Key takeaway: The rate of change compares how much one quantity changes to how much another changes, and for a linear function it is constant and equals the slope.
Grade 88.F.B.4Functions and Graphs
Definition of Rate of Change
The rate of change measures how fast an output grows or shrinks as the input moves. Pay per hour, miles per gallon and gallons drained per minute are all examples. It always comes with units: the output’s unit divided by the input’s unit.
Read it from a table by dividing a change in the output by the matching change in the input, using any two rows. If every pair of rows gives the same answer, the relationship is linear and the value is constant. A positive result means the output increases, a negative one means it decreases. On a graph, this ratio is the slope of the line.
Formula and notation
Using the variables \(x\) (input) and \(y\) (output):
\[\text{rate of change} = \frac{\Delta y}{\Delta x} = \frac{y_2 – y_1}{x_2 – x_1}\]
The symbol \(\Delta\) means “change in.” State units, such as gallons per minute.
Key facts
- A constant rate means the graph is a straight line.
- A negative value signals that the output falls as the input grows.
- The units of the answer are output units per input unit.
- A table that does not give equal ratios shows a nonlinear change.
Rate of Change example
A water tank at a car wash holds 90 gallons at 0 minutes, 78 gallons at 4 minutes, 66 at 8 minutes and 54 at 12 minutes. Find the rate of change and decide whether it is constant.
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From 0 to 4 minutes: \(\dfrac{78 – 90}{4 – 0} = \dfrac{-12}{4} = -3\).
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From 4 to 8 minutes: \(\dfrac{66 – 78}{8 – 4} = \dfrac{-12}{4} = -3\).
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From 8 to 12 minutes: \(\dfrac{54 – 66}{12 – 8} = -3\).
The rate is constant at -3 gallons per minute: the tank loses 3 gallons every minute.
Common mistake
Watch out
Tess says the tank drains at 12 gallons per minute, because 90 minus 78 is 12.
Why it is wrong
That 12 gallons took 4 minutes, so it is a change, not a rate. A rate divides by the time: \(\frac{-12}{4} = -3\) gallons per minute. Always divide the change in the output by the change in the input, and attach the units.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Functions and Graphs