Constant Rate of Change: Meaning, Formula and Examples

A constant rate of change means y changes by the same amount for each equal increase in x. Its value is the slope: m = (y₂ − y₁)/(x₂ − x₁), where x₂ ≠ x₁. For a linear function y = mx + b, m is constant. A positive rate means y increases, a negative rate means y decreases, and a zero rate means y stays the same.

How to find a constant rate of change

1. Identify the input and output. Usually x is the input and y is the output. Note their units.
2. Choose two points with different x-values. Label them (x₁, y₁) and (x₂, y₂).
3. Subtract in the same order. Calculate y₂ − y₁ and x₂ − x₁.
4. Divide: m = (y₂ − y₁)/(x₂ − x₁). State the result in y-units per x-unit.
5. Check other intervals when available. Matching rates across the table support a linear pattern. One pair alone gives an average rate; it does not prove the rate is constant everywhere.

For a graph of a linear function, choose two readable points on the line and calculate rise/run. A horizontal line has rate 0. A vertical line has an undefined slope because its change in x is 0; it is not a function of x.

Finding the rate from a table with unequal intervals

Suppose a table lists these points:

From x = 1 to x = 3, the rate is (13 − 7)/(3 − 1) = 6/2 = 3. From x = 3 to x = 6, it is (22 − 13)/(6 − 3) = 9/3 = 3. The y-differences, 6 and 9, differ because the x-intervals differ. Compare the ratios of changes, not just the y-differences.

These points fit y = 3x + 4. If the problem states that the relationship is linear, its rate is 3 throughout. A few table values alone cannot establish what happens between or beyond the listed points.

Constant rate of change examples

A positive rate: distance traveled

At a steady speed, a vehicle travels 60 miles in 2 hours. Its distance changes from 0 to 60 miles while time changes from 0 to 2 hours, so m = (60 − 0)/(2 − 0) = 30 miles per hour. Without the steady-speed assumption, this calculation gives the average speed for those 2 hours.

A negative rate: water leaving a tank

A tank has 48 liters after 2 minutes and 30 liters after 5 minutes. If it drains at a constant rate, m = (30 − 48)/(5 − 2) = −18/3 = −6 liters per minute. The negative sign means the amount remaining decreases; the tank loses 6 liters each minute while this model applies.

A zero rate: an unchanged value

The points (2, 9) and (5, 9) give m = (9 − 9)/(5 − 2) = 0. A constant function y = 9 has a horizontal graph. A zero rate is different from an undefined rate.

Is constant rate of change the same as constant of proportionality?

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A proportional relationship has the form y = kx. Its graph passes through (0, 0), and its constant rate of change is k. A linear function y = mx + b can have a constant rate even when b ≠ 0; then it is not proportional.

For example, y = 3x + 4 has constant rate 3, but y/x is not constant: at x = 1, y/x = 7; at x = 3, y/x = 13/3. Use Δy/Δx for the rate of change. Use y/x for a constant of proportionality only when the relationship is proportional and x ≠ 0.

A changing rate can produce a curve. For y = x², the points (1, 1), (2, 4) and (3, 9) give rates 3 and 5 across equal x-intervals, so the rate is not constant.

Practice: calculate and interpret each rate

1. A linear function passes through (−2, 5) and (4, 17). Find its rate.
2. A table contains (0, 8), (2, 14) and (5, 23). Do the listed intervals have the same rate?
3. A tank drains steadily from 80 liters to 56 liters in 6 minutes. Find the signed rate of change.
4. Find the rate for y = −2x + 7. Is this relationship proportional?
5. A table contains (1, 2), (2, 4) and (3, 8). Is its rate constant?

Continue with graphing linear equations, or use the Algebra 1 learning center to choose a related lesson or practice resource.

References: OpenStax, linear functions and direct variation. The numerical examples and practice on this page are original.

How to spot a constant rate in a table, graph, equation or word problem

The same idea shows up in four forms. Use the check that matches what you are given.

A quick graph check: pick any two points on the line, find the rise over the run, then repeat with two other points. If the line is straight, you get the same number both times. For the line through (0, 2) and (4, 10), the rate is (10 − 2) ÷ (4 − 0) = 8 ÷ 4 = 2. The point (2, 6) is also on that line, and (6 − 2) ÷ (2 − 0) = 2 again.

Frequently Asked Questions

What is a constant rate of change?

A constant rate of change means the output changes by the same amount every time the input increases by the same amount. A phone plan that adds $10 for every extra gigabyte has a constant rate of $10 per gigabyte. Its graph is a straight line, and the rate is the line’s slope.

How do you find the constant rate of change?

Pick two points, subtract the y-values and the x-values in the same order, and divide: rate = (y₂ − y₁) ÷ (x₂ − x₁). For a table with (2, 9) and (5, 18), the rate is (18 − 9) ÷ (5 − 2) = 9 ÷ 3 = 3. Then check a third pair; if every pair gives 3, the rate really is constant.

What is the constant rate of change for the equation y = 3x + 5?

The constant rate of change is 3. In the form y = mx + b, the rate of change is m, the number multiplied by x. Each time x goes up by 1, y goes up by 3: at x = 0, y = 5; at x = 1, y = 8; at x = 2, y = 11. The 5 is the starting value, or y-intercept, not the rate.

Becca has $5 and earns $2 for chores each day. What is the constant rate of change?

The constant rate of change is $2 per day. The $5 is Becca’s starting amount, and the $2 she adds every day is the rate. Her money follows y = 2x + 5, where x is the number of days. After 4 days she has 2 × 4 + 5 = $13.

I have 10 Starbursts and eat 1 each day. What is the constant rate of change?

The constant rate of change is −1 Starburst per day. The rate is negative because the amount goes down by the same number each day. The situation fits y = −x + 10, where x is days. After 6 days, 10 − 6 = 4 candies are left, and they run out on day 10.

What function has a constant rate of change?

A linear function has a constant rate of change. Any function written as y = mx + b changes by m for every 1-unit step in x, so its graph is a straight line. Quadratic and exponential functions do not: for y = 2^x, the outputs 1, 2, 4, 8 grow by 1, then 2, then 4, so the rate keeps changing.