The constant rate of change is the amount one quantity changes for every one-unit increase in another quantity, and it stays the same no matter which two points you use to calculate it. You find it using the formula: (y₂ − y₁) ÷ (x₂ − x₁). If you’ve worked with slope before, this is the same thing. A relationship with a constant rate of change is always linear, meaning its graph is a straight line.
The Formula
The constant rate of change between two points is calculated as rise over run:
Rate of change = (y₂ − y₁) ÷ (x₂ − x₁)
Here, (x₁, y₁) and (x₂, y₂) are any two points in the relationship. The top of the fraction (y₂ − y₁) measures how much the output changes, and the bottom (x₂ − x₁) measures how much the input changes. What makes a rate of change “constant” is that you’ll get the exact same number no matter which pair of points you pick. If different pairs give you different results, the rate of change is variable, and the relationship isn’t linear.
Finding It from a Table
When you’re given a table of x and y values, pick any two rows and plug the numbers into the formula. For example, suppose a table shows these values:
- x = 3, y = 150
- x = 5, y = 250
- x = 7, y = 350
Using the first and third rows: (350 − 150) ÷ (7 − 3) = 200 ÷ 4 = 50. The constant rate of change is 50, meaning y increases by 50 for every 1-unit increase in x.
To confirm the rate is truly constant, check another pair. Using the first and second rows: (250 − 150) ÷ (5 − 3) = 100 ÷ 2 = 50. Same answer. If every pair of rows produces the same rate, you have a constant rate of change. If even one pair gives a different value, the relationship is not linear and doesn’t have a constant rate.
Finding It from a Graph
On a graph, a constant rate of change shows up as a straight line. To calculate it, pick any two points on the line and count the rise (vertical change) and the run (horizontal change) between them. The rate of change equals rise ÷ run.
For instance, if your two points are (2, 4) and (6, 16), the rise is 16 − 4 = 12 and the run is 6 − 2 = 4. That gives you 12 ÷ 4 = 3. The line goes up 3 units for every 1 unit it moves to the right. A line that slopes upward from left to right has a positive rate of change, and a line that slopes downward has a negative one.
If the graph is curved at all, the rate of change is not constant. Curves mean the steepness is shifting from point to point, so different pairs of points will give different slopes.
Finding It from an Equation
When a linear equation is written in slope-intercept form, y = mx + b, the constant rate of change is simply m, the coefficient in front of x. There’s no calculation needed beyond identifying that number. In y = 3x + 7, the rate of change is 3. In y = −0.5x + 12, it’s −0.5.
If the equation isn’t in that form yet, rearrange it so y is alone on one side. For example, 2y = 8x + 10 becomes y = 4x + 5 after dividing everything by 2, making the constant rate of change 4.
Finding It in Word Problems
Word problems often signal a constant rate of change with phrases like “per hour,” “per mile,” “for every,” or “at a steady rate.” Any time two quantities are described with a “per” unit, you’re looking at a rate. Kilometers per hour, dollars per item, and gallons per minute are all rates of change.
To find the rate, identify two data points the problem gives you and use the same formula. Say a car travels 120 miles in 2 hours and 300 miles in 5 hours at a constant speed. The rate of change is (300 − 120) ÷ (5 − 2) = 180 ÷ 3 = 60 miles per hour. The units come from dividing the units of the y-quantity (miles) by the units of the x-quantity (hours).
When a problem states that something changes “at a constant rate,” it’s telling you the relationship is linear. That means you only need two data points to find the rate, and it will hold true across the entire scenario.
Positive, Negative, and Zero Rates
A positive constant rate of change means y increases as x increases. A car moving forward, a bank account earning steady interest, or a plant growing at a fixed pace per week all have positive rates.
A negative constant rate of change means y decreases as x increases. A candle losing half an inch of height per hour or a phone battery draining 5% per minute are negative rates. On a graph, this looks like a line going downhill from left to right.
A zero rate of change means y doesn’t change at all regardless of x. The graph is a perfectly horizontal line. If a swimming pool holds 500 gallons and nothing is added or removed, the water level over time has a rate of change of zero.