Precalculus · CED Unit 1: Polynomial and Rational Functions · 14 min read · Updated 2026-05-11
Rates of change (average and over equal intervals) — AP Precalculus
AP Precalculus · CED Unit 1: Polynomial and Rational Functions · 14 min read
1. Core Definition of Average Rate of Change★☆☆☆☆⏱ 3 min
Average rate of change measures how much a function changes per unit input across a specified interval, unlike instantaneous rate of change which describes change at a single point. Geometrically, it equals the slope of the secant line connecting two points on the function's graph, and is a foundational skill for nearly all AP Precalculus topics.
Exam tip: Reversing the order of subtraction for both numerator and denominator gives the same result, but always match the order: left-to-right for both to avoid sign errors.
2. Average Rate of Change Over Equal Intervals★★☆☆☆⏱ 3 min
Equal intervals are consecutive subintervals with identical width $h = \Delta x$. When working with equal intervals, we can use a time-saving shortcut for comparing the magnitude of average rates: since $h$ is constant, $\frac{f(x+h)-f(x)}{h}$ is directly proportional to $f(x+h)-f(x)$, so you can compare differences directly without dividing by $h$. This shortcut only works for equal intervals, never for unequal widths.
Exam tip: If an FRQ asks for explicit average rate values, always show the full division by $\Delta x$, even when $h=1$.
3. Higher-Order Differences and the Constant nth Difference Property★★★☆☆⏱ 5 min
For polynomials over equal intervals, we can use finite differences to find the degree of an unknown polynomial from tabular data, a skill unique to AP Precalculus. A first difference $\Delta^1 f(x) = f(x+h) - f(x)$ is the difference between consecutive function values. A second difference is the difference between consecutive first differences, and this pattern continues for higher-order differences.
Exam tip: You need at least two identical nth differences to confirm they are constant, which requires at least $n+2$ total points to confirm a degree $n$ polynomial.
4. Concept Check: Exam-Style Practice★★☆☆☆⏱ 3 min
Common Pitfalls
Why: Mismatched order of subtraction leads to sign errors
Why: A single value cannot confirm the difference is constant across all intervals
Why: The shortcut of skipping division only works for equal interval widths
Why: Average rate over the whole interval does not describe behavior at every point
Why: The formula for unit intervals is memorized, but the $h^n$ term is required for non-unit interval widths