Integrated Math 1 / Unit 3: Functions
3.5 Average Rate of Change
Notes
The average rate of change of $f$ from $x = a$ to $x = b$ is $\displaystyle \frac{f(b) - f(a)}{b - a}$. It is the slope of the line through the two points.
- Include units: output units per input unit (like dollars per year).
- A negative rate means the output is decreasing.
- Example: $f(x) = x^2$ from $x = 1$ to $x = 3$: $\displaystyle \frac{9 - 1}{3 - 1} = 4$.
- Do not divide by the number of data points; divide by the change in input.
Practice
Find the average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 3$.
$\displaystyle \frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4$.
A table shows distance $d$ = 12 miles at time $t$ = 0 hours and $d$ = 40 miles at $t$ = 4 hours. What is the average rate of change?
$\displaystyle \frac{40 - 12}{4 - 0} = \frac{28}{4} = 7$.
A town's population was 5000 in 2020 and 4400 in 2023. What is the average rate of change per year?
$\displaystyle \frac{4400 - 5000}{2023 - 2020} = \frac{-600}{3} = -200$.
Ready for more? Take the Unit 3 test.