Constant Rate of Change

A rate of change describes how one quantity changes compared with another quantity.

Rates of change appear everywhere in mathematics, science, and everyday life. They can describe:

• how far a car travels each hour
• how much money is earned each day
• how quickly water fills a tank
• how temperature changes with time
• how a plant grows each week
• how much fuel a vehicle uses
• how the cost of an item changes with quantity

A constant rate of change occurs when one quantity changes by the same amount for every equal change in another quantity.

For example, if a cyclist travels 15 km every hour, the distance increases at a constant rate of:

15 km/h

What Is a Rate?

A rate compares two quantities that have different units.

Examples include:

• 60 kilometres per hour
• $18 per hour
• 5 litres per minute
• 3 centimetres per year
• 20 words per minute
• 4°C per hour

The word per is important.

It tells us that one quantity is being compared with another.

For example:

60 km per hour

means:

60 kilometres for every 1 hour

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What Is Rate of Change?

A rate of change tells us how much the output changes when the input changes.

The basic relationship is:

rate of change = change in output / change in input

Using x and y:

rate of change = change in y / change in x

or:

rate of change = Δy/Δx

The Greek letter Δ (delta) means change in.

So:

Δy = change in y

and:

Δx = change in x

Therefore:

rate of change = Δy/Δx

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Calculating Change

To calculate a change, subtract the earlier value from the later value.

change = final value − initial value

For example, if distance increases from 20 km to 50 km:

change in distance = 50 − 20

= 30 km

If time increases from 2 hours to 4 hours:

change in time = 4 − 2

= 2 hours

Therefore:

rate of change = 30 km / 2 h

= 15 km/h

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Rate of Change and Slope

Rate of change is closely connected to the slope of a graph.

For a straight line:

slope = change in y / change in x

Therefore:

slope = rate of change

You may also see the slope formula written as:

m = (y₂ − y₁)/(x₂ − x₁)

where:

• (x₁, y₁) is one point
• (x₂, y₂) is another point
• m is the slope

Another common way to remember this is:

slope = rise/run

where:

rise = vertical change

and:

run = horizontal change

Worked Example: Rate of Change

A car travels:

120 km in 2 hours

and:

300 km in 5 hours.

Calculate its rate of change in distance.

Change in distance:

300 − 120 = 180 km

Change in time:

5 − 2 = 3 h

Rate of change:

180/3

= 60 km/h

The car's distance is increasing at:

60 km per hour

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Positive Rates of Change

A rate of change is positive when the output increases as the input increases.

For example:

Each hour, the distance increases by:

20 km

Therefore:

rate of change = +20 km/h

The positive sign indicates that distance is increasing.

On a graph, a positive rate of change produces a line that rises from left to right.

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Negative Rates of Change

Rates of change can also be negative.

Consider the temperature of a cooling drink:

Every 5 minutes, temperature changes by:

50 − 60 = −10°C

The rate of change is:

−10/5

= −2°C/min

The negative sign tells us that temperature is decreasing.

On a graph, a negative rate of change produces a line that falls from left to right.

Zero Rate of Change

A rate of change can also be zero.

Consider:

The water level does not change.

Change in water level:

50 − 50 = 0 cm

Therefore:

rate of change = 0 cm/min

A zero rate of change appears as a horizontal line on a graph.

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What Makes a Rate Constant?

A rate is constant when the same change occurs for every equal interval.

Consider:

Every time increases by:

2 s

Distance increases by:

10 m

Therefore:

rate of change = 10/2

= 5 m/s

The same rate occurs throughout the table.

Therefore, the relationship has a:

constant rate of change of 5 m/s

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Identifying Constant Rate from a Table

Consider:

Look at the changes.

x increases by:

+1 each time

y increases by:

+4 each time

Therefore:

rate of change = 4/1

= 4

Since this rate is the same throughout the table, the rate of change is constant.

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Be Careful When x Does Not Increase by 1

Students sometimes look only at the changes in y.

That can give the wrong answer.

Consider:

y increases by:

6

But x increases by:

3

Therefore:

rate of change = 6/3

= 2

The rate is 2, not 6.

Always calculate:

change in y / change in x

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Unequal Intervals Can Still Have a Constant Rate

The x-values do not need to be equally spaced.

Consider:

From 0 to 1:

rate = 12/1 = 12 km/h

From 1 to 3:

rate = (36 − 12)/(3 − 1)

= 24/2

= 12 km/h

From 3 to 7:

rate = (84 − 36)/(7 − 3)

= 48/4

= 12 km/h

The intervals are different sizes, but the rate remains:

12 km/h

Therefore, the rate of change is constant.

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Identifying a Non-Constant Rate

Consider:

Calculate the changes in distance:

0 → 4 = +4

4 → 10 = +6

10 → 18 = +8

18 → 28 = +10

Since time increases by 1 second each time, the rates are:

4 m/s
6 m/s
8 m/s
10 m/s

The rate is changing.

Therefore:

the rate of change is not constant.

This could describe an object that is speeding up.

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Constant Rate and Straight-Line Graphs

A constant rate of change produces a straight-line graph.

Why?

Because every equal horizontal change produces the same vertical change.

For example, suppose:

rate of change = 3

For every increase of 1 in x:

y increases by 3.

The pattern is consistent.

This produces a straight line.

Linear Relationships

A relationship with a constant rate of change is called a linear relationship.

Linear relationships can often be written as:

y = mx + b

where:

• y = output
• x = input
• m = rate of change or slope
• b = initial value or y-intercept

The value m tells us how much y changes for every increase of 1 in x.

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Example: Interpreting y = mx + b

Consider:

y = 4x + 7

The coefficient of x is:

m = 4

Therefore, the rate of change is:

4

The value:

b = 7

means that when x = 0:

y = 7

So the relationship begins at 7 and increases by 4 for every increase of 1 in x.

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Rate of Change Is Not the Starting Value

This distinction is important.

Consider:

y = 5x + 20

The rate of change is:

5

The starting value is:

20

These numbers have different meanings.

If this equation represented a taxi fare:

20 might represent the starting fee.

5 might represent the additional cost per kilometre.

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Worked Example: Taxi Fare

Suppose a taxi charges:

$4 starting fee

plus:

$2.50 per kilometre

The total cost can be represented by:

C = 2.5d + 4

where:

• C = cost
• d = distance travelled

The rate of change is:

$2.50/km

This means:

for every additional kilometre travelled, the fare increases by $2.50.

The $4 is the initial cost, not the rate of change.

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Comparing Rates of Change

Rates allow us to compare situations.

Suppose two cyclists travel at constant speeds.

Cyclist A

Travels 36 km in 2 hours.

Rate:

36/2 = 18 km/h

Cyclist B

Travels 50 km in 2 hours.

Rate:

50/2 = 25 km/h

Since:

25 km/h > 18 km/h

Cyclist B has the greater rate of change in distance.

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Comparing Rates from Tables

Consider two machines filling containers.

Machine A

Rate:

16/2 = 8 L/min

Machine B

Rate:

30/3 = 10 L/min

Therefore, Machine B fills at a greater rate.

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Comparing Positive Rates

Suppose two lines have rates:

Line A: +2

Line B: +6

Both are increasing.

But Line B increases more rapidly.

On a graph, Line B would generally appear steeper.

For positive slopes:

larger slope → steeper increase

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Comparing Negative Rates

Suppose:

Line A: −2

Line B: −7

Both are decreasing.

But Line B decreases more rapidly because its rate has the greater magnitude.

For every increase of 1 in x:

Line A decreases by 2.

Line B decreases by 7.

Therefore, Line B has the steeper downward slope.

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Units Matter

A rate of change should normally include appropriate units.

Suppose:

distance is measured in kilometres

and:

time is measured in hours.

Then:

rate of change = kilometres/hours

= km/h

If volume is measured in litres and time in minutes:

L/min

If temperature is measured in degrees Celsius and time in seconds:

°C/s

Units tell us what the rate actually means.

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Units Come from the Axes

If a graph has:

y-axis: Distance (m)

and:

x-axis: Time (s)

then the rate of change has units:

m/s

because:

rate = change in distance / change in time

If the graph has:

y-axis: Cost ($)

and:

x-axis: Number of Tickets

then the rate has units:

$/ticket

Always examine what each variable represents.

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Rate of Change from Two Points

Suppose a line passes through:

(2, 7)

and:

(6, 19)

Use:

m = (y₂ − y₁)/(x₂ − x₁)

Substitute:

m = (19 − 7)/(6 − 2)

= 12/4

= 3

Therefore, the rate of change is:

3

This means y increases by 3 for every increase of 1 in x.

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The Order of Subtraction

When using:

m = (y₂ − y₁)/(x₂ − x₁)

you must subtract in the same order.

Correct:

m = (19 − 7)/(6 − 2)

You could also reverse both:

m = (7 − 19)/(2 − 6)

= −12/−4

= 3

Both work.

But you cannot reverse only one.

Incorrect:

(19 − 7)/(2 − 6)

= 12/−4

= −3

The numerator and denominator must use the points in the same order.

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Rate of Change from a Graph

To calculate rate from a straight-line graph:

1. Choose two clear points on the line.
2. Find the vertical change.
3. Find the horizontal change.
4. Divide.

rate of change = vertical change / horizontal change

or:

rate of change = rise/run

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

This reduces measurement error.

Interpreting a Rate in Context

Calculating a number is only part of the task.

We should also explain what the number means.

Suppose:

rate = 12 L/min

A strong interpretation is:

The volume increases by 12 litres for every minute that passes.

Suppose:

rate = −3°C/h

Interpretation:

The temperature decreases by 3°C every hour.

Suppose:

rate = $8/day

Interpretation:

The cost increases by $8 for each additional day.

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Positive, Negative, and Zero Rates

The sign of a rate gives useful information.

The sign tells us the direction of change.

The size tells us how quickly the change occurs.

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Real-World Example: Water Tank

Suppose a tank contains:

Every 2 minutes:

water increases by 30 L

Rate:

30/2

= 15 L/min

Interpretation:

The amount of water in the tank increases by 15 litres every minute.

Notice that the tank begins with 50 L.

The starting amount and rate are different quantities.

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Real-World Example: Draining a Tank

Now suppose:

Every 5 minutes:

water changes by:

175 − 200 = −25 L

Rate:

−25/5

= −5 L/min

Interpretation:

The amount of water decreases by 5 litres every minute.

The negative sign is meaningful.

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Real-World Example: Temperature

Suppose a freezer cools from:

20°C at 0 minutes

to:

−10°C at 15 minutes.

Change in temperature:

−10 − 20

= −30°C

Change in time:

15 − 0

= 15 min

Rate:

−30/15

= −2°C/min

Interpretation:

The temperature decreases by 2°C each minute.

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Real-World Example: Earnings

A student earns $72 for 6 hours of work.

Rate:

72/6

= $12/h

If the hourly rate remains the same, then:

2 hours → $24
4 hours → $48
6 hours → $72
8 hours → $96

This is a constant rate of change.

Real-World Example: Fuel Consumption

Suppose a vehicle uses 24 L of fuel while travelling 300 km.

A useful rate is:

24/300

= 0.08 L/km

This means the vehicle uses an average of:

0.08 litres per kilometre

We could also express the relationship differently:

300/24

= 12.5 km/L

This means:

12.5 kilometres per litre

Both are rates, but they answer different questions.

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The Direction of a Rate Matters

Consider:

60 km/h

This compares:

distance / time

But:

1/60 h/km

compares:

time / distance

These are related, but they mean different things.

Always ask:

What quantity is changing compared with what other quantity?

The order matters.

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Constant Rate and Proportional Relationships

Some constant-rate relationships are also proportional relationships.

Example:

d = 5t

If t = 0:

d = 0

The graph passes through:

(0, 0)

This is both:

• linear
• proportional

But consider:

C = 5t + 20

This still has a constant rate of change of 5.

However, when t = 0:

C = 20

Therefore, the graph does not pass through the origin.

It is linear but not proportional.

So:

Every proportional relationship has a constant rate of change.

But:

Not every constant-rate linear relationship is proportional.

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Rate of Change and Calculus

Constant rate of change provides an important foundation for calculus.

For a straight line, the rate of change is the same everywhere.

For a curved graph, the rate may change continuously.

This leads to an important question:

How can we determine the rate of change at one exact point on a curve?

Calculus answers this using the derivative.

So there is an important progression:

constant rate → average rate → instantaneous rate → derivative

Constant Rate Versus Changing Rate

Consider a car travelling at exactly:

20 m/s

Its position increases by:

20 m every second.

The position-time graph is a straight line.

The rate of change is constant.

Now imagine a car accelerating.

It might travel:

5 m during one second

10 m during the next

15 m during the next

Its position is changing at an increasing rate.

The position-time graph becomes curved.

This distinction becomes extremely important in calculus.

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Worked Example: Table

Determine whether the rate of change is constant.

From the first pair to the second:

Δy = 15 − 9 = 6

Δx = 4 − 2 = 2

Rate:

6/2 = 3

Next interval:

(21 − 15)/(6 − 4)

= 6/2

= 3

Next:

(27 − 21)/(8 − 6)

= 6/2

= 3

The rate is always 3.

Therefore:

the relationship has a constant rate of change.

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Worked Example: Non-Constant Rate

Consider:

Rates:

(5 − 2)/(2 − 1) = 3

(10 − 5)/(3 − 2) = 5

(17 − 10)/(4 − 3) = 7

Since:

3 ≠ 5 ≠ 7

the rate of change is not constant.

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Worked Example: Comparing Two Rates

Company A charges:

$30 for 3 hours

Company B charges:

$56 for 8 hours

Company A:

30/3 = $10/h

Company B:

56/8 = $7/h

Therefore:

Company A's cost increases at a greater rate.

Notice that this statement compares the rates, not necessarily the total cost in every possible situation. Starting fees could change the total-price comparison.

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Common Mistakes

Using y/x instead of change in y/change in x

The rate of change between two points is:

Δy/Δx

not automatically:

y/x

Looking only at changes in y

You must also consider how much x changed.

Assuming x always increases by 1

Always check the table.

Forgetting units

A rate such as 12 is much less informative than:

12 km/h

Confusing starting value with rate

In:

y = 4x + 20

the rate is 4, not 20.

Ignoring negative signs

A rate of:

−5°C/h

means the temperature is decreasing.

Thinking a larger negative number means faster decrease

Compare magnitude.

−8 represents a faster decrease than −3 because the quantity changes by 8 units rather than 3 units per interval.

Subtracting coordinates in different orders

Keep the subtraction order consistent.

Assuming every straight line passes through the origin

A straight line can have a nonzero starting value.

Assuming every linear relationship is proportional

A proportional relationship must pass through the origin.

Assuming any changing quantity has a constant rate

Many real systems have changing rates.

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Key Terms

Rate — A comparison of two quantities with different units.

Rate of change — The amount one quantity changes relative to another.

Constant rate of change — A rate that remains the same throughout a relationship.

Variable rate of change — A rate that changes over the interval being considered.

Input — The independent quantity, commonly represented by x.

Output — The dependent quantity, commonly represented by y.

Change — The difference between a final value and an initial value.

Delta (Δ) — A symbol meaning "change in."

Δx — Change in the x-variable.

Δy — Change in the y-variable.

Slope — The rate of change of a straight line.

Rise — The vertical change between two points.

Run — The horizontal change between two points.

Linear relationship — A relationship with a constant rate of change that produces a straight-line graph.

Positive rate of change — A rate where the output increases as the input increases.

Negative rate of change — A rate where the output decreases as the input increases.

Zero rate of change — A rate where the output remains unchanged.

Initial value — The output value when the input is zero.

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

Proportional relationship — A constant-rate relationship that passes through the origin.

Unit rate — A rate expressed for one unit of another quantity.

Independent variable — A variable used as the input of a relationship.

Dependent variable — A variable whose value depends on the independent variable.

magnitude — The size or absolute value of a quantity.

Derivative — In calculus, a quantity describing instantaneous rate of change.

Instantaneous rate of change — The rate at one particular point or instant.

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Key Takeaways

• A rate compares two quantities.
• A rate of change describes how one quantity changes relative to another.
• The basic formula is:

rate of change = change in output / change in input

• In x and y notation:

rate of change = Δy/Δx

• For a straight line, rate of change is the same as slope.
• Slope can be calculated using:

m = (y₂ − y₁)/(x₂ − x₁)

• A constant rate of change means the same rate occurs throughout the relationship.
• Constant rates produce straight-line graphs.
• Relationships with constant rates of change are linear.
• A positive rate indicates an increasing quantity.
• A negative rate indicates a decreasing quantity.
• A zero rate indicates no change.
• The magnitude of a rate tells us how rapidly the quantity changes.
• Units are essential when interpreting rates.
• The units of a rate come from output units divided by input units.
• A table can show a constant rate even when the input intervals are unequal.
• Always compare Δy with Δx rather than looking only at changes in y.
• The starting value is different from the rate of change.
• In y = mx + b, m is the rate of change and b is the initial value.
• A proportional relationship has a constant rate and passes through the origin.
• A linear relationship can have a constant rate without being proportional.
• Real-world rates include speed, wages, flow rate, temperature change, growth, and cost per item.
• Comparing rates allows us to compare how quickly different quantities change.
• A changing rate usually produces a curved rather than straight graph.
• Constant rate of change provides an important foundation for understanding instantaneous rate of change in calculus.

A useful question to ask whenever you see a rate is:

"How much does the output change for each unit of input?"

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Check Your Understanding

1. Define rate of change in your own words.

2. What does Δ mean?

3. Write the formula for rate of change using Δx and Δy.

4. A car travels 180 km in 3 hours. Calculate its rate.

5. Interpret your answer to Question 4 in a complete sentence.

6. A plant grows from 12 cm to 32 cm in 5 weeks. Calculate its average rate of growth.

7. A temperature decreases from 30°C to 18°C in 4 hours. Calculate the rate of change.

8. Why is the answer to Question 7 negative?

9. Determine the rate of change:

10. Is the rate in Question 9 constant? Explain.

11. Determine whether this relationship has a constant rate:

12. Explain how you know.

13. Find the rate of change between:

(3, 8)

and:

(7, 20)

14. A line has a rate of change of −4. What does the negative sign tell you?

15. What does a zero rate of change look like on a graph?

16. What does a constant rate of change look like on a graph?

17. For:

y = 7x + 3

identify:

a. the rate of change
b. the initial value

18. For:

C = 12t + 50

explain what 12 could mean if C represents cost in dollars and t represents time in hours.

19. Why is 50 not the rate of change in Question 18?

20. Machine A fills a tank at 8 L/min. Machine B fills at 12 L/min. Which has the greater rate of change?

21. Compare the rates:

−3°C/min

and:

−8°C/min

Which represents the more rapid decrease?

22. Determine whether this table shows a constant rate:

23. Explain why unequal x-intervals do not automatically mean the rate is non-constant.

24. A water tank contains 300 L and loses 15 L every minute. State the rate of change, including its sign and units.

25. Explain the difference between a rate of change and an initial value.

26. Explain the difference between a linear relationship and a proportional relationship.

27. Give three real-world examples of constant rates of change.

28. A cyclist's distance-time graph is a straight line with slope 6. If distance is measured in kilometres and time in hours, interpret the slope.

29. Explain why rate of change is an important idea for calculus.

30. Challenge: Two delivery companies use these pricing models:

Company A:

C = 3d + 12

Company B:

C = 4.5d + 5

where C is cost in dollars and d is distance in kilometres.

For each company:

a. identify the rate of change
b. identify the initial value
c. explain what each rate means
d. determine which company's price increases more quickly with distance
e. calculate the cost of a 10 km delivery
f. explain why comparing only the rates is not enough to determine which company is cheaper for every possible delivery distance.

Worked Example: Slope from Two Points

Find the slope between:

(2, 3)

and:

(6, 11)

Use:

m = (y₂ − y₁)/(x₂ − x₁)

Substitute:

m = (11 − 3)/(6 − 2)

= 8/4

= 2

Therefore:

slope = 2

Interpretation:

For every increase of 1 in x, y increases by 2.

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Another Worked Example

Find the slope between:

(3, 5)

and:

(7, 9)

m = (9 − 5)/(7 − 3)

= 4/4

= 1

Therefore:

m = 1

The line rises one unit for every one unit it moves to the right.

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The Order of Subtraction Matters

Suppose the points are:

(2, 4) and (5, 10)

We could calculate:

m = (10 − 4)/(5 − 2)

= 6/3

= 2

We can also reverse the points:

m = (4 − 10)/(2 − 5)

= −6/−3

= 2

Both methods work.

The important rule is:

Use the same point first in both the numerator and denominator.

Do not write:

(10 − 4)/(2 − 5)

because you have changed the order in only one part of the calculation.

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Determining Slope from a Graph

To determine slope from a graph:

1. Choose two clear points on the line.
2. Determine the vertical change between them.
3. Determine the horizontal change between them.
4. Calculate rise/run.

It is usually best to choose points where the line passes exactly through grid intersections.

For example, suppose a line passes through:

(1, 2) and (4, 8)

Rise:

8 − 2 = 6

Run:

4 − 1 = 3

Therefore:

m = 6/3

= 2

The Four Types of Slope

There are four important possibilities:

• positive slope
• negative slope
• zero slope
• undefined slope

Being able to recognize these quickly from a graph is essential.

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Positive Slope

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

For example:

m = 3

means that for every increase of 1 in x:

y increases by 3.

Another example:

m = 1/2

means that for every increase of 2 in x:

y increases by 1.

Both slopes are positive.

A positive slope represents an increasing relationship.

Positive Fractional Slopes

A slope does not have to be a whole number.

Consider:

m = 2/5

This means:

rise = 2

run = 5

So for every 5 units moved to the right, the line rises 2 units.

The line still rises from left to right, but less steeply than a line with slope 1 or 2.

---

Negative Slope

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

For example:

m = −3

means that for every increase of 1 in x:

y decreases by 3.

We could think of this as:

rise = −3

run = 1

Therefore:

m = −3/1

= −3

A negative slope represents a decreasing relationship.

Worked Example: Negative Slope

Find the slope between:

(2, 10)

and:

(6, 2)

m = (2 − 10)/(6 − 2)

= −8/4

= −2

Therefore:

m = −2

Interpretation:

For every increase of 1 in x, y decreases by 2.

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Zero Slope

A horizontal line has a slope of zero.

Suppose two points are:

(2, 5)

and:

(8, 5)

Calculate:

m = (5 − 5)/(8 − 2)

= 0/6

= 0

The y-value does not change.

Therefore:

horizontal line → zero slope

A horizontal line can be written in the form:

y = constant

For example:

y = 5

has a slope of zero.

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Why Is a Horizontal Slope Zero?

Think about:

rise/run

For a horizontal line:

rise = 0

The run is nonzero.

Therefore:

0/run = 0

The line has no vertical change, so its slope is zero.

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Undefined Slope

A vertical line has an undefined slope.

Suppose two points are:

(4, 2)

and:

(4, 8)

Calculate:

m = (8 − 2)/(4 − 4)

= 6/0

Division by zero is undefined.

Therefore:

vertical line → undefined slope

A vertical line can be written as:

x = constant

For example:

x = 4

has an undefined slope.

Zero Slope vs Undefined Slope

These two are commonly confused.

Horizontal Line

rise = 0

run ≠ 0

Therefore:

m = 0

Vertical Line

rise ≠ 0

run = 0

Therefore:

m = undefined

A useful memory aid is:

Horizontal → zero

Vertical → undefined

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Comparing Slopes

Slope tells us both:

• direction
• steepness

Consider:

m = 1

and:

m = 4

Both slopes are positive.

But the line with:

m = 4

rises more quickly and is therefore steeper.

Now consider:

m = −2

and:

m = −6

Both slopes are negative.

The line with slope −6 falls more steeply.

The absolute value of the slope helps describe steepness.

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Slope and Rate of Change

Slope is not just a feature of a graph.

It represents a rate of change.

Suppose a distance-time graph has:

m = 20 km/h

This means:

distance increases by 20 kilometres for every hour that passes.

Suppose a temperature-time graph has:

m = −3°C/h

This means:

temperature decreases by 3°C every hour.

The meaning of slope depends on the quantities shown on the axes.

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Units of Slope

The units of slope come from:

vertical-axis units / horizontal-axis units

Suppose:

y-axis = distance in kilometres

x-axis = time in hours

Then:

slope units = km/h

If:

y-axis = temperature in °C

x-axis = time in minutes

then:

slope units = °C/min

If:

y-axis = cost in dollars

x-axis = number of tickets

then:

slope units = $/ticket

Always include units when interpreting a real-world slope.

---

Slope in Distance-Time Graphs

Consider a person walking at a constant speed.

Calculate the slope:

m = (20 − 0)/(4 − 0)

= 20/4

= 5 km/h

The slope represents the person's speed.

A steeper distance-time graph represents a greater speed.

Slope in a Temperature Graph

Suppose a drink cools from:

90°C at 0 minutes

to:

60°C at 10 minutes.

Slope:

m = (60 − 90)/(10 − 0)

= −30/10

= −3°C/min

Interpretation:

The temperature decreases by an average of 3°C each minute over this interval.

The negative slope tells us the quantity is decreasing.

---

Slope in a Cost Graph

Suppose a rental company charges according to:

Calculate:

m = (40 − 25)/(2 − 1)

= 15/1

= $15/h

The slope means:

The cost increases by $15 for each additional hour.

Notice that the slope does not necessarily tell us the starting cost.

---

Slope and Linear Equations

A linear equation is often written:

y = mx + b

where:

• m = slope
• b = y-intercept

For example:

y = 3x + 5

The slope is:

m = 3

The y-intercept is:

b = 5

These have different meanings.

The slope describes the rate of change.

The y-intercept describes the value of y when:

x = 0

---

Example: Interpreting y = mx + b

Suppose:

C = 4d + 10

where:

C = delivery cost in dollars

d = distance in kilometres

The slope is:

4

Its units are:

$/km

Therefore:

The delivery cost increases by $4 for every additional kilometre.

The 10 represents the starting cost:

$10

Slope and starting value should not be confused.

---

Finding Slope from an Equation

When an equation is already written as:

y = mx + b

the slope can be read directly from the coefficient of x.

Example:

y = 7x − 4

Slope:

m = 7

Example:

y = −2x + 8

Slope:

m = −2

Example:

y = (1/3)x + 5

Slope:

m = 1/3

---

When the Equation Is Not in y = mx + b Form

Sometimes we must rearrange the equation.

Consider:

2x + y = 8

Solve for y:

y = −2x + 8

Therefore:

m = −2

Another example:

3x + 2y = 10

Subtract 3x:

2y = −3x + 10

Divide by 2:

y = −(3/2)x + 5

Therefore:

m = −3/2

---

Slope from a Table

We can also calculate slope from a table.

Consider:

Use any two points.

For example:

(1, 5)

and:

(3, 11)

m = (11 − 5)/(3 − 1)

= 6/2

= 3

Check another interval:

m = (17 − 11)/(5 − 3)

= 6/2

= 3

The slope is constant.

Therefore, the table represents a linear relationship.

---

Tables with Unequal Intervals

Consider:

The changes in y are not the same:

+6
+12
+9

But the x-intervals are also different.

Calculate the slopes.

First interval:

(12 − 6)/(3 − 1)

= 6/2

= 3

Second:

(24 − 12)/(7 − 3)

= 12/4

= 3

Third:

(33 − 24)/(10 − 7)

= 9/3

= 3

The slope is still constant.

Therefore, the relationship is linear.

---

Slope and Straight Lines

Every non-vertical straight line has a constant slope.

This means we can choose any two different points on the line and obtain the same slope.

For example, if one pair gives:

m = 2

then every other pair of points on that same straight line will also give:

m = 2

This is one of the defining features of a linear relationship.

---

Curved Graphs

A curved graph does not generally have one constant slope.

Its steepness changes.

For example, a moving car may accelerate.

Its position-time graph may become increasingly steep.

We can still calculate the average slope between two points:

average rate of change = Δy/Δx

But the slope at one exact point requires a more advanced idea.

This leads directly to calculus.

Slope and Calculus

For a straight line, finding slope is easy because the slope is constant.

But suppose the graph is curved.

Its slope may be:

small at one point,

larger at another,

zero at another,

and negative somewhere else.

Calculus allows us to determine the slope at one exact point on a curve.

That slope is called the instantaneous rate of change.

The mathematical tool used to find it is the derivative.

So slope provides an important bridge from algebra into calculus.

---

Real-World Slope: Roads

Road slopes are often described using a grade or percentage.

Suppose a road rises:

8 m

for every:

100 m horizontally.

Its slope is:

8/100

= 0.08

As a percentage:

0.08 × 100%

= 8%

So the road has an:

8% grade

The sign of the slope depends on the direction of travel.

---

Real-World Slope: Ramps

Ramps are designed with slope in mind.

A ramp that rises too quickly over a short horizontal distance is steep.

A ramp with the same rise spread over a longer horizontal distance is gentler.

For example:

Ramp A:

rise = 1 m
run = 5 m

m = 1/5

= 0.20

Ramp B:

rise = 1 m
run = 10 m

m = 1/10

= 0.10

Ramp A is steeper because:

0.20 > 0.10

Real-World Slope: Roofs

Roof designers also consider slope.

A roof's slope affects:

• water drainage
• snow accumulation
• construction materials
• usable roof space
• appearance

A steeper roof has a larger ratio of vertical rise to horizontal run.

This is another example of the same mathematical idea being used in a practical setting.

---

Real-World Slope: Science

Slope appears throughout science.

Position-Time Graph

Slope represents:

velocity

Velocity-Time Graph

Slope represents:

acceleration

Mass-Volume Graph

Slope can represent:

density

Force-Extension Graph

Under suitable conditions, slope can represent:

spring stiffness

Voltage-Current Relationships

Depending on which variable is placed on each axis, slope can represent electrical quantities related to resistance or conductance.

This is why graph interpretation is so important in science.

---

The Meaning Depends on the Axes

Suppose a graph compares:

distance on the y-axis

with:

time on the x-axis

Slope means:

change in distance / change in time

which is speed or velocity magnitude in an appropriate context.

But if we reverse the axes:

time on the y-axis

and:

distance on the x-axis

then slope becomes:

change in time / change in distance

The numerical meaning changes.

Always check:

What is on the y-axis?

What is on the x-axis?

---

Worked Example: Science Graph

A spring experiment produces two points:

Extension = 2 cm, Force = 4 N

Extension = 6 cm, Force = 12 N

If force is on the y-axis and extension is on the x-axis:

m = (12 − 4)/(6 − 2)

= 8/4

= 2 N/cm

Interpretation:

The force increases by 2 N for each additional centimetre of extension over this interval.

---

Worked Example: Negative Real-World Slope

A container is draining.

At 2 minutes:

Volume = 90 L

At 8 minutes:

Volume = 60 L

Calculate:

m = (60 − 90)/(8 − 2)

= −30/6

= −5 L/min

Interpretation:

The volume decreases by 5 litres every minute.

The negative sign communicates important information.

---

Worked Example: Zero Slope

A parked vehicle remains 200 m from a reference point for 5 minutes.

Its position-time graph is horizontal.

Change in position:

Δy = 0

Therefore:

m = 0/time

= 0 m/min

The zero slope tells us that its position is not changing.

---

Choosing Good Points from a Graph

When calculating slope visually:

• choose points clearly on the line
• use grid intersections when possible
• choose points reasonably far apart
• read coordinates carefully
• calculate rise and run
• include the correct sign

Choosing points farther apart can sometimes reduce the effect of small graph-reading errors.

But both points must lie on the actual line.

---

A Useful Visual Check

Before calculating, predict the sign.

Ask:

Does the line rise, fall, stay flat, or stand vertically as I move left to right?

Then predict:

Rises → positive

Falls → negative

Horizontal → zero

Vertical → undefined

After calculating, compare your result with your prediction.

If a line clearly falls from left to right but you calculate a positive slope, check your subtraction.

---

Common Mistakes

Using run/rise

Slope is:

rise/run

not:

run/rise.

Mixing the subtraction order

If you calculate:

y₂ − y₁

you must also calculate:

x₂ − x₁.

Forgetting negative signs

A falling line has a negative slope.

Saying a horizontal line has undefined slope

A horizontal line has:

m = 0

Saying a vertical line has zero slope

A vertical line has:

undefined slope

because the run is zero.

Dividing by zero

Division by zero is undefined.

Looking only at steepness

Slope describes both steepness and direction.

Assuming every positive slope equals 1

Any positive value can represent a positive slope.

Thinking −2 is steeper than −5

The slope −5 has the greater magnitude and represents a steeper line.

Confusing slope with the y-intercept

In:

y = 4x + 7

the slope is 4.

The y-intercept is 7.

Forgetting units in real-world problems

A slope of:

5 km/h

communicates more information than simply writing 5.

Ignoring the axes

The meaning and units of slope depend on which quantities are plotted.

Using only one point

Slope describes change between points. One point alone is not enough to determine the slope of an arbitrary line.

---

Key Terms

Slope — A measure of the steepness and direction of a line.

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

Rise — The vertical change between two points.

Run — The horizontal change between two points.

Slope formula — The formula m = (y₂ − y₁)/(x₂ − x₁).

Delta (Δ) — A symbol meaning "change in."

Δx — Change in the horizontal variable.

Δy — Change in the vertical variable.

Positive slope — A slope that rises from left to right.

Negative slope — A slope that falls from left to right.

Zero slope — The slope of a horizontal line.

Undefined slope — The slope of a vertical line.

Linear relationship — A relationship represented by a straight line and having a constant rate of change.

Constant slope — A slope that remains the same throughout a straight line.

Steepness — How rapidly a line rises or falls.

Magnitude of slope — The absolute value of the slope, useful for comparing steepness.

Coordinate — An ordered pair (x, y) describing a point's location.

x-coordinate — The horizontal position of a point.

y-coordinate — The vertical position of a point.

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

Slope-intercept form — The linear equation form y = mx + b.

Initial value — The y-value when x = 0.

Horizontal line — A line with zero slope.

Vertical line — A line with undefined slope.

Grade — A way of expressing slope as a percentage, often used for roads.

Average rate of change — The slope between two points on a graph.

Instantaneous rate of change — The rate of change at one exact point, studied using calculus.

Derivative — A calculus concept used to determine instantaneous rate of change.

---

Key Takeaways

• Slope describes the steepness and direction of a line.
• Slope is also a rate of change.
• The symbol m is commonly used for slope.
• Slope can be calculated using:

m = rise/run

• It can also be written:

m = Δy/Δx

• From two points:

m = (y₂ − y₁)/(x₂ − x₁)

• The subtraction order must remain consistent.
• A positive slope rises from left to right.
• A negative slope falls from left to right.
• A horizontal line has zero slope.
• A vertical line has undefined slope.
• Vertical slopes are undefined because they require division by zero.
• A larger absolute slope generally represents a steeper line.
• Straight lines have constant slopes.
• Slope can be calculated from coordinates, graphs, tables, and equations.
• In y = mx + b, m represents slope.
• The y-intercept is not the same as the slope.
• The units of slope are:

y-axis units / x-axis units

• Real-world slopes must be interpreted in context.
• Slope can represent speed, acceleration, cost per unit, temperature change, density, and many other rates.
• Road and ramp steepness can also be described using slope.
• The meaning of slope depends on which quantities appear on the axes.
• Curved graphs generally have changing slopes.
• The slope between two points on a curve represents an average rate of change.
• Calculus extends the idea of slope to find the instantaneous rate of change at a single point.
• The derivative is fundamentally connected to slope.

A useful visual reminder is:

↗ Positive

↘ Negative

→ Zero

↑ Undefined

And the most important calculation to remember is:

Slope = change in y / change in x

---

Check Your Understanding

1. Define slope in your own words.

2. What two features of a line does slope describe?

3. Write the slope formula using rise and run.

4. Write the slope formula using Δy and Δx.

5. Find the slope between:

(1, 3) and (5, 11)

6. Find the slope between:

(2, 9) and (6, 17)

7. Find the slope between:

(3, 12) and (7, 4)

8. Is the slope in Question 7 positive or negative?

9. Find the slope between:

(2, 6) and (8, 6)

10. Explain why the answer to Question 9 is zero.

11. Find the slope between:

(5, 2) and (5, 10)

12. Explain why the slope in Question 11 is undefined.

13. What type of slope does a horizontal line have?

14. What type of slope does a vertical line have?

15. A line rises 12 units while moving 4 units to the right. Find its slope.

16. A line falls 15 units while moving 5 units to the right. Find its slope.

17. Which is steeper: a line with slope 2 or a line with slope 6?

18. Which is steeper: a line with slope −3 or a line with slope −8?

19. Identify the slope of:

y = 5x + 2

20. Identify the slope of:

y = −4x + 7

21. Identify the slope of:

y = (2/3)x − 5

22. Rearrange and determine the slope:

2x + y = 10

23. Determine the slope represented by the table:

24. A distance-time graph has a slope of 12 km/h. Explain what this means.

25. A temperature-time graph has a slope of −2°C/min. Explain what this means.

26. A water-volume graph has a slope of 8 L/min. What does the slope tell you?

27. A road rises 6 m for every 100 m of horizontal distance. Calculate its slope and percentage grade.

28. Explain why the units of slope depend on the axes.

29. Explain the connection between slope and rate of change.

30. Challenge: A delivery company's cost is represented by:

C = 3.5d + 8

where:

C = cost in dollars

d = distance in kilometres

a. Identify the slope.
b. State the units of the slope.
c. Explain the meaning of the slope.
d. Identify the initial value.
e. Explain why the initial value is not the slope.
f. Calculate the cost for a 12 km delivery.
g. Sketch what the graph should generally look like.
h. Explain why the line has a positive slope.
i. Describe what would happen to the graph if the rate increased from $3.50/km to $5.00/km.
j. Explain how this example connects slope to the idea of rate of change.

Writing Linear Equations

A linear equation describes a relationship in which one variable changes at a constant rate compared with another.

One of the most useful forms is:

y = mx + b

where:

• m = slope or rate of change
• b = y-intercept or initial value

Linear equations can be written from many kinds of information, including:

• two points
• a slope and y-intercept
• a table of values
• a graph
• a real-world situation

A useful overall strategy is:

Find the rate → Find the initial value → Write the equation → Verify

---

What Information Defines a Line?

A straight line can be determined in several ways.

For example, we can determine a unique nonvertical line if we know:

• two different points
• its slope and one point
• its slope and y-intercept

A table or real-world description can also provide this information.

Once we know:

m

and:

b

we can write:

y = mx + b

---

Writing an Equation from Slope and y-Intercept

Suppose:

slope = 3

y-intercept = 5

Use:

y = mx + b

Substitute:

m = 3

b = 5

Therefore:

y = 3x + 5

This is the simplest situation because both values needed for slope-intercept form are already known.

---

Negative Slope and Intercept

Suppose:

m = −2

b = 7

Then:

y = −2x + 7

The negative slope means the line decreases as x increases.

The y-intercept means the line crosses the y-axis at:

(0, 7)

---

Negative y-Intercept

Suppose:

m = 4

b = −6

Then:

y = 4x − 6

The graph:

• rises from left to right
• crosses the y-axis at (0, −6)

---

Zero y-Intercept

Suppose:

m = 5

and the line passes through the origin.

The origin is:

(0, 0)

Therefore:

b = 0

and:

y = 5x

An equation of the form:

y = mx

represents a direct proportional relationship.

---

Writing an Equation from Two Points

Suppose a line passes through:

(2, 5)

and:

(6, 13)

We do not yet know m or b.

First find the slope:

m = (y₂ − y₁)/(x₂ − x₁)

Substitute:

m = (13 − 5)/(6 − 2)

m = 8/4

m = 2

Now use:

y = mx + b

and substitute one of the points.

Using:

(2, 5)

gives:

5 = 2(2) + b

5 = 4 + b

Therefore:

b = 1

The equation is:

y = 2x + 1

---

The Two-Point Strategy

When given two points:

(x₁, y₁) and (x₂, y₂)

use:

Find m → Find b → Write y = mx + b → Verify

First:

m = (y₂ − y₁)/(x₂ − x₁)

Then substitute m and one known point into:

y = mx + b

to determine b.

Finally, write the complete equation.

---

Another Two-Point Example

Find the equation through:

(1, 4)

and:

(5, 12)

Find the slope:

m = (12 − 4)/(5 − 1)

m = 8/4

m = 2

Now use:

y = 2x + b

Substitute:

(1, 4)

4 = 2(1) + b

4 = 2 + b

b = 2

Therefore:

y = 2x + 2

---

Verify Using the Second Point

We used:

(1, 4)

to find b.

Now check:

(5, 12)

in:

y = 2x + 2

Substitute:

12 = 2(5) + 2

12 = 12 ✓

Therefore the equation matches both points.

---

Two Points with a Negative Slope

Find the equation through:

(1, 8)

and:

(4, 2)

Slope:

m = (2 − 8)/(4 − 1)

m = −6/3

m = −2

Now:

y = −2x + b

Use:

(1, 8)

8 = −2(1) + b

8 = −2 + b

Therefore:

b = 10

Equation:

y = −2x + 10

Check:

2 = −2(4) + 10

2 = 2 ✓

---

Be Consistent When Finding Slope

Suppose the points are:

(2, 7) and (5, 16)

Correct:

m = (16 − 7)/(5 − 2)

You can also reverse both:

m = (7 − 16)/(2 − 5)

Both give:

m = 3

But do not reverse only the numerator:

(7 − 16)/(5 − 2)

because this changes the sign incorrectly.

The point order must be consistent.

---

Fractional Slopes

Suppose a line passes through:

(2, 3)

and:

(6, 5)

Slope:

m = (5 − 3)/(6 − 2)

m = 2/4

m = 1/2

Use:

y = (1/2)x + b

Substitute:

3 = (1/2)(2) + b

3 = 1 + b

b = 2

Therefore:

y = (1/2)x + 2

Fractional slopes are completely normal.

---

Points Containing Negative Coordinates

Find the equation through:

(−2, 1)

and:

(2, 9)

Slope:

m = (9 − 1)/(2 − (−2))

m = 8/4

m = 2

Now:

y = 2x + b

Use:

(−2, 1)

1 = 2(−2) + b

1 = −4 + b

Therefore:

b = 5

Equation:

y = 2x + 5

---

Writing an Equation from Slope and One Point

Suppose:

m = 3

and the line passes through:

(4, 14)

Start:

y = mx + b

Substitute:

14 = 3(4) + b

14 = 12 + b

Therefore:

b = 2

Equation:

y = 3x + 2

---

Point-Slope Form

Another useful form is:

y − y₁ = m(x − x₁)

This is called point-slope form.

It is especially useful when you know:

• the slope
• one point

For example, if:

m = 3

and:

(x₁, y₁) = (4, 14)

then:

y − 14 = 3(x − 4)

Expand:

y − 14 = 3x − 12

Add 14:

y = 3x + 2

The same equation is obtained.

---

Which Method Should You Use?

If you know m and b directly:

Use y = mx + b.

If you know m and one point:

You can:

• substitute into y = mx + b to find b, or
• use point-slope form

If you know two points:

• calculate m first
• then use either method

Choose the method that makes the algebra easiest.

---

Writing an Equation from a Table

Consider:

Look at the change in y.

Each time x increases by 1:

y increases by 2

Therefore:

m = 2

When:

x = 0

we have:

y = 3

Therefore:

b = 3

Equation:

y = 2x + 3

---

Finding Slope from a Table

Slope is:

m = change in y / change in x

Suppose:

From the first row to the second:

change in x = 2

change in y = 6

Therefore:

m = 6/2

m = 3

---

Finding the Intercept When x = 0 Is Missing

In the previous table:

we know:

m = 3

But the table does not show x = 0.

Use one point:

(2, 7)

in:

y = mx + b

7 = 3(2) + b

7 = 6 + b

Therefore:

b = 1

Equation:

y = 3x + 1

---

Tables with Larger x-Intervals

Do not assume that the change in y is automatically the slope.

Consider:

The y-values increase by:

20

but the x-values increase by:

10

Therefore:

m = 20/10

m = 2

The slope is 2, not 20.

---

Tables with Decreasing Values

Consider:

Changes:

Δx = 2

Δy = −4

Therefore:

m = −4/2

m = −2

Since:

x = 0 → y = 20

we know:

b = 20

Equation:

y = −2x + 20

---

How Do We Know a Table Is Linear?

A table represents a linear relationship when it has a constant rate of change.

For example:

Each time x increases by 1, y increases by 3.

Therefore the rate of change is constant.

The relationship is linear.

---

A Nonlinear Table

Consider:

The changes in y are:

+3, +5, +7

The rate of change is not constant.

Therefore these values do not represent a linear relationship.

Not every table should be modeled with:

y = mx + b

---

Unequal x-Intervals

Constant differences in y are only useful when the x-intervals are equal.

Suppose:

First interval:

m = (11 − 5)/(3 − 1)

m = 6/2

m = 3

Second interval:

m = (23 − 11)/(7 − 3)

m = 12/4

m = 3

The rate of change is constant.

Therefore the table is linear even though the x-values are not equally spaced.

---

Writing an Equation from a Graph

To write an equation from a graph:

1. identify two clear points
2. calculate the slope
3. identify the y-intercept or calculate it
4. write y = mx + b
5. check that the equation matches the graph

If the graph clearly crosses the y-axis at an integer value, that point is often the easiest starting point.

---

Example from a Graph

Suppose a graph passes through:

(0, 4)

and:

(3, 10)

Slope:

m = (10 − 4)/(3 − 0)

m = 6/3

m = 2

Since:

(0, 4)

is the y-intercept:

b = 4

Therefore:

y = 2x + 4

---

Graphs Without an Obvious y-Intercept

Suppose a graph clearly passes through:

(2, 5)

and:

(6, 13)

but the y-intercept is not easy to read.

Calculate:

m = (13 − 5)/(6 − 2)

m = 2

Then use:

(2, 5)

to find b:

5 = 2(2) + b

b = 1

Therefore:

y = 2x + 1

You do not need to estimate the intercept from the graph.

---

Verifying an Equation Against a Graph

Suppose you think a graph represents:

y = 3x − 2

Check several features.

Check the y-Intercept

The equation predicts:

(0, −2)

Does the graph cross the y-axis there?

Check the Slope

The equation predicts:

m = 3 = 3/1

Does the line rise 3 units for every 1 unit to the right?

Check Another Point

If the graph contains:

(2, 4)

substitute:

4 = 3(2) − 2

4 = 4 ✓

The equation is consistent with the graph.

---

Why Checking More Than One Feature Matters

Suppose two lines both cross the y-axis at:

(0, 5)

One might be:

y = 2x + 5

and another:

y = −3x + 5

Checking only the y-intercept would not distinguish them.

Similarly, two parallel lines can have the same slope but different intercepts.

A good verification checks both:

slope and intercept

or:

at least two points

---

Modeling Real Situations

Many real-world situations have:

• an initial amount
• a constant rate of change

These naturally produce linear equations.

The general model is:

output = rate × input + initial value

This is the same structure as:

y = mx + b

---

Taxi Fare

A taxi charges:

• $4 starting fee
• $2.50 per kilometre

Let:

C = total cost

d = distance in kilometres

The rate is:

m = 2.5

The initial value is:

b = 4

Therefore:

C = 2.5d + 4

---

Savings

A student already has $80 and saves $15 each week.

Let:

S = total savings

w = number of weeks

Initial value:

80

Rate:

15 dollars/week

Therefore:

S = 15w + 80

After 10 weeks:

S = 15(10) + 80

S = 230

---

Temperature

A liquid begins at 90°C and cools by 4°C each minute.

Let:

T = temperature

t = time in minutes

The rate is negative:

m = −4

Initial temperature:

b = 90

Therefore:

T = −4t + 90

---

Filling a Tank

A tank contains 120 L of water and receives water at 8 L/min.

Let:

V = volume

t = time

Initial amount:

120 L

Rate:

8 L/min

Equation:

V = 8t + 120

---

Draining a Tank

A tank contains 500 L and drains at 20 L/min.

The initial value is:

500

The rate is:

−20 L/min

Therefore:

V = −20t + 500

The negative slope represents decreasing volume.

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Constant-Speed Motion

A cyclist begins 10 km from a reference point and travels away from it at 18 km/h.

Let:

d = distance from the reference point

t = time in hours

Then:

d = 18t + 10

Slope:

18 km/h

Intercept:

10 km

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Modeling from Two Measurements

Suppose an experiment records:

At 2 seconds:

temperature = 24°C

At 6 seconds:

temperature = 36°C

Assuming a linear relationship, use the points:

(2, 24)

and:

(6, 36)

Slope:

m = (36 − 24)/(6 − 2)

m = 12/4

m = 3

So:

T = 3t + b

Use:

(2, 24)

24 = 3(2) + b

b = 18

Therefore:

T = 3t + 18

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Interpreting the Model

For:

T = 3t + 18

the slope:

3°C/s

means temperature increases by 3°C each second.

The intercept:

18°C

means the model predicts:

T = 18°C when t = 0

Even if the experiment did not measure t = 0 directly, the equation allows us to determine the model's predicted initial value.

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Interpolation

Suppose a linear model is based on measurements at:

x = 2

and:

x = 10

Using the equation to predict a value at:

x = 6

is called interpolation.

The prediction is being made inside the range of known data.

Interpolation is often reasonably reliable when the relationship is genuinely linear.

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Extrapolation

Using the same model to predict a value at:

x = 100

would be extrapolation.

This prediction is far outside the measured range.

A linear pattern may not continue indefinitely, so extrapolated predictions should be treated more cautiously.

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Linear Models Have Limits

An equation may accurately model a situation only over a certain range.

For example:

V = −20t + 500

models a draining tank.

Eventually:

V = 0

After that, the formula would predict negative water volume.

That is mathematically possible in the equation but physically impossible.

Real-world models have appropriate domains.

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Writing Equations from Verbal Descriptions

Look for two pieces of information:

Rate of Change

Words such as:

• per
• each
• every
• increases by
• decreases by

often indicate slope.

Initial Value

Words such as:

• starts with
• initially
• base fee
• fixed charge
• already has

often indicate the intercept.

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Example from Words

A phone plan costs $25 per month plus $0.10 per text message.

Let:

C = total cost

t = number of text messages

Rate:

$0.10/text

Initial amount:

$25

Therefore:

C = 0.10t + 25

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Writing an Equation from a Pattern

Suppose:

The cost increases by:

$4 per item

Therefore:

m = 4

Use:

(1, 7)

to find b:

7 = 4(1) + b

b = 3

Equation:

C = 4n + 3

This means there is a $3 fixed cost plus $4 per item.

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Equations from Horizontal Lines

Suppose two points are:

(2, 5)

and:

(8, 5)

Slope:

m = (5 − 5)/(8 − 2)

m = 0

Therefore:

y = 5

This is a linear equation representing a horizontal line.

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Vertical Lines

Suppose the points are:

(4, 2)

and:

(4, 9)

The slope calculation would require:

(9 − 2)/(4 − 4)

which involves division by zero.

The slope is undefined.

The equation is:

x = 4

This is a linear equation, but it cannot be written in slope-intercept form:

y = mx + b

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Parallel Lines

Suppose one line is:

y = 3x + 5

Any different line parallel to it must have the same slope:

m = 3

For example:

y = 3x − 2

The slopes are equal, but the intercepts are different.

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Writing a Parallel Line

Find the equation of a line parallel to:

y = 2x + 7

and passing through:

(3, 1)

Parallel lines have the same slope.

Therefore:

m = 2

Use:

y = 2x + b

Substitute:

1 = 2(3) + b

1 = 6 + b

b = −5

Therefore:

y = 2x − 5

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Comparing Equations and Graphs

Suppose a graph:

• crosses the y-axis at 4
• decreases 2 units for every 1 unit right

Then:

m = −2

b = 4

Equation:

y = −2x + 4

If someone proposes:

y = 2x + 4

the intercept is correct, but the slope has the wrong sign.

The graph itself provides evidence that the equation is incorrect.

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Checking with Substitution

Suppose you believe the equation through:

(3, 7)

and:

(5, 11)

is:

y = 2x + 1

Check the first point:

7 = 2(3) + 1

7 = 7 ✓

Check the second:

11 = 2(5) + 1

11 = 11 ✓

Both points satisfy the equation.

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Checking with Slope

The points:

(3, 7)

and:

(5, 11)

have slope:

m = (11 − 7)/(5 − 3)

m = 4/2

m = 2

The proposed equation:

y = 2x + 1

also has slope 2.

This provides another check.

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A Complete Example

Find the equation through:

(−1, 6)

and:

(3, −2)

Find the slope

m = (−2 − 6)/(3 − (−1))

m = −8/4

m = −2

Find the intercept

Use:

y = −2x + b

Substitute:

(−1, 6)

6 = −2(−1) + b

6 = 2 + b

b = 4

Write the equation

y = −2x + 4

Verify

Using:

(3, −2)

−2 = −2(3) + 4

−2 = −6 + 4

−2 = −2 ✓

Therefore:

y = −2x + 4

is correct.

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A Reliable Strategy

When writing a linear equation, use:

Identify → Calculate → Construct → Verify → Interpret

Identify

Determine what information is given.

Calculate

Find the slope if necessary.

Construct

Find the intercept and write the equation.

Verify

Check known points, the table, or the graph.

Interpret

If the equation models a real situation, explain what its slope and intercept mean.

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Common Mistakes

Mistake: Using only one point to determine a line without knowing the slope.

Many different lines can pass through one point.

---

Mistake: Mixing the order of points in the slope formula.

Keep subtraction order consistent.

---

Mistake: Assuming the change in y is always the slope.

Slope is:

change in y / change in x

---

Mistake: Assuming the first table value is the y-intercept.

The y-intercept occurs when:

x = 0

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Mistake: Forgetting to calculate b after finding m.

Finding the slope does not give the complete equation.

---

Mistake: Using a point's y-coordinate as b when x ≠ 0.

Only a point with x = 0 directly gives the y-intercept.

---

Mistake: Ignoring negative signs.

A decreasing relationship has a negative slope.

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Mistake: Assuming every table is linear.

Check for a constant rate of change first.

---

Mistake: Using y = mx + b for a vertical line.

A vertical line has equation:

x = constant

---

Mistake: Failing to verify the equation.

Substitute known points back into the final equation.

---

Mistake: Checking only the intercept against a graph.

An equation must match both the slope and intercept.

---

Mistake: Ignoring the meaning of the variables.

In real-world models, interpret the rate, initial value, and appropriate domain.

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Key Terms

Linear equation — equation representing a constant rate of change whose graph is a straight line.

Slope-intercept form — equation written as y = mx + b.

Slope — rate of change of y compared with x.

Rate of change — ratio of the change in one variable to the change in another.

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

Initial value — value of the dependent variable when the independent variable equals zero.

Slope formula — m = (y₂ − y₁)/(x₂ − x₁).

Point-slope form — equation written as y − y₁ = m(x − x₁).

Ordered pair — coordinate written (x, y).

Linear relationship — relationship with a constant rate of change.

Constant rate of change — rate that remains the same throughout a relationship.

Direct proportion — linear relationship of the form y = mx that passes through the origin.

Model — mathematical representation of a real situation.

Interpolation — using a model to predict within the range of known data.

Extrapolation — using a model to predict beyond the range of known data.

Domain — set of input values for which a relationship or model is meaningful.

Verification — checking that an equation satisfies known information.

Horizontal line — line with slope zero.

Vertical line — line with undefined slope.

Parallel lines — lines with equal slopes that do not intersect.

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Key Takeaways

• Linear equations describe relationships with constant rates of change.
• Slope-intercept form is y = mx + b.
• The slope m represents the rate of change.
• The intercept b represents the value of y when x = 0.
• If slope and intercept are known, substitute them directly into y = mx + b.
• Two distinct points determine one straight line unless the line is vertical.
• When given two points, calculate the slope first.
• Use m = (y₂ − y₁)/(x₂ − x₁).
• Keep the order of subtraction consistent.
• After finding m, substitute one point into y = mx + b to determine b.
• Point-slope form can also be used when a slope and point are known.
• Tables represent linear relationships when the rate of change is constant.
• Do not confuse change in y with slope when x changes by more than 1.
• If x = 0 appears in a table, its y-value gives the y-intercept directly.
• If x = 0 is missing, use another point to calculate b.
• Graphs can provide two points, a slope, or a y-intercept.
• Verify a graph equation by checking both slope and intercept or by checking multiple points.
• Real-world linear models often contain an initial value and a constant rate.
• Words such as "per" and "each" often indicate slope.
• Words such as "initial," "starting," and "base fee" often indicate the intercept.
• A negative slope represents a decreasing relationship.
• A zero slope represents a horizontal line.
• A vertical line cannot be written in ordinary slope-intercept form.
• Linear models may only be appropriate over a limited domain.
• Interpolation is usually safer than extrapolation.
• Always verify a final equation using the information originally provided.

Check Your Understanding

1. Write an equation with slope 4 and y-intercept 7.
2. Write an equation with slope −3 and y-intercept 5.
3. Write an equation with slope 2 and y-intercept 0.
4. Find the slope between (2, 5) and (6, 13).
5. Find the equation of the line through (2, 5) and (6, 13).
6. Find the equation through (1, 8) and (4, 2).
7. Find the equation through (−2, 1) and (2, 9).
8. A line has slope 5 and passes through (2, 13). Find its equation.
9. A line has slope −2 and passes through (4, −1). Find its equation.
10. Determine the equation represented by the table: (0, 4), (1, 7), (2, 10), (3, 13).
11. Determine the equation represented by: (2, 9), (4, 15), (6, 21), (8, 27).
12. Determine whether the table (1, 2), (2, 4), (3, 8), (4, 16) represents a linear relationship. Explain.
13. Explain why the change in y alone does not always equal the slope.
14. A graph passes through (0, −3) and (4, 5). Write its equation.
15. Explain how you could verify that y = 2x − 3 matches a particular graph.
16. A taxi charges $5 initially and $2.20 per kilometre. Write a linear equation for the total cost C after travelling d kilometres and interpret both numbers in your equation.
17. A tank contains 300 L and drains at 12 L/min. Write an equation for its volume V after t minutes. State a reasonable domain for the model.
18. At 3 seconds an experiment records 14°C, and at 7 seconds it records 30°C. Assuming a linear relationship, write an equation relating temperature T to time t.
19. A line is parallel to y = 4x + 2 and passes through (3, 5). Find its equation and verify that the slopes are equal.
20. Create a real-world situation that can be modeled by y = −5x + 80. Explain what the slope and intercept represent, and describe how you would verify that a graph correctly represents your equation.