Average Rate of Change
The slope of the secant line between two points.
About this topic
The average rate of change of a function over an interval is the change in output divided by the change in input: (f(b) − f(a))/(b − a). It is the slope of the secant line joining two points on the graph.
It tells you how fast something changed on average, such as the average speed over a trip. For linear functions it is constant, while for curves it depends on the interval. It leads naturally to the instantaneous rate of change, the derivative.
Worked example: a car's position is 40 km at t = 1 hour and 130 km at t = 3 hours. The average rate of change is (130 − 40)/(3 − 1) = 45 km per hour. For f(x) = x² between x = 1 and x = 4, it is (16 − 1)/3 = 5, the slope of the secant line. Between x = 1 and x = 2 it is only 3, so the curve gets steeper.
Average rates of change describe population growth per year, the cooling of a drink per minute, or a stock price gain per day. Do not confuse the average rate with the rate at a single instant: the car may have gone faster and slower during the trip. Be careful with subtraction order, which must be the same in numerator and denominator, and include units such as dollars per month, which give the answer meaning. Shrinking the interval toward zero leads to the derivative.
Key ideas
- (f(b) − f(a))/(b − a)
- Constant for lines, varies for curves
- Leads to the derivative
Quick check
Use cases: where Average Rate of Change helps
- Homework: review the key idea behind Average Rate of Change before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Average Rate of Change without notes.
- Visual learning: play with the interactive lab to see how Average Rate of Change changes when you adjust its inputs.
- Teaching a friend: explain Average Rate of Change in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Average Rate of Change?
The slope of the secant line between two points.
What will I learn in Average Rate of Change?
(f(b) − f(a))/(b − a). Constant for lines, varies for curves. Leads to the derivative.
Is Average Rate of Change part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Average Rate of Change lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Average Rate of Change?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.