reduce to one variable — substitute constraints into the objective
- Optimize — find the vertex (quadratic), or critical points
- Answer the question — with units, in context
Example 1 — Maximize Area (Classic)
A 100-foot fence will enclose three sides of a rectangular garden (the wall forms the fourth side). What dimensions maximize the area?
Let $x$ = width (two sides); $L$ = length.
Constraint: $2x + L = 100 \implies L = 100 - 2x$
Objective: $A(x) = x(100 - 2x) = -2x^2 + 100x$
Vertex: $x = \dfrac{100}{4} = 25$ ft; $L = 50$ ft; $A = 1250$ ft²
Example 2 — Minimizing the Sum
The product of two positive numbers is 144. Minimize their sum.
$xy = 144 \implies y = 144/x$
$S(x) = x + 144/x$
Vertex (this is not a quadratic — use reasoning): minimum of $S$ occurs where the two terms are equal:
$x = 144/x \implies x^2 = 144 \implies x = 12$
Both numbers = 12; sum = 24. (AM-GM inequality: min sum given fixed product is when numbers are equal.)
Example 3 — Geometry + Algebra
An open box is made from an $12 \times 12$ square of cardboard by cutting equal squares from the corners and folding up the sides. What cut size $x$ maximizes volume?
After cutting: length = width = $12 - 2x$; height = $x$
$V(x) = x(12-2x)^2 = x(144 - 48x + 4x^2) = 4x^3 - 48x^2 + 144x$
Domain: $0 < x < 6$
$V'(x) = 12x^2 - 96x + 144 = 12(x^2 - 8x + 12) = 12(x-2)(x-6)$
Critical points: $x = 2$ (in domain) and $x = 6$ (endpoint)