1. Objectives and prerequisites

Prerequisite: evaluate a function and read two points from its graph. You will learn to calculate a rate of change, interpret it as a secant slope, keep track of units, and compare intervals.

2. Definition: output change per input change

For two distinct inputs a and b, the rate of change of f from a to b is (f(b)-f(a))/(b-a). The numerator Δy=f(b)-f(a) is the output change; the denominator Δx=b-a is the input change. We require b≠a to avoid division by zero.

3. Geometric meaning: secant slope

Points A=(a,f(a)) and B=(b,f(b)) lie on the graph. The line through A and B is a secant. Its slope is exactly Δy/Δx, the rate of change. A positive rate gives an upward secant, a negative rate a downward secant, and zero a horizontal secant.

4. Units and representations

The units are output units divided by input units: km/h, °C/h, €/unit, and so on. The rate can be calculated from a formula or a table and estimated from a graph by reading two points. Graph readings may be rounded, so they can be less precise than calculations from exact values.

5. Worked example

For f(x)=x² from 1 to 3, f(1)=1 and f(3)=9. Thus Δy=8, Δx=2, and the rate of change is 8/2=4. Geometrically, the secant through (1,1) and (3,9) has slope 4.

6. Practice

6.1. Calculate with a square function

For f(x)=x², calculate the rate of change from x=1 to x=4. Show f(1), f(4), Δx, Δy, and then the quotient Δy/Δx.

Guided correction. f(1)=1 and f(4)=16; Δy=15 and Δx=3; the rate is 5.

6.2. Compare a temperature change

A temperature is 12 °C at 8:00, 18 °C at 11:00, and 20 °C at 15:00. Calculate the average rate of change over 8:00–11:00 and 11:00–15:00 in °C/h. Compare the intervals and explain why the averages do not prove that temperature changed at a constant rate.

Success criteria. Both quotients use the correct elapsed times, units are explicit, and the conclusion distinguishes an interval average from instantaneous change.

6.3. Compare two intervals without copying a formula

For g(x)=x²+1, calculate the rate of change on [-1,2] and on [2,3]. Compare the results and use the two secant lines to explain what the difference says about how g changes.

Success criteria. The justification rebuilds both quotients from g-values, compares the average slopes correctly, and does not call them derivatives.

7. Summary

A rate of change is a quotient of changes: Δy/Δx. It describes average change between two distinct inputs, equals the slope of the corresponding secant line, and carries meaningful units.

8. Curriculum sources

• French Ministry of Education — 2026 mathematics specialty curriculum for Première générale, annex p. 360, “Dérivation”: official bulletin.
• California Department of Education — F-IF.6, calculate and interpret average rate of change: California Content Standards.

9. Revision note

Lesson prepared on 25 September 2026 from two official references from different countries. Numerical examples were recomputed; text and translations were prepared with AI. No measured learning gain is claimed.