Algebra
Relations and functions
Average rate of change
20 practice questions
2 video lessons
Theory + worked examples
Theory
The average rate of change of a function between \(x=a\) and \(x=b\) is:
\[\dfrac{f(b)-f(a)}{b-a},\]
the slope of the secant line joining the two points.
For a straight line the rate is constant — it is the slope.
The rate is the slope of the secant line.
The average-rate-of-change formula.
Average rate of change:
\[\dfrac{\Delta y}{\Delta x}=\dfrac{f(b)-f(a)}{b-a}\]
It has units of \(y\)-units per \(x\)-unit.
How to find the average rate of change
- Find \(f(a)\) and \(f(b)\).
- Compute the change in \(y\): \(f(b)-f(a)\).
- Divide by the change in \(x\): \(b-a\).
- Attach units if in context.
Example 1 — From a function
For \(f(x)=x^2\), find the average rate of change from \(x=1\) to \(x=4\).
Solution
Use \(\dfrac{f(4)-f(1)}{4-1}\).
| \(=\) | \(\dfrac{16-1}{4-1}\) | |
| \(=\) | \(\dfrac{15}{3}=5\) |
Example 2 — From a table
A table gives \(f(2)=7\) and \(f(6)=19\). Find the average rate of change.
Solution
Divide the change in \(y\) by the change in \(x\).
| \(\dfrac{19-7}{6-2}\) | \(=\) | \(\dfrac{12}{4}=3\) |
Example 3 — Linear function
What is the average rate of change of \(y=2x+1\)?
Solution
For a line it equals the slope everywhere.
| \(\text{rate}\) | \(=\) | \(2\) |
Example 4 — With units
Distance goes from \(10\) m to \(70\) m in \(3\) s. Find the average speed.
Solution
Rate of change of distance is speed.
| \(\dfrac{70-10}{3}\) | \(=\) | \(20\ \text{m/s}\) |
Common pitfalls
Subtract in the same order on top and bottom.
It is change in \(y\) over change in \(x\), not the reverse.
For a curve it varies; for a line it is constant.
Frequently asked questions
What is the average rate of change?
The change in \(y\) divided by the change in \(x\) between two points.
How does it relate to slope?
It is the slope of the secant line joining the two points.
What is it for a straight line?
Constant — equal to the line's slope.
Does it have units?
Yes — \(y\)-units per \(x\)-unit, like meters per second.
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