Average Rate of Change
Aligned to F-IF.B.6 — Common Core State Standards for Mathematics.
What this lesson teaches
The average rate of change of a function over an interval [a, b] is the change in output divided by the change in input: (f(b) − f(a)) / (b − a). It measures how fast the quantity changes on average.
Worked examples
Example 1
For f(x) = x^2, find the average rate of change from x = 1 to x = 4. Compute (f(4) − f(1)) / (4 − 1) = (16 − 1) / 3 = 15 / 3 = 5.
Example 2
Average rate of change is the slope of the line through two points on the curve, not the curve's steepness at a point.
Example 3
For a linear function the average rate of change is the same over every interval. That is what makes it linear.
Practice questions
- Find the average rate of change of f(x) = 3x + 2 from x = 0 to x = 5.
- A table shows f(2) = 7 and f(6) = 19. Find the average rate of change.
- For f(x) = x^2, find the average rate of change from x = 2 to x = 3.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Tasa Promedio de Cambio
La tasa promedio de cambio de una función en un intervalo [a, b] es el cambio en la salida dividido entre el cambio en la entrada: (f(b) − f(a)) / (b − a). Mide qué tan rápido cambia la cantidad en promedio.
Ejemplo 1: Para f(x) = x^2, halla la tasa promedio de cambio de x = 1 a x = 4. Calcula (f(4) − f(1)) / (4 − 1) = (16 − 1) / 3 = 15 / 3 = 5.
Ejemplo 2: La tasa de cambio promedio es la pendiente de la recta que une dos puntos de la curva, no la inclinación de la curva en un punto.
Ejemplo 3: Para una función lineal la tasa de cambio promedio es la misma en todo intervalo. Eso es lo que la hace lineal.
More Functions lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required