The average rate of change over an interval is the slope of the line joining the two endpoints.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The calculation
It is the change in output over the change in input: (f(b) − f(a)) / (b − a).
It depends on the interval
A curve has no single rate of change. Ask over which interval, or the question is incomplete.
What the sign says
Positive means the function ended higher than it started. It does not mean it rose the whole way.
The slope between two points
The average rate of change of f over [a, b] is (f(b) − f(a))/(b − a) — the slope of the secant line joining those two points. For a straight line it is the same over every interval; for a curve it depends on which interval you pick.
Say what it measures
Its units are output units per input unit, so it always has a contextual reading: metres per second, dollars per unit, degrees per hour. AP questions routinely ask for that interpretation, not just the number.
Constant average rate means linear
A function whose average rate of change is the same over every interval of equal length is linear. Comparing rates across several intervals is the standard test, and a single mismatch rules linearity out.
Where it leads
Shrinking the interval towards a single point turns the average rate into the instantaneous rate — the derivative. This lesson is the last step before calculus, which is why it opens the course.
Step 2: Try It Yourself
Move the two points and watch the secant line follow.
- Point(-2, 0)
- Second point(3, 5)
- Slope between them1
Slide the second point towards the first. The slope between them approaches the slope of the curve at that point.
Step 3: Watch an Example
One step at a time.
Watch Priya Compare Two Intervals
For f(x) = x², Priya finds the average rate of change on [0, 2] and on [−2, 0].
- Step 1
On [0, 2]: f(2) − f(0) is 4 − 0, over 2 − 0.
Step 4: Your Turn
Practice makes it stick.
The Interval
Problem 1 of 2
For f(x) = x², what is the average rate of change on [1, 4]?
Reading It Off the Graph
Problem 2 of 2
The secant shown joins x = −2 and x = 3 on y = x² − 4. What is its slope?
- Point(-2, 0)
- Second point(3, 5)
- Slope between them1
a
b
c
First point
Second point
Rates Across Intervals
1 of 8
f(x) = x². Average rate of change on [2, 5]?
2 of 8
f(x) = 3x + 1. Average rate of change on [0, 10]?
3 of 8
f(x) = x². Average rate of change on [−3, 3]?
4 of 8
f(x) = x³. Average rate of change on [0, 2]?
5 of 8
Select every statement that must be true when the average rate of change on [a, b] is 0.
6 of 8
Put these steps in the order you would carry them out.
- 1Subtract to get the change in output.
- 2Subtract to get the change in input.
- 3Divide output change by input change.
- 4Work out f(a) and f(b).
7 of 8
f(x) = −x². Average rate of change on [1, 3]?
8 of 8
Does a positive average rate of change guarantee the function increased throughout the interval?
Step 5: Quick Check
Show what you know.
Question 1 of 1
f(x) = x². Average rate of change on [3, 6]?
What You Learned
- The average rate of change is the slope of the secant between the endpoints.
- It is (f(b) − f(a)) / (b − a), and it depends on the interval chosen.
- It compares endpoints only — it says nothing about what happened in between.