Rate of Change
In short: Rate of change tells how much one quantity changes for each single unit of another, found by dividing the change in the output by the change in the input.
8th GradeFunctions and Graphs8.F.B.4
What is Rate of Change?
A rate of change compares two changes: how much the output moved compared with how much the input moved. A van that goes from 110 to 285 miles in 5 hours changes position by 35 miles each hour.
The formula is \(\frac{\Delta y}{\Delta x} = \frac{y_2 – y_1}{x_2 – x_1}\). If the ratio is the same for every pair of points, the relationship is linear and the rate equals the slope. If it differs, the relationship is nonlinear. A negative result means the output shrinks as the input grows.
How to spot it
- A table lists times and measurements, and the question is how fast the measurement changes.
- The unit of the answer contains per, such as miles per hour or dollars per month.
- A graph is a line and the question wants the change in y for each step of 1 in x.
- A quantity grows or decays, like a tank level or a temperature, and a rate is requested.
The method
- Choose two data points and write the inputs and outputs separately.
- Subtract the outputs for the numerator and the inputs for the denominator, in the same order.
- Divide and attach units: output units per input unit.
- Test a different pair; if the quotient matches, the rate is constant.
Rate of Change: solved example
Problem
A delivery van is 110 miles from its depot after 2 hours and 285 miles away after 7 hours. Find the rate of change of its distance.
Solution
Change in distance: \(285 – 110 = 175\) miles. Change in time: \(7 – 2 = 5\) hours.
Divide: \(\frac{175}{5} = 35\). The distance changes at 35 miles per hour.
Try it
Question 1
A tank holds 12 gallons at the 3-minute mark and 44 gallons at the 11-minute mark. Find the rate of change.
Show the answer
\(\frac{44 – 12}{11 – 3} = \frac{32}{8} = 4\) gallons per minute.
Question 2
The temperature is \(71^{\circ}\) F at 4 pm and \(53^{\circ}\) F at 10 pm. What is the rate of change?
Show the answer
\(\frac{53 – 71}{6} = -3\) degrees per hour; it is falling.
Rate of Change vs. Slope
Rate of change is the broad idea: the change in one quantity per change in another, with units attached. Slope is the same number read on a straight-line graph, as rise over run, often without units. On a curve the rate of change differs from place to place, while slope is a single value for the whole line.
Keep going
Lessons
Worksheets and quizzes
- Slope and Rate of Change – Worksheet
- Slope and Rate of Change – Quiz
- Linear Functions as Models – Worksheet
- Linear Functions as Models – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Functions and Graphs