A situation shows a constant rate of change when one quantity increases or decreases by the same amount every time the other quantity changes by one unit. Driving at a steady 60 miles per hour, earning $15 for every hour worked, or filling a tank at two-thirds of a liter per second are all constant rate of change situations. The defining feature is that the relationship between the two quantities forms a straight line when graphed.
What Makes a Rate of Change “Constant”
A rate of change compares how much one quantity (the output) changes relative to another quantity (the input). It becomes constant when that ratio stays the same no matter which two points you pick. If you earn $15 per hour, the rate of change between hour 1 and hour 3 is the same as between hour 5 and hour 10. You can check this with the formula: (y₂ − y₁) / (x₂ − x₁). When every pair of points gives the same result, the rate of change is constant.
This constant ratio is also called the slope. In the equation y = mx + b, the value of m is your constant rate of change. Each time x increases by 1, y increases by exactly m units, every single time, with no exceptions.
Situations That Show a Constant Rate of Change
The easiest way to recognize these situations is to ask: does the same input always produce the same increase in output? Here are common examples.
- Steady driving speed. A van traveling at 60 miles per hour covers 60 miles for every hour that passes. Its position is a linear function of time: distance = 60t + starting position. The slope of that line, 60, is the constant velocity.
- Hourly wages. A worker earning $18 per hour makes $18 for the first hour, $18 for the second, and $18 for every hour after that. Total pay = 18 × hours worked.
- Filling a container at a fixed flow rate. A tank receiving two-thirds of a liter every second has its volume described by V = (2/3)t. After 3 seconds you have 2 liters, after 6 seconds you have 4 liters, and the ratio never changes.
- Flat-rate pricing. A phone plan charging $0.10 per text message adds the same cost for every additional message, making total cost a linear function of texts sent.
- Walking at a steady pace. Someone walking 1/20 of a kilometer per minute covers the same distance in every one-minute interval. The unit rate, 1/20 km per minute, is the constant rate of change.
Situations That Do Not Show a Constant Rate
Recognizing what isn’t constant helps just as much. A ball thrown into the air speeds up and slows down because gravity changes its velocity over time. The distance it covers each second is different from the last, so the rate of change varies. Compound interest works the same way: your balance grows by a percentage of itself, meaning the actual dollar increase gets larger each period. On a graph, both of these produce curves rather than straight lines.
If you calculate (y₂ − y₁) / (x₂ − x₁) for two different pairs of points and get different numbers, the rate of change is not constant.
How to Spot It on a Graph
A constant rate of change always produces a straight line. The slope of that line is the rate. A steeper line means a larger rate of change, and a line going downward means the output is decreasing at a constant rate. If the graph curves at any point, the rate of change is varying.
On a position vs. time graph, a straight line means constant velocity. On a velocity vs. time graph, a straight line means constant acceleration. The key principle is the same in both cases: a straight line equals a constant slope.
How to Spot It in a Table
When you’re given a table of values, check whether the output changes by the same amount for each equal step in the input. For example:
- x = 0, y = 5
- x = 1, y = 8
- x = 2, y = 11
- x = 3, y = 14
Every time x goes up by 1, y goes up by 3. That’s a constant rate of change of 3. You could write the equation as y = 3x + 5. Now compare that to a table where y increases by 2, then 4, then 8. Those unequal jumps signal a varying rate.
How to Spot It in an Equation
Any equation in the form y = mx + b represents a constant rate of change. The number multiplied by x (the coefficient m) is the rate. It doesn’t matter what b is; b just shifts the starting point. So y = 4x + 10 has a constant rate of 4, and y = −2x + 7 has a constant rate of −2, meaning the output drops by 2 for every 1-unit increase in x.
If the equation contains x² , x³, or x in a denominator or exponent, the rate of change is not constant. Those terms create curves, which means the slope is different at every point.
Unit Rate, Slope, and Constant of Proportionality
You may see different names for the same idea depending on the context. In a word problem about speed or price, the constant rate of change is usually called the unit rate: miles per hour, dollars per item, liters per second. On a graph, it’s called the slope. In a proportional relationship like y = kx (where b = 0), it’s called the constant of proportionality.
There’s one subtle distinction worth knowing. In a proportional relationship, the two quantities themselves are proportional, and the line passes through the origin. In a non-proportional linear relationship like y = 3x + 5, the changes are still proportional to each other (y always increases by 3 when x increases by 1), but the quantities aren’t proportional because the line doesn’t pass through zero. Both situations have a constant rate of change. The difference is only in whether there’s a starting value other than zero.