Mathfolis

Rates of Change

Unit 1 · Polynomial and Rational Functions

What AP Precalc asks here

The average rate of change (AROC) of a function f on the interval [a, b] is (f(b) − f(a)) / (b − a) — the slope of the secant line through (a, f(a)) and (b, f(b)). Topic 1.2 sits at the front of Unit 1 because every later concept (concavity, end behavior, transformations) leans on the idea that a function's output changes per unit input. Instantaneous rate of change is previewed only as 'the limit of AROC as the interval shrinks' — the derivative is AP Calculus territory and out of scope here.

Average rate of change

AROC of f over [a, b]
AROC=f(b)−f(a)b−a\text{AROC} = \frac{f(b) - f(a)}{b - a}
Geometric meaning
AROC=slope of the secant line through (a,f(a)) and (b,f(b))\text{AROC} = \text{slope of the secant line through } (a, f(a)) \text{ and } (b, f(b))

Sign of AROC

Positive
f(b)>f(a)  ⟹  net increase on [a,b]f(b) > f(a) \;\Longrightarrow\; \text{net increase on } [a, b]
Zero
f(a)=f(b)  ⟹  no net changef(a) = f(b) \;\Longrightarrow\; \text{no net change}
Negative
f(b)<f(a)  ⟹  net decrease on [a,b]f(b) < f(a) \;\Longrightarrow\; \text{net decrease on } [a, b]
AP Tip: AROC is just a slope. If the function is given as a table or as two points on a graph, you don't need a formula — pick the two outputs, subtract, and divide by the difference of the inputs.
Caution: Always include units in a real-world AROC: 'people per year', '°F per day'. A bare number is the most common distractor — and on free-response items, missing units cost you the point.

AP problem types

Type 1

Direct AROC computation

You are given an explicit formula for f(x) and an interval [a, b]. Evaluate f at both endpoints, subtract, and divide by (b − a). The hardest part is keeping the arithmetic clean — especially when f involves fractions, radicals, or negative inputs.

Example 1
Compute the average rate of change of f(x)=x2+1f(x) = x^2 + 1 over the interval [1,4][1, 4]. (A) 33 (B) 55 (C) 1515 (D) 1717
Example 2
Compute the average rate of change of f(x)=1xf(x) = \dfrac{1}{x} over the interval [1,4][1, 4]. (A) −14-\dfrac{1}{4} (B) −13-\dfrac{1}{3} (C) 14\dfrac{1}{4} (D) 34\dfrac{3}{4}

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Type 2

AROC from a table or graph

When the function is presented as a table of values or as two labeled points on a graph, you do not need a formula for f — just pick the two outputs at the requested inputs and apply the slope formula directly.

Example 3
The temperature TT (in °F) at noon on five consecutive days is given in the table: Day 1: 50, Day 2: 52, Day 3: 60, Day 4: 65, Day 5: 64. What is the average rate of change of TT from Day 1 to Day 5? (A) 2.82.8 °F per day (B) 3.53.5 °F per day (C) 5.05.0 °F per day (D) 1414 °F per day
Example 4
A line on the graph of ff passes through the points (2,−1)(2, -1) and (5,8)(5, 8). What is the average rate of change of ff over the interval [2,5][2, 5]? (A) 13\dfrac{1}{3} (B) 33 (C) 77 (D) 99

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Type 3

AROC in a real-world context

Translate a short scenario (population growth, cost, distance) into an AROC computation. Identify the two endpoint values and the input interval, apply the formula, and report the result with the correct units. AP Precalc free-response items deduct for missing units.

Example 5
A town's population was 8,0008{,}000 in 2010 and 13,00013{,}000 in 2020. What was the average rate of change of the population, in people per year, over that decade? (A) 250250 people per year (B) 500500 people per year (C) 1,0001{,}000 people per year (D) 5,0005{,}000 people per year
Example 6
Suppose f(2)=5f(2) = 5 and the average rate of change of ff over the interval [2,7][2, 7] is 33. What is f(7)f(7)? (A) 88 (B) 1515 (C) 2020 (D) 3535

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Rates of Change | AP Precalculus — Mathfolis