Mathfolis

Rates of Change in Linear and Quadratic Functions

Unit 1 · Polynomial and Rational Functions

What AP Precalc asks here

Linear and quadratic functions have distinctive rate-of-change patterns. A linear function has constant first differences over equally spaced inputs, and a constant AROC. A quadratic has constant second differences, and its AROC over [p, p + h] is linear in p — secant slopes change at a constant rate across the parabola. The College Board uses this to test family identification from tables and to set up the bridge to AP Calc concepts.

Linear

Form
f(x)=mx+bf(x) = mx + b
AROC
AROC of f on [a,b]=m\text{AROC of } f \text{ on } [a, b] = m

Quadratic

Form
f(x)=ax2+bx+cf(x) = ax^2 + bx + c
AROC over [p, p + h]
a(2p+h)+ba(2p + h) + b
Second difference (with Δ\Delta x = 1)
Δ2y=2a\Delta^2 y = 2a
Caution: The difference test only works when inputs are equally spaced. If the table jumps by uneven amounts, compute AROCs over each interval instead.
Type 1

Identify family from differences

Compute first differences. If constant, f is linear. If not constant, compute second differences. If those are constant, f is quadratic.

Example 1
Given the table f(0) = 5, f(1) = 6, f(2) = 9, f(3) = 14, f(4) = 21, identify f's family. (A) Linear (B) Quadratic (C) Exponential (D) Cannot be determined

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Type 2

AROC of a quadratic over intervals

Compute AROC of a quadratic over a sequence of equally spaced unit intervals. The resulting AROCs form a linear (arithmetic) progression — the quadratic AROC formula a(2p + h) + b shows why.

Example 2
For f(x)=x2f(x) = x^2, compute the average rate of change over [2,3][2, 3]. (A) 33 (B) 55 (C) 66 (D) 99

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Type 3

Build a linear or quadratic from data

Set up a small system with the unknown coefficients and solve. For a quadratic, three points give three equations in a, b, c.

Example 3
A quadratic g passes through (0,3)(0, 3), (1,5)(1, 5), and (2,11)(2, 11). Find g. (A) g(x)=2x2+3g(x) = 2x^2 + 3 (B) g(x)=x2+x+3g(x) = x^2 + x + 3 (C) g(x)=3x2+3g(x) = 3x^2 + 3 (D) g(x)=2x2+x+3g(x) = 2x^2 + x + 3

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Rates of Change in Linear and Quadratic Functions | AP Precalculus — Mathfolis