Rate of Change: Slope With Units Attached

Quick answer

The average rate of change measures how fast one quantity changes compared with another: the change in output divided by the change in input. It is the slope formula with units attached. If a car travels 150 miles in 3 hours, the rate of change is 50 miles per hour. For a linear function it is the same everywhere; for a curve it depends on the interval you pick.

What you'll learn

Calculate an average rate of change from a table, graph or function
State the units of a rate of change and interpret its sign
Explain why a linear function has a constant rate of change

The idea

A rate of change compares how much one quantity changes with how much another changes:

\text{average rate of change} = \frac{\text{change in output}}{\text{change in input}} = \frac{\Delta y}{\Delta x}

The symbol $\Delta$ is a capital Greek delta and means “change in”. $\Delta y$ is read “delta $y$ ” and means the amount $y$ changed by.

Between two points $(x_1, y_1)$ and $(x_2, y_2)$ this is

\frac{y_2 - y_1}{x_2 - x_1}

which is the slope formula, unchanged.

So why a second name?

Because the same number answers two different questions, and the wording follows the question.

Slope answers “how steep is this line on the grid?” It is a geometric fact about a picture.

Rate of change answers “how fast is this changing?” It is a statement about the world, and it comes with units.

m = 50 \qquad\text{versus}\qquad 50 \text{ miles per hour}

The second is the more useful sentence, and the units are what make it so. Whenever the axes mean something, say the rate with its units.

What the sign tells you

Sign	Meaning
positive	the output increases as the input increases
negative	the output decreases
zero	the output does not change at all

A negative rate is not an error. A draining tank, a cooling drink, and a shrinking debt all have negative rates of change, and the negative is carrying real information.

Why a linear function has one rate everywhere

Take $f(x) = mx + b$ and step from any input $x$ to $x + h$ :

f(x + h) - f(x) = \big(m(x+h) + b\big) - \big(mx + b\big) = mh

Dividing by the change in input, which is $h$ :

\frac{mh}{h} = m

The $x$ cancelled and the $h$ cancelled. The answer depends on neither where you started nor how far you went — it is always $m$ .

That is what makes a function linear. A curve fails this test: the same calculation on $f(x) = x^2$ leaves an $x$ behind, so the rate depends on where you look.

Worked examples

Example 5 — Reading a table

A plant is measured weekly. Is its growth at a constant rate?

Week	$0$	$2$	$4$	$6$
Height (cm)	$5$	$11$	$17$	$23$

Compute each interval:

\frac{11 - 5}{2 - 0} = 3 \qquad \frac{17 - 11}{4 - 2} = 3 \qquad \frac{23 - 17}{6 - 4} = 3

Every interval gives $3$ cm per week, so the growth is at a constant rate and the relationship is linear. Since the height at week $0$ is $5$ , the function is $h(w) = 3w + 5$ .

Common mistakes

Practice problems

A runner covers $12$ km in $1$ hour and $30$ km in $3$ hours. Find the average rate of change with units.

Hint

Change in distance over change in time.

Answer

$9$ km per hour

Full solution

$\tfrac{30 - 12}{3 - 1} = \tfrac{18}{2} = 9$ , so the rate is $9$ kilometres per hour.
A candle is $20$ cm tall after $1$ hour and $14$ cm after $4$ hours. Find the rate of change.

Answer

$-2$ cm per hour

Full solution

$\tfrac{14 - 20}{4 - 1} = \tfrac{-6}{3} = -2$ .

The candle shortens by $2$ cm each hour, and the negative sign records the shortening.
Find the average rate of change of $f(x) = 5x + 2$ between $x = 1$ and $x = 4$ .

Answer

$5$

Full solution

$f(1) = 7$ and $f(4) = 22$ , so $\tfrac{22 - 7}{4 - 1} = \tfrac{15}{3} = 5$ .

This matches the coefficient of $x$ , as it must for a linear function.
Find the average rate of change of $f(x) = x^2 + 1$ between $x = 0$ and $x = 4$ .

Answer

$4$

Full solution

$f(0) = 1$ and $f(4) = 17$ , so $\tfrac{17 - 1}{4 - 0} = \tfrac{16}{4} = 4$ .
Using the same function $f(x) = x^2 + 1$ , find the rate of change between $x = 4$ and $x = 6$ , and say why it differs.

Hint

Compare with the previous answer, then think about the shape of the graph.

Answer

$10$ , because the curve is steeper further right.

Full solution

$f(4) = 17$ and $f(6) = 37$ , so $\tfrac{37 - 17}{6 - 4} = \tfrac{20}{2} = 10$ .

The earlier interval gave $4$ . A parabola steepens as $x$ grows, so no single rate describes the whole curve — only the interval you name.
Does this table show a constant rate of change?

$x$ $1$ $3$ $5$
$y$ $4$ $10$ $18$

Answer

No.

Full solution

First interval: $\tfrac{10 - 4}{3 - 1} = 3$ .

Second interval: $\tfrac{18 - 10}{5 - 3} = 4$ .

The rates differ, so the relationship is not linear.
A pool drains at a constant rate. After $5$ minutes it holds $900$ litres, and after $20$ minutes it holds $300$ litres. Find the rate and predict the volume at $10$ minutes.

Hint

Find the rate first, then step forward from a known reading.

Answer

$-40$ litres per minute; $700$ litres at $10$ minutes.

Full solution

$\tfrac{300 - 900}{20 - 5} = \tfrac{-600}{15} = -40$ litres per minute.

From $900$ litres at $5$ minutes, five more minutes removes $5 \times 40 = 200$ litres, leaving $700$ litres at $10$ minutes.
Two phone plans are compared. Plan A costs $40 for 2 GB and $70 for 5 GB. Plan B costs $30 for 2 GB and $66 for 5 GB. Which has the lower rate per gigabyte, and what else should you check?

Hint

Compute each rate, then think about what happens at zero gigabytes.

Answer

Plan A is cheaper per gigabyte at $10 versus $12, but Plan B starts lower.

Full solution

Plan A: $\tfrac{70 - 40}{5 - 2} = \tfrac{30}{3} = 10$ dollars per gigabyte.

Plan B: $\tfrac{66 - 30}{5 - 2} = \tfrac{36}{3} = 12$ dollars per gigabyte.

Plan A has the lower rate. But the rate is only the slope — the fixed fee is the intercept. Plan A costs $40 - 2(10) = 20$ before any data, while Plan B costs $30 - 2(12) = 6$ . So Plan B is cheaper for light use and Plan A wins once usage grows. Comparing rates alone would have missed that.

$x$	$1$	$3$	$5$
$y$	$4$	$10$	$18$

Frequently asked questions

Is rate of change the same thing as slope?

It is the same calculation. Slope usually describes a line on a grid; rate of change describes what that number means when the axes carry units, like dollars per hour. Same arithmetic, different emphasis.

Why is it called the AVERAGE rate of change?

Because over a curved graph the steepness varies, so one number can only summarise the interval as a whole. For a straight line the average equals the rate everywhere, so the word adds nothing there.

What do the units mean?

They are always output units per input unit. If y is litres and x is minutes, the rate is litres per minute. Reading the units off the axes is the fastest way to check you divided the right way round.

What to learn next

Linear vs Nonlinear Functions: How to Tell Them ApartFour reliable tests for deciding whether a relationship is linear — from its equation, its table, its graph, or its rate of change.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.HSF.IF.B.6Interpreting FunctionsCalculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
CCSS.MATH.CONTENT.8.EE.B.5Expressions and EquationsGraph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.

Quick answer

What you'll learn

The idea

So why a second name?

What the sign tells you

Why a linear function has one rate everywhere

Worked examples

Common mistakes

Practice problems

Frequently asked questions

Is rate of change the same thing as slope?

Why is it called the AVERAGE rate of change?

What do the units mean?

What to learn next

Standards alignment

More Algebra 1 lessons