Lesson 4: Average Rate of Change of a Function

What does average rate of change represent in the context of a function?

How is average rate of change related to slope?

How can we calculate the average rate of change using function notation?

How does the average rate of change describe how a quantity changes over time?

How can we interpret the meaning of average rate of change in real-world situations?

How does average rate of change differ from instantaneous rate of change?

Average rate of change

Interval

Slope

Secant line

Change in y

Change in x

Function notation

Difference quotient

Rate of change

HSF-IF.B.6

Calculate and interpret the average rate of change of a function over a specified interval. Estimate the rate of change from a graph.

HSF-IF.B.4

Interpret key features of graphs in terms of quantities.

HSF-IF.C.7

Graph functions and show key features including intercepts and slopes.

Lesson Description

Students will learn to calculate the average rate of change of a function over a given interval using both numerical values and graphical representations.

Students will:

Evaluate functions using function notation

Compute the change in output values

Divide by the change in input values

Interpret the result as the slope of a secant line

Students will also analyze graphs to estimate rates of change and interpret them in real-world contexts such as speed, growth, or cost changes.
Purpose

The purpose of this lesson is to develop students’ understanding of how functions describe changing quantities.

This concept serves as an important bridge between:

algebraic reasoning

graphical interpretation

future calculus ideas involving derivatives and instantaneous rate of change

Understanding average rate of change also strengthens students’ ability to interpret real-world data and models.

Primary Level: DOK 2 – Skills and Concepts

Support for Developing Learners

Review slope from Algebra I before introducing rate of change.

Provide step-by-step guided examples.

Use visual graphs showing secant lines.

Use structured graphic organizers showing the formula.

On-Level Strategies

Students compute average rate of change from tables, graphs, and equations.

Pair students to interpret real-world examples.

Practice translating word problems into function notation.

Enrichment / Extension

Compare average rate of change over smaller and smaller intervals.

Explore how rates of change behave for different types of functions (linear, quadratic, polynomial).

Connect to instantaneous rate of change as a preview of calculus.

Exit slips and quiz

textbook

• Desmos Graphing Calculator

• TI-83 / TI-84 Graphing Calculators

• SmartBoard interactive graph demonstrations