Slope and Rate of Change

Slope describes how quickly one variable changes compared with another. On a graph, slope tells us how steep a line is and whether it rises, falls, or remains horizontal as we move from left to right.

Slope is also a rate of change.

For example, if a cyclist travels 15 km farther from home every hour, the rate of change is:

15 km/h

If distance is plotted against time, the graph has a slope of:

15

Slope connects graphs with real-world rates such as:

• speed
• cost per item
• pay per hour
• temperature change per minute
• water level change per second
• population change per year

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Understanding Rate of Change

A rate of change compares the change in one quantity with the change in another.

It can be written as:

rate of change = change in output / change in input

For coordinates:

slope = change in y / change in x

The symbol commonly used for slope is:

m

Therefore:

m = Δy/Δx

The symbol Δ means change in.

So:

• Δy = change in y
• Δx = change in x

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Rise and Run

Slope is often described using:

slope = rise/run

The rise is the vertical change.

The run is the horizontal change.

Therefore:

m = rise/run

For example, if a line rises 6 units while moving 3 units to the right:

m = 6/3

m = 2

This means y increases by 2 units for every increase of 1 unit in x.

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Calculating Slope from Two Points

If two points are known:

(x₁, y₁) and (x₂, y₂)

the slope can be calculated using:

m = (y₂ − y₁)/(x₂ − x₁)

This formula compares the vertical change with the horizontal change.

A useful way to remember it is:

change in y over change in x

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Worked Example: Finding Slope

Suppose a line passes through:

(2, 3) and (6, 11)

Calculate the changes.

Change in y:

11 − 3 = 8

Change in x:

6 − 2 = 4

Therefore:

m = 8/4

m = 2

The slope is:

2

This means y increases by 2 units for every increase of 1 unit in x.

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Keeping the Point Order Consistent

When using the slope formula, the subtraction order must remain consistent.

Using:

(2, 3) and (6, 11)

we calculated:

m = (11 − 3)/(6 − 2)

But we could reverse both:

m = (3 − 11)/(2 − 6)

m = −8/−4

m = 2

Both methods give the same answer.

The mistake would be mixing the order:

(11 − 3)/(2 − 6)

because this would incorrectly produce:

8/−4 = −2

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Slope from a Table

Consider:

Each time x increases by:

1

y increases by:

3

Therefore:

m = 3/1

m = 3

The slope is the constant rate of change.

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Slope When x Changes by More Than One

Consider:

From the first two rows:

Δy = 19 − 7 = 12

Δx = 5 − 2 = 3

Therefore:

m = 12/3

m = 4

The slope is 4, not 12.

This is why we calculate:

Δy/Δx

rather than looking only at the change in y.

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Unequal Intervals

Consider:

The changes in x are not equal, so simply comparing consecutive y-values is not enough.

Between the first two points:

m = (13 − 7)/(3 − 1)

m = 6/2

m = 3

Between the next two:

m = (28 − 13)/(8 − 3)

m = 15/5

m = 3

The rate remains constant, so the relationship is linear.

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Positive Slope

A line has a positive slope when it rises from left to right.

As x increases:

y increases

For example:

y = 3x + 2

has:

m = 3

This is a positive slope.

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Interpreting Positive Slope

Suppose a savings account increases by:

$25 each week

Then:

slope = 25 dollars/week

The positive slope means the amount of money increases as time passes.

A positive slope does not simply mean that the y-values themselves are positive.

It means that y increases as x increases.

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Negative Slope

A line has a negative slope when it falls from left to right.

As x increases:

y decreases

For example:

y = −4x + 20

has:

m = −4

The negative sign describes the direction of change.

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Interpreting Negative Slope

Suppose a tank loses:

8 L of water each minute

The rate of change is:

−8 L/min

The negative sign means the volume is decreasing.

If:

V = 100 − 8t

then the slope is:

−8

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Zero Slope

A horizontal line has a slope of:

0

For example:

y = 5

Consider two points:

(2, 5)

and:

(8, 5)

Calculate:

m = (5 − 5)/(8 − 2)

m = 0/6

m = 0

There is no vertical change.

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Interpreting Zero Slope

A zero slope means the dependent variable remains constant as the independent variable changes.

For example, suppose a machine maintains a temperature of:

20°C

for 30 minutes.

On a temperature-time graph, the line would be horizontal.

Its slope would be:

0°C/min

The temperature is not changing.

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Undefined Slope

A vertical line has an undefined slope.

Consider:

(4, 2)

and:

(4, 9)

Calculate:

m = (9 − 2)/(4 − 4)

m = 7/0

Division by zero is undefined.

Therefore, the slope is:

undefined

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Why Vertical Lines Have Undefined Slope

For a vertical line:

Δx = 0

There is vertical change but no horizontal change.

Because slope requires:

Δy/Δx

we would have to divide by zero.

Division by zero is undefined.

Therefore, every vertical line has an undefined slope.

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The Four Types of Slope

Recognizing these visually is an important graphing skill.

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Comparing Positive Slopes

Consider:

Line A: m = 2

Line B: m = 5

Both slopes are positive.

Line B has the greater slope because:

5 > 2

For the same increase in x, Line B increases more quickly.

Line B therefore appears steeper.

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Comparing Negative Slopes

Consider:

Line A: m = −2

Line B: m = −6

Both are negative, so both decrease from left to right.

Line B changes more rapidly because its slope has the larger absolute value:

|−6| > |−2|

However, numerically:

−2 > −6

This distinction matters.

Line B is steeper, but Line A has the greater numerical slope.

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Steepness and Absolute Value

The steepness of a line is related to the absolute value of its slope.

Compare:

m = 1

m = 4

The line with m = 4 is steeper.

Compare:

m = −2

m = −7

The line with m = −7 is steeper because:

|−7| = 7

which is greater than:

|−2| = 2

The sign tells us the direction.

The absolute value tells us how rapidly the line changes.

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Fractional Slopes

Slope does not have to be a whole number.

Suppose:

m = 2/3

This means:

• rise 2
• run 3

For every increase of 3 units in x, y increases by 2 units.

Equivalent rates include:

4/6

6/9

8/12

All simplify to:

2/3

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Negative Fractional Slopes

Suppose:

m = −3/4

This can be interpreted as:

• move 4 units right
• move 3 units down

because:

rise = −3

run = 4

It could also be represented as:

rise = 3

run = −4

Both describe the same slope.

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Finding Slope from a Graph

To calculate slope from a graph:

1. choose two clear points on the line
2. identify their coordinates
3. calculate the vertical change
4. calculate the horizontal change
5. divide Δy by Δx
6. simplify

Whenever possible, choose points that lie exactly on grid intersections.

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Using a Slope Triangle

Suppose two points on a line are:

(1, 2) and (5, 8)

Horizontal change:

5 − 1 = 4

Vertical change:

8 − 2 = 6

This creates a right triangle with:

rise = 6

run = 4

Therefore:

m = 6/4

m = 3/2

A slope triangle makes the rise and run visible on the graph.

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Different Points Give the Same Slope

For a straight line, any two distinct points produce the same slope.

Suppose points on a line include:

(0, 1)

(2, 5)

(4, 9)

Using the first two:

m = (5 − 1)/(2 − 0)

m = 4/2

m = 2

Using the second and third:

m = (9 − 5)/(4 − 2)

m = 4/2

m = 2

This constant slope is what makes the relationship linear.

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Slope and Linear Equations

Recall slope-intercept form:

y = mx + b

where:

m = slope

b = y-intercept

For:

y = 4x + 7

the slope is:

m = 4

For:

y = −3x + 10

the slope is:

m = −3

For:

y = 8

the slope is:

m = 0

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Slope and Direct Variation

Direct variation has the form:

y = kx

The constant of variation k is also the slope.

For example:

y = 6x

has:

k = 6

and:

m = 6

The output increases by 6 units for every increase of 1 unit in the input.

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Slope as a Unit Rate

Slope often represents a unit rate.

Suppose:

The slope is:

m = (140 − 70)/(2 − 1)

m = 70/1

m = 70 km/h

The units of slope are:

y-units per x-unit

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Units of Slope

Always consider the quantities on the axes.

If:

x = time in seconds

y = distance in metres

then slope has units:

metres/second

If:

x = number of items

y = cost in dollars

then slope has units:

dollars/item

If:

x = time in minutes

y = temperature in °C

then slope has units:

°C/min

Units help explain what the slope actually means.

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Real-World Example: Constant Speed

Suppose a car's distance is represented by:

d = 80t

where:

d = distance in kilometres

t = time in hours

The slope is:

80

In context:

80 km/h

This means the car travels 80 additional kilometres for every additional hour.

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Real-World Example: Hourly Pay

Suppose:

P = 22h + 50

where:

P = total pay in dollars

h = hours worked

The slope is:

22

This means:

$22/hour

The $50 is the starting amount, but it does not affect the rate of change.

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Real-World Example: Temperature

Suppose a substance cools according to:

T = 90 − 4t

where:

T = temperature in °C

t = time in minutes

The slope is:

−4°C/min

This means the temperature decreases by 4°C every minute.

The negative slope communicates the decrease.

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Real-World Example: Draining Water

A tank contains water according to:

V = 600 − 25t

where:

V = volume in litres

t = time in minutes

Slope:

−25 L/min

This means the tank loses 25 litres of water each minute.

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Real-World Example: Cost

Suppose a printing company charges:

C = 0.20p + 5

where:

C = cost in dollars

p = number of pages

Slope:

0.20 dollars/page

The slope tells us the additional cost for each page.

The $5 is a fixed starting charge.

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Interpreting Slope in Context

A complete interpretation should state:

1. what is changing
2. how much it changes
3. for each unit of what other quantity
4. whether it increases or decreases

Instead of saying:

"The slope is −5."

a stronger interpretation is:

"The water level decreases by 5 centimetres for every hour that passes."

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Comparing Rates from Two Tables

Car A

Slope:

m = 120/2

m = 60 km/h

Car B

Slope:

m = 210/3

m = 70 km/h

Therefore, Car B has the greater rate of change.

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Comparing Rates from Equations

Consider:

A: y = 3x + 8

B: y = 5x − 2

The slopes are:

A: 3

B: 5

Therefore, B increases more quickly.

The intercepts do not affect the comparison of their rates of change.

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Comparing Rates from Different Representations

Suppose:

Relationship A

y = 4x + 10

Relationship B

For A:

m = 4

For B:

m = (12 − 2)/(2 − 0)

m = 10/2

m = 5

Therefore, Relationship B has the greater rate of change.

This demonstrates why slope allows us to compare relationships even when they are presented differently.

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Comparing Rates from Graphs

When comparing two graphs, visual steepness can provide a quick estimate.

However, calculating slope is more reliable.

Choose two points from each graph and calculate:

m = Δy/Δx

This is especially important if:

• the slopes are close
• the graph uses unusual intervals
• the axes use different scales

Do not judge slope only by appearance.

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Graph Scale Can Be Misleading

Two lines may appear to have different steepness simply because the graphs use different scales.

For example:

one graph might use:

1 square = 1 unit

while another uses:

1 square = 100 units

Therefore, visual steepness alone is not enough when comparing separate graphs.

Calculate the actual rates.

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Slope Does Not Depend on the Intercept

Consider:

y = 3x + 2

and:

y = 3x + 10

Both have:

m = 3

Their graphs are different because their y-intercepts differ.

However, they have the same rate of change.

The lines are parallel.

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Parallel Lines

Two non-vertical lines with the same slope but different intercepts are parallel.

For example:

y = 4x + 2

and:

y = 4x − 7

both have:

m = 4

They rise at the same rate and never intersect.

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Horizontal and Vertical Lines

Horizontal lines have equations such as:

y = 5

Their slope is:

0

Vertical lines have equations such as:

x = 5

Their slope is:

undefined

This distinction is fundamental when analyzing graphs.

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Determining a Missing Coordinate Using Slope

Suppose the slope between:

(2, 5) and (6, y)

is:

3

Use:

3 = (y − 5)/(6 − 2)

3 = (y − 5)/4

Multiply by 4:

12 = y − 5

y = 17

So the second point is:

(6, 17)

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Determining a Missing x-Coordinate

Suppose the slope between:

(2, 4) and (x, 16)

is:

3

Then:

3 = (16 − 4)/(x − 2)

3 = 12/(x − 2)

Multiply:

3(x − 2) = 12

x − 2 = 4

x = 6

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Average Rate of Change

For a straight line, the rate of change is constant.

For a relationship that is not linear, the rate may change.

We can still calculate the average rate of change between two points:

average rate of change = (change in y)/(change in x)

For example, suppose a moving object is at:

(2, 10)

and later at:

(6, 34)

Average rate:

m = (34 − 10)/(6 − 2)

m = 24/4

m = 6

This tells us the average change over that interval, even if the rate was not constant throughout.

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Slope and Direction

The sign of the slope communicates direction:

m > 0

The relationship increases.

m < 0

The relationship decreases.

m = 0

The relationship remains constant.

m undefined

The graph is vertical.

This allows us to understand a graph quickly even before calculating exact values.

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A Reliable Strategy for Finding Slope

When given two points:

1. Write both points clearly.
2. Label them (x₁, y₁) and (x₂, y₂).
3. Calculate Δy.
4. Calculate Δx.
5. Divide Δy by Δx.
6. Simplify the result.
7. Determine whether the slope is positive, negative, zero, or undefined.
8. Include units if the graph has context.
9. Interpret the result in words.

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A Reliable Strategy for Comparing Slopes

When comparing relationships:

1. Identify how each relationship is represented.
2. Find the slope of each one.
3. Make sure the units are compatible.
4. Compare the numerical values.
5. Compare absolute values if discussing steepness.
6. Identify whether each relationship increases, decreases, or remains constant.
7. Interpret the comparison in context.

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Common Mistakes

Using change in x over change in y

Slope is:

Δy/Δx

not:

Δx/Δy

Mixing subtraction order

If you calculate:

y₂ − y₁

you must also calculate:

x₂ − x₁

in the same point order.

Looking only at the change in y

A change of 12 in y does not mean the slope is 12 unless x changes by exactly 1.

Thinking every negative slope is smaller in steepness

A slope of −8 is steeper than −2 even though −8 is numerically smaller.

Thinking zero slope and undefined slope are the same

Horizontal line → zero slope.

Vertical line → undefined slope.

Saying a negative slope means y is negative

A negative slope means y decreases as x increases. The actual y-values may be positive or negative.

Ignoring units

A slope of 20 is much more meaningful when written as:

20 km/h

or:

$20/hour

Using the intercept as the rate of change

In:

y = 5x + 12

the slope is 5, not 12.

Comparing graph steepness without checking the scales

Different axis scales can make visual comparisons misleading.

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Key Terms

Slope — measure of the rate of change of y compared with x.

Rate of change — amount one variable changes relative to another.

Rise — vertical change between two points.

Run — horizontal change between two points.

Δy — change in the y-values.

Δx — change in the x-values.

Positive slope — slope describing a line that rises from left to right.

Negative slope — slope describing a line that falls from left to right.

Zero slope — slope of a horizontal line.

Undefined slope — slope of a vertical line.

Slope triangle — right triangle used to show rise and run on a graph.

Unit rate — rate expressed for one unit of the independent variable.

Linear relationship — relationship with a constant rate of change.

y-intercept — point where a graph crosses the y-axis.

Parallel lines — lines in the same plane that do not intersect; non-vertical parallel lines have equal slopes.

Average rate of change — change in output divided by change in input over a specified interval.

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Key Takeaways

• Slope measures the rate of change between two variables.
• Slope can be written as m = Δy/Δx.
• Slope is also described as rise/run.
• From two points, use m = (y₂ − y₁)/(x₂ − x₁).
• The subtraction order must remain consistent.
• Positive slope means y increases as x increases.
• Negative slope means y decreases as x increases.
• Zero slope represents a horizontal line.
• Undefined slope represents a vertical line.
• Larger absolute slope means a steeper line.
• The sign of slope describes the direction of change.
• The absolute value of slope describes how rapidly the quantities change.
• For a linear relationship, any two distinct points on the line produce the same slope.
• In y = mx + b, m represents slope.
• The y-intercept does not determine the rate of change.
• Non-vertical parallel lines have the same slope.
• Slope often represents a meaningful real-world unit rate.
• The units of slope are y-units per x-unit.
• Tables, graphs, equations, and verbal descriptions can all be used to determine slope.
• Graph scales must be considered before comparing apparent steepness.
• For non-linear relationships, slope between two points represents an average rate of change over that interval.
• A complete real-world interpretation of slope explains what changes, by how much, and for each unit of the other variable.

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Check Your Understanding

1. Define slope in your own words and explain why it is considered a rate of change.
2. Calculate the slope between (2, 4) and (6, 12).
3. Calculate the slope between (−3, 7) and (5, −1). State whether the slope is positive or negative.
4. Find the slope between (−4, 3) and (2, 3). Explain what the result tells you about the graph.
5. Find the slope between (6, −2) and (6, 9). Explain why the slope has this result.
6. Determine the slope represented by the table:

7. A line has a rise of −12 and a run of 4. Determine its slope and describe the direction of the line.
8. Compare slopes 3, 7, and −5. Which is numerically greatest? Which represents the steepest line?
9. Compare slopes −2 and −8. Which slope is numerically greater? Which line would be steeper?
10. A car travels from 40 km to 220 km from a starting point between t = 1 hour and t = 4 hours. Calculate its rate of change and include appropriate units.
11. A tank contains 500 L at t = 0 minutes and 350 L at t = 6 minutes. Calculate the slope and interpret it in context.
12. Relationship A is represented by y = 4x + 9. Relationship B passes through (2, 5) and (6, 25). Determine which relationship has the greater rate of change.
13. A worker's total pay is represented by P = 24h + 75. Identify the slope and y-intercept. Explain what each value means in context.
14. The slope between (3, 7) and (9, y) is 4. Determine the missing y-coordinate.
15. Two students examine separate graphs. Line A appears steeper than Line B, so one student concludes that Line A must have the greater rate of change. Explain why this conclusion may be incorrect, describe how graph scales can affect appearance, and explain how the students could calculate and compare the actual rates.