To find the rate of change on a table, pick any two rows and divide the difference in the output values (y) by the difference in the input values (x). The formula is (y₂ − y₁) ÷ (x₂ − x₁). That single calculation gives you the rate of change between those two points, and if the table represents a linear relationship, you’ll get the same result no matter which two rows you choose.
The Formula and How to Use It
The rate of change tells you how much the output changes for every one unit increase in the input. It’s the same concept as slope. Written out:
Rate of change = (y₂ − y₁) ÷ (x₂ − x₁)
Here’s a quick example. Suppose your table looks like this:
- x = 1, y = 5
- x = 3, y = 11
- x = 5, y = 17
Pick two rows. Using the first and second: (11 − 5) ÷ (3 − 1) = 6 ÷ 2 = 3. The rate of change is 3, meaning y increases by 3 for every 1 unit increase in x. You can verify with the second and third rows: (17 − 11) ÷ (5 − 3) = 6 ÷ 2 = 3. Same answer, which confirms the relationship is linear.
When the X Values Aren’t Evenly Spaced
Tables don’t always increase by neat, equal steps. You might see x values of 2, 5, 12, and 20. The formula works exactly the same way. You’re always subtracting one y from the other, then dividing by the difference in the corresponding x values. The uneven spacing is already accounted for because you’re dividing by the actual gap between the x values, not assuming the gap is 1.
For example, if x goes from 5 to 12 and y goes from 30 to 51, the rate of change is (51 − 30) ÷ (12 − 5) = 21 ÷ 7 = 3. The key is to always subtract in the same direction for both x and y. If you subtract the first row’s y from the second row’s y, do the same order for x.
Checking Whether the Rate Is Constant
If you calculate the rate of change between every consecutive pair of rows and get the same number each time, the table represents a linear relationship. The rate of change is constant, and that number is the slope of the line.
If the rate of change differs between pairs, the data is non-linear. You might be looking at a curve, an exponential pattern, or something else entirely. In that case, what you’re calculating between any two rows is the average rate of change over that interval, not a single slope that describes the whole table.
Here’s a non-linear example:
- x = 1, y = 2
- x = 2, y = 4
- x = 3, y = 8
Between rows 1 and 2: (4 − 2) ÷ (2 − 1) = 2. Between rows 2 and 3: (8 − 4) ÷ (3 − 2) = 4. The rates don’t match, so this table is not linear. Each rate you calculated is the average rate of change for that specific interval.
Labeling Your Answer With Units
When your table has real-world context, the rate of change carries units. You find the units by putting the y column’s unit over the x column’s unit. If x is measured in hours and y is measured in miles, the rate of change is in miles per hour. If x is the number of items and y is the cost in dollars, the rate of change is dollars per item.
This labeling matters because it tells you what the number actually means. A rate of change of 3 on its own is abstract. A rate of change of 3 miles per hour, or 3 dollars per bicycle, gives you something you can interpret and use. Always check the column headers of your table to determine the correct units.
A Full Worked Example
Suppose you’re given a table showing the relationship between time (in minutes) and water level (in gallons) in a tank:
- 0 minutes, 50 gallons
- 4 minutes, 38 gallons
- 10 minutes, 20 gallons
- 15 minutes, 5 gallons
Step 1: Pick two rows. Let’s start with the first and second. Subtract the y values: 38 − 50 = −12. Subtract the x values in the same order: 4 − 0 = 4. Divide: −12 ÷ 4 = −3.
Step 2: Check another pair. Using rows 2 and 3: (20 − 38) ÷ (10 − 4) = −18 ÷ 6 = −3. Same result. Check rows 3 and 4: (5 − 20) ÷ (15 − 10) = −15 ÷ 5 = −3. The rate of change is constant at −3.
Step 3: Attach units. The y column is gallons and the x column is minutes, so the rate of change is −3 gallons per minute. The negative sign tells you the water level is decreasing. The tank is draining at 3 gallons every minute.
Notice that the time intervals in this table were uneven (4 minutes, then 6, then 5), but the formula handled it without any extra steps. As long as you divide by the actual difference in x, the spacing doesn’t matter.