AP Precalc 1.3 Linear & Quadratic RatesAP Precalc 1.3 Linear & Quadratic Rates

A P P R E C A L C U L U S · UNIT1

Rates of Change in Linear and Quadratic

Functions

Topic 1.3 Study Guide

In one sentence: linear functions have a constant rate of change, and quadratics have a rate of change that changes at a constant rate.

Linear functions

AROC = slope, on every interval, no matter the length.

The rate of change of the AROCs over equal-length intervals is 0.

y = −7x + 10 on any interval: AROC = −7.

Quadratic functions

AROC is not constant.

Over consecutive equal-length intervals, the AROCs change by a constant amount. This is the signature of a quadratic.

For f(x) = ax² + bx + c with interval length h, the AROC changes by 2ah each step. For h = 1 that is 2a. Example. f(x) = x² + 3x on unit intervals: AROC on [0,1] = 4, [1,2] = 6, [2,3] = 8. It increases by 2 each time (2a = 2).

Extending a pattern

AROC is 1 on [−3, 0] and 5 on [0, 3]. Intervals have length 3 and the AROC rose by 4, so on [3, 6] it is 5 + 4 = 9. To go backward, subtract the constant change instead.

Concavity from a table

Compute the AROC over equal-length intervals and see which way it moves.

AROCs are… Concavity

increasing Concave up

decreasing Concave down

AP Precalculus Unit 1 · 1.3 1 of 2

Example. x: 5, 10, 15, 20, 25 and h(x): 100, 75, 40, 0, −50. AROCs: −5, −7, −8, −10 (decreasing). Justify: concave down because the rate of change is decreasing over equal-length intervals. Traps. (1) Only compare AROCs over equal-length intervals. (2) "Increasing function" and "increasing AROC" are different ideas. (3) Negative AROCs getting more negative means decreasing AROC, so concave down.

Test-prep pattern

If p(x) = g(x + 1) − g(x) and p(x) = 2 for all x, then g changes by a constant 2 per unit. So g is linear with positive slope: it is always increasing, but it is not concave up.

Quick check

1. Rate of change of the AROCs for y = 13x − 10? 0

2. f(x) = 3x² + 2x, unit intervals: AROCs change by? 6

3. Quadratic with AROC 2 on [0,5] and 6 on [5,10]: AROC on [10,15]? 10

AP Precalculus Unit 1 · 1.3 2 of 2