Review of the Coordinate Plane

The coordinate plane is a two-dimensional system used to locate points using numbers. It allows us to describe exactly where an object or point is positioned relative to two perpendicular number lines.

The two number lines are called the:

• x-axis — the horizontal axis
• y-axis — the vertical axis

The axes intersect at a special point called the origin.

The coordinate plane is used throughout mathematics and science for graphing functions, displaying data, describing motion, mapping locations, and analyzing relationships between variables.

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The x-Axis

The x-axis is the horizontal number line.

Values increase as you move to the right and decrease as you move to the left.

For example:

• x = 5 is 5 units to the right of the origin
• x = −5 is 5 units to the left of the origin

The x-axis itself can be described by:

y = 0

Every point on the x-axis has a y-coordinate of zero.

Examples include:

(4, 0)

(−7, 0)

(12, 0)

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The y-Axis

The y-axis is the vertical number line.

Values increase as you move upward and decrease as you move downward.

For example:

• y = 4 is 4 units above the origin
• y = −4 is 4 units below the origin

The y-axis can be described by:

x = 0

Every point on the y-axis has an x-coordinate of zero.

Examples include:

(0, 6)

(0, −3)

(0, 10)

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The Origin

The x-axis and y-axis intersect at the origin.

The coordinates of the origin are:

(0, 0)

The origin is the reference point from which other positions are measured.

At the origin:

x = 0

and:

y = 0

The origin does not belong to any quadrant.

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Ordered Pairs

A point on the coordinate plane is represented by an ordered pair:

(x, y)

The first number gives the horizontal position.

The second number gives the vertical position.

For example:

(4, 3)

means:

• move 4 units to the right
• move 3 units upward

The order matters.

(4, 3) and (3, 4) represent different points.

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x Comes Before y

A useful rule is:

x first, y second

For:

(−5, 2)

x = −5

y = 2

Start at the origin.

Move:

5 units left

then:

2 units up

This locates the point (−5, 2).

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Horizontal Movement

The x-coordinate tells you how far to move horizontally.

If x is:

• positive → move right
• negative → move left
• zero → remain on the y-axis

Examples:

(6, 2) → 6 units right

(−6, 2) → 6 units left

(0, 2) → no horizontal movement

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Vertical Movement

The y-coordinate tells you how far to move vertically.

If y is:

• positive → move up
• negative → move down
• zero → remain on the x-axis

Examples:

(3, 5) → 5 units up

(3, −5) → 5 units down

(3, 0) → no vertical movement

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Plotting an Ordered Pair

To plot:

(3, 4)

begin at:

(0, 0)

Then:

1. move 3 units right
2. move 4 units up
3. mark the point

The x-coordinate is always used first.

A common phrase is:

"Across first, then up or down."

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Plotting Negative Coordinates

Consider:

(−4, −2)

Start at the origin.

The x-coordinate is −4:

move 4 units left

The y-coordinate is −2:

move 2 units down

Mark the point.

The point lies in the lower-left part of the coordinate plane.

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The Four Quadrants

The x-axis and y-axis divide the coordinate plane into four regions called quadrants.

They are numbered using Roman numerals:

• Quadrant I
• Quadrant II
• Quadrant III
• Quadrant IV

The quadrants are numbered counterclockwise, beginning in the upper-right region.

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Quadrant I

Quadrant I is the upper-right region.

In Quadrant I:

x is positive

y is positive

Sign pattern:

(+, +)

Examples:

(2, 5)

(7, 3)

(1, 9)

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Quadrant II

Quadrant II is the upper-left region.

In Quadrant II:

x is negative

y is positive

Sign pattern:

(−, +)

Examples:

(−3, 5)

(−8, 2)

(−1, 7)

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Quadrant III

Quadrant III is the lower-left region.

In Quadrant III:

x is negative

y is negative

Sign pattern:

(−, −)

Examples:

(−4, −6)

(−2, −9)

(−7, −1)

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Quadrant IV

Quadrant IV is the lower-right region.

In Quadrant IV:

x is positive

y is negative

Sign pattern:

(+, −)

Examples:

(3, −4)

(8, −2)

(1, −6)

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Remembering the Quadrant Signs

The signs follow this pattern:

Beginning in Quadrant I and moving counterclockwise:

(+,+) → (−,+) → (−,−) → (+,−)

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Points on the Axes

A point located directly on an axis is not in a quadrant.

For example:

(5, 0)

lies on the x-axis.

(0, −4)

lies on the y-axis.

(0, 0)

is the origin.

None of these points belongs to Quadrants I, II, III, or IV.

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Identifying the Quadrant Without Plotting

You can determine the quadrant simply by looking at the signs.

Consider:

(−7, 4)

x is negative.

y is positive.

Sign pattern:

(−, +)

Therefore:

Quadrant II

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Another Example

Consider:

(6, −9)

x is positive.

y is negative.

Sign pattern:

(+, −)

Therefore:

Quadrant IV

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Determining Coordinates from a Graph

Sometimes a point is already plotted and you need to determine its coordinates.

To read the point:

1. determine its horizontal position
2. record this as x
3. determine its vertical position
4. record this as y
5. write the coordinates as (x, y)

For example, suppose a point is:

• 4 units left of the origin
• 3 units above the origin

Then:

x = −4

y = 3

The point is:

(−4, 3)

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Reading the Scale Carefully

Not every coordinate plane uses intervals of 1.

An axis might increase by:

2, 5, 10, 100, or another amount.

For example, suppose the x-axis is labelled:

0, 10, 20, 30, 40

A point at the third marked interval to the right of zero may represent:

x = 30

not x = 3.

Always check the scale before reading coordinates.

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Plotting Several Points

Suppose we need to plot:

A = (2, 3)

B = (−3, 4)

C = (−4, −2)

D = (5, −3)

Their quadrants are:

Each point has a different combination of coordinate signs.

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Comparing Points

Consider:

A = (3, 5)

B = (3, −2)

The points have the same x-coordinate.

Therefore, they are positioned on the same vertical line.

Now consider:

C = (−4, 6)

D = (2, 6)

These points have the same y-coordinate.

Therefore, they are positioned on the same horizontal line.

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Same x-Coordinate

Points with the same x-coordinate line up vertically.

For example:

(4, 6)

(4, 2)

(4, −3)

all have:

x = 4

They lie on the vertical line:

x = 4

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Same y-Coordinate

Points with the same y-coordinate line up horizontally.

For example:

(−5, 3)

(0, 3)

(7, 3)

all have:

y = 3

They lie on the horizontal line:

y = 3

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Reflecting Across the x-Axis

Consider the point:

(3, 4)

Reflecting it across the x-axis changes the sign of the y-coordinate.

Therefore:

(3, 4) → (3, −4)

The x-coordinate stays the same.

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Reflecting Across the y-Axis

Reflecting:

(3, 4)

across the y-axis changes the sign of the x-coordinate.

Therefore:

(3, 4) → (−3, 4)

The y-coordinate stays the same.

These patterns can help strengthen understanding of coordinate signs.

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Reflecting Across Both Axes

If both coordinate signs change:

(3, 4) → (−3, −4)

The point moves to the opposite side of the origin.

Likewise:

(−5, 2) → (5, −2)

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Coordinates and Direction

Coordinates can describe position relative to a reference point.

Suppose the origin represents a person's starting position.

Let:

• positive x = east
• negative x = west
• positive y = north
• negative y = south

Then:

(4, 3)

could represent:

4 km east and 3 km north of the starting point

while:

(−2, −5)

could represent:

2 km west and 5 km south

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Real-World Context: Maps

Coordinate systems are useful for locating objects on maps.

Suppose a park entrance is defined as:

(0, 0)

A playground is at:

(5, 2)

This could mean the playground is:

• 5 grid units east
• 2 grid units north

A pond at:

(−3, 4)

would be:

• 3 grid units west
• 4 grid units north

Coordinates provide a precise way to communicate location.

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Real-World Context: Games

Many computer games use coordinate systems to track the positions of:

• players
• objects
• enemies
• targets
• buildings

If a character moves from:

(2, 3)

to:

(5, 3)

the y-coordinate remains unchanged.

The character has moved horizontally.

The change in x is:

5 − 2 = 3

So the character moved 3 units to the right.

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Real-World Context: Science

Coordinate planes are widely used in science.

For example, a graph might represent:

• time on the x-axis
• distance on the y-axis

A point:

(4, 20)

could mean:

At 4 seconds, the object has travelled 20 metres.

The meaning of a coordinate depends on the quantities represented by the axes.

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Real-World Context: Temperature

Suppose:

x = time in hours

y = temperature in °C

The point:

(3, −5)

means:

After 3 hours, the temperature was −5°C.

In a real graph, the signs of coordinates describe values, not necessarily physical directions.

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Coordinates Need Units and Labels

The point:

(4, 10)

has little real-world meaning without context.

It could represent:

• 4 hours and 10 kilometres
• 4 seconds and 10 metres
• 4 items and $10
• 4 weeks and 10 centimetres

Always examine the axis labels and units when interpreting coordinates.

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Independent and Dependent Variables

When a coordinate plane is used to graph a relationship, the x-axis often represents the independent variable.

The y-axis often represents the dependent variable.

For example:

x = hours worked

y = money earned

The point:

(5, 90)

could mean:

After working 5 hours, a person earned $90.

This connects coordinate geometry with functions and linear relationships.

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Coordinates in a Table

A table can represent points on a coordinate plane.

Each row creates an ordered pair:

(−2, 4)

(0, 2)

(2, 0)

(4, −2)

These points can then be plotted on a coordinate plane.

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Tables, Equations, and Coordinate Points

Suppose:

y = 2x + 1

A table might contain:

Each row corresponds to a point:

(−2, −3)

(−1, −1)

(0, 1)

(1, 3)

(2, 5)

Plotting these points reveals the graph of the relationship.

This is why understanding the coordinate plane is essential for graphing equations.

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Quadrants and Linear Graphs

A line may pass through several quadrants.

For example, consider:

y = x

Positive x-values produce positive y-values, so those points lie in Quadrant I.

Negative x-values produce negative y-values, so those points lie in Quadrant III.

The line also passes through the origin.

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Coordinates with Decimals and Fractions

Coordinates do not have to be whole numbers.

Examples include:

(2.5, 4)

(−1.5, 3.2)

(1/2, −3)

To plot these accurately, use the scale of the graph to locate values between marked numbers.

For example:

(2.5, 4)

lies halfway between x = 2 and x = 3, and at y = 4.

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Distance Along an Axis

Suppose:

A = (−3, 4)

B = (5, 4)

Because both points have the same y-coordinate, the distance between them is horizontal.

Calculate the difference in x-values:

5 − (−3) = 8

Therefore, the points are:

8 units apart

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Vertical Distance

Suppose:

C = (2, −4)

D = (2, 5)

Both points have the same x-coordinate.

Their vertical distance is:

5 − (−4) = 9

Therefore:

C and D are 9 units apart.

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Moving Between Coordinates

Suppose an object begins at:

(−2, 3)

It moves:

• 5 units right
• 4 units down

Moving right changes x:

−2 + 5 = 3

Moving down changes y:

3 − 4 = −1

The new position is:

(3, −1)

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Describing Movement

Suppose an object moves from:

(1, 2)

to:

(6, 5)

Horizontal change:

6 − 1 = 5

Vertical change:

5 − 2 = 3

Therefore, the movement is:

5 units right and 3 units up

This idea will later connect to slope, vectors, and displacement.

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Determining an Unknown Coordinate

Suppose point A is:

(3, 7)

Point B lies directly below A.

Because the points are vertically aligned, B must also have:

x = 3

If B is 5 units below A:

y = 7 − 5

y = 2

Therefore:

B = (3, 2)

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A Reliable Strategy for Plotting Points

When plotting an ordered pair:

1. Identify x.
2. Identify y.
3. Start at the origin.
4. Move horizontally according to x.
5. Move vertically according to y.
6. Mark the point.
7. Label the point if required.
8. Check the signs and scale.

Remember:

x first, y second

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A Reliable Strategy for Reading Coordinates

When reading a point from a graph:

1. Locate the point.
2. Trace vertically to determine x.
3. Trace horizontally to determine y.
4. Check the scale of each axis.
5. Determine whether each coordinate is positive, negative, or zero.
6. Write the ordered pair as (x, y).
7. If the graph has context, include the meaning and units.

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Common Mistakes

Reversing x and y

The point:

(3, 7)

means x = 3 and y = 7, not the other way around.

Moving vertically first and losing track of the coordinates

A reliable habit is:

horizontal x first, vertical y second.

Forgetting negative directions

Negative x means left.

Negative y means down.

Numbering the quadrants clockwise

Quadrants are numbered counterclockwise, starting in the upper-right.

Putting axis points into quadrants

Points on either axis are not located in a quadrant.

Calling (0, 0) Quadrant I

The origin does not belong to any quadrant.

Ignoring the graph's scale

One grid interval does not always equal one unit.

Assuming the axes always represent x and y without context

Real-world graphs may use labels such as time, distance, cost, temperature, or population.

Giving coordinates without interpreting them

In a contextual graph, explain what each coordinate represents and include units.

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Key Terms

Coordinate plane — two-dimensional system used to locate points using horizontal and vertical coordinates.

Cartesian plane — another name for the coordinate plane.

x-axis — horizontal axis of the coordinate plane.

y-axis — vertical axis of the coordinate plane.

Origin — point (0, 0) where the two axes intersect.

Ordered pair — pair of coordinates written in the form (x, y).

x-coordinate — first value in an ordered pair, describing horizontal position.

y-coordinate — second value in an ordered pair, describing vertical position.

Quadrant — one of the four regions created by the x-axis and y-axis.

Quadrant I — region where x and y are both positive.

Quadrant II — region where x is negative and y is positive.

Quadrant III — region where x and y are both negative.

Quadrant IV — region where x is positive and y is negative.

Scale — value represented by each interval along an axis.

Coordinate — numerical value describing a position relative to an axis.

Plot — place a point at its correct position on a coordinate plane.

Independent variable — input variable, often represented on the horizontal axis.

Dependent variable — output variable, often represented on the vertical axis.

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Key Takeaways

• The coordinate plane uses two perpendicular number lines to locate points.
• The x-axis is horizontal.
• The y-axis is vertical.
• The axes intersect at the origin, (0, 0).
• Coordinates are written as ordered pairs in the form (x, y).
• The x-coordinate is always written first.
• Positive x-values are to the right of the origin.
• Negative x-values are to the left.
• Positive y-values are above the origin.
• Negative y-values are below.
• Quadrants are numbered counterclockwise beginning in the upper-right.
• Quadrant I has sign pattern (+, +).
• Quadrant II has sign pattern (−, +).
• Quadrant III has sign pattern (−, −).
• Quadrant IV has sign pattern (+, −).
• Points on an axis do not belong to any quadrant.
• The origin does not belong to any quadrant.
• Points with the same x-coordinate are vertically aligned.
• Points with the same y-coordinate are horizontally aligned.
• Always check the scale before reading or plotting coordinates.
• In real-world graphs, coordinates represent meaningful pairs of quantities.
• Axis labels and units are necessary for interpreting points correctly.
• Tables, equations, and graphs can all represent collections of ordered pairs.
• Understanding coordinates provides a foundation for graphing functions, linear equations, transformations, vectors, and data.

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Check Your Understanding

1. What are the coordinates of the origin?
2. Explain the difference between the x-axis and the y-axis.

3. State the quadrant containing each point:

   A = (4, 7)
   B = (−3, 8)
   C = (−6, −2)
   D = (5, −9)

4. Explain how you would plot the point (−4, 6).
5. Explain how you would plot the point (3, −7).
6. Does the point (0, 5) belong to a quadrant? Explain your answer.
7. Does the point (−8, 0) belong to a quadrant? Where is it located?
8. A point is 5 units left of the origin and 3 units below the origin. Write its coordinates and identify its quadrant.
9. Point A is at (−2, 6). Reflect A across the y-axis. What are the coordinates of the reflected point?
10. Point B is at (4, −5). Reflect B across the x-axis. What are the coordinates of the reflected point?
11. An object begins at (−3, 2) and moves 7 units right and 5 units down. Determine its new coordinates.
12. Points P = (−5, 4) and Q = (3, 4) lie on the same horizontal line. Determine the distance between them.
13. A graph uses time in seconds on the x-axis and distance in metres on the y-axis. Explain the meaning of the point (6, 42).
14. A map uses the origin as a town centre, with positive x representing east and positive y representing north. Interpret the location (−4, 7) in words.
15. A student plots (−3, 5) by moving 3 units up and 5 units left. Explain the student's error, give the correct plotting procedure, and identify the quadrant containing the point.

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.

---

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

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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."

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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)

---

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

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

Slope-Intercept Form

Slope-intercept form is one of the most useful ways to write the equation of a straight line because it immediately shows two important features:

y = mx + b

where:

• m = slope
• b = y-intercept
• x = independent variable
• y = dependent variable

The slope tells us the rate of change. The y-intercept tells us the value of y when x = 0.

For example:

y = 3x + 5

has:

• slope = 3
• y-intercept = 5

This means the line crosses the y-axis at (0, 5) and rises 3 units for every 1 unit it moves to the right.

---

Understanding y = mx + b

The equation:

y = mx + b

contains two pieces of information that completely determine a non-vertical straight line.

The value m determines:

• direction
• steepness
• rate of change

The value b determines:

• where the line crosses the y-axis
• the starting value when x = 0

For:

y = 4x − 2

m = 4

b = −2

Therefore, the line has:

slope = 4

and:

y-intercept = (0, −2)

---

Identifying the Slope

In slope-intercept form, the slope is the coefficient of x.

Consider:

y = 7x + 3

The coefficient of x is:

7

Therefore:

m = 7

The line rises 7 units for every increase of 1 unit in x.

---

Identifying a Negative Slope

Consider:

y = −3x + 8

The coefficient of x is:

−3

Therefore:

m = −3

The line decreases by 3 units for every increase of 1 unit in x.

The graph falls from left to right.

---

Identifying a Fractional Slope

Consider:

y = (2/5)x + 4

The slope is:

m = 2/5

This can be interpreted as:

rise = 2

run = 5

So from one point on the line, move:

• 5 units right
• 2 units up

to find another point.

---

Identifying the y-Intercept

The y-intercept is the constant term in:

y = mx + b

Consider:

y = 5x + 9

The y-intercept is:

b = 9

Therefore, the line crosses the y-axis at:

(0, 9)

Remember that the y-intercept is a point on the y-axis, so its coordinate is:

(0, b)

---

A Negative y-Intercept

Consider:

y = 2x − 6

This can be viewed as:

y = 2x + (−6)

Therefore:

b = −6

The y-intercept is:

(0, −6)

The line crosses the y-axis below the origin.

---

Why b Gives the y-Intercept

Consider:

y = mx + b

At the y-axis:

x = 0

Substitute:

y = m(0) + b

y = b

Therefore, the line crosses the y-axis at:

(0, b)

This explains why b is called the y-intercept.

---

Reading Slope and Intercept Quickly

Consider:

y = 6x + 4

m = 6

b = 4

Consider:

y = −2x + 9

m = −2

b = 9

Consider:

y = (1/3)x − 7

m = 1/3

b = −7

Once an equation is in slope-intercept form, these values can usually be identified immediately.

---

When the Slope Is Not Written as a Fraction

Any whole number can be written as a fraction over 1.

For:

y = 4x + 2

the slope is:

4 = 4/1

This means:

rise = 4

run = 1

For:

y = −5x + 3

we can write:

−5 = −5/1

This can be interpreted as:

• move 1 right
• move 5 down

---

When There Is No Number Written Before x

Consider:

y = x + 4

The coefficient of x is understood to be:

1

Therefore:

m = 1

b = 4

Similarly:

y = −x + 6

means:

y = −1x + 6

Therefore:

m = −1

b = 6

---

When There Is No Constant Term

Consider:

y = 3x

This can be written as:

y = 3x + 0

Therefore:

m = 3

b = 0

The y-intercept is:

(0, 0)

The line passes through the origin.

This is also a direct variation.

---

Horizontal Lines

Consider:

y = 5

This can be thought of as:

y = 0x + 5

Therefore:

m = 0

b = 5

The graph is a horizontal line through:

(0, 5)

Every point on the line has y = 5.

---

Vertical Lines

A vertical line such as:

x = 4

cannot be written in slope-intercept form.

Why?

A vertical line has an undefined slope.

Slope-intercept form:

y = mx + b

requires a defined value of m.

Therefore, vertical lines are an important exception.

---

Graphing Using Slope-Intercept Form

Suppose:

y = 2x + 3

Identify:

m = 2

b = 3

First plot the y-intercept:

(0, 3)

Then interpret:

m = 2/1

From (0, 3):

• move 1 unit right
• move 2 units up

This gives:

(1, 5)

Repeat:

(2, 7)

Draw a straight line through the points.

---

The Slope Triangle

A slope triangle shows the vertical and horizontal changes between two points.

For:

m = 3/2

we can use:

rise = 3

run = 2

Starting from the y-intercept:

• move 2 units right
• move 3 units up

This creates another point on the line.

The same pattern can be repeated because a linear relationship has a constant slope.

---

Graphing a Positive Slope

Consider:

y = 3x − 4

Identify:

m = 3 = 3/1

b = −4

Plot:

(0, −4)

Then move:

• 1 right
• 3 up

This gives:

(1, −1)

Again:

• 1 right
• 3 up

This gives:

(2, 2)

The graph rises from left to right.

---

Graphing a Negative Slope

Consider:

y = −2x + 5

Identify:

m = −2 = −2/1

b = 5

Plot:

(0, 5)

Then move:

• 1 right
• 2 down

This gives:

(1, 3)

Repeat:

(2, 1)

The line falls from left to right.

---

Graphing a Fractional Slope

Consider:

y = (3/4)x − 2

Identify:

m = 3/4

b = −2

Plot the y-intercept:

(0, −2)

Use the slope:

rise = 3

run = 4

Move:

• 4 right
• 3 up

This gives:

(4, 1)

Repeat the pattern to find more points.

---

Moving in the Opposite Direction

A slope of:

3/4

can also be represented by:

−3/−4

So instead of moving:

4 right and 3 up

you can move:

4 left and 3 down.

Starting from:

(0, −2)

this produces:

(−4, −5)

Both methods remain on the same line.

---

Graphing a Negative Fractional Slope

Consider:

y = −(2/3)x + 4

Plot:

(0, 4)

Since:

m = −2/3

you can move:

• 3 right
• 2 down

giving:

(3, 2)

Or move:

• 3 left
• 2 up

giving:

(−3, 6)

All three points lie on the same line.

---

A Reliable Graphing Method

For an equation in slope-intercept form:

1. identify m
2. identify b
3. plot the y-intercept (0, b)
4. write m as rise/run
5. use the rise and run to locate another point
6. repeat if useful
7. check the points
8. draw a straight line through them
9. extend the line in both directions when appropriate

---

Writing an Equation from Slope and Intercept

Suppose:

slope = 4

y-intercept = 7

Start with:

y = mx + b

Substitute:

m = 4

b = 7

Therefore:

y = 4x + 7

---

Another Example

Suppose:

slope = −3

y-intercept = 5

Substitute into:

y = mx + b

The equation is:

y = −3x + 5

---

Writing an Equation with a Fractional Slope

Suppose:

m = 2/3

b = −4

Then:

y = (2/3)x − 4

This equation completely describes the line.

---

Determining an Equation from a Graph

To determine the equation of a line from a graph:

1. identify the y-intercept
2. choose another clear point on the line
3. calculate the slope
4. substitute m and b into y = mx + b
5. check the equation using another point

For example, suppose a graph crosses the y-axis at:

(0, 2)

and contains:

(3, 8)

The slope is:

m = (8 − 2)/(3 − 0)

m = 6/3

m = 2

The intercept is:

b = 2

Therefore:

y = 2x + 2

---

When the y-Intercept Is Negative

Suppose a graph crosses the y-axis at:

(0, −5)

and contains:

(2, 1)

Calculate slope:

m = (1 − (−5))/(2 − 0)

m = 6/2

m = 3

Therefore:

y = 3x − 5

---

Finding the Equation from Two Points

Sometimes two points are known but the y-intercept is not immediately given.

Suppose the line passes through:

(2, 7)

and:

(5, 16)

First calculate slope:

m = (16 − 7)/(5 − 2)

m = 9/3

m = 3

Now use:

y = 3x + b

Substitute (2, 7):

7 = 3(2) + b

7 = 6 + b

b = 1

Therefore:

y = 3x + 1

---

Checking the Equation

Use the second point:

(5, 16)

Substitute:

16 = 3(5) + 1

16 = 15 + 1

16 = 16

The equation is verified.

---

Determining an Equation from a Table

Consider:

The change in y is:

+3

for each:

+1 in x

Therefore:

m = 3

When x = 0:

y = 4

Therefore:

b = 4

The equation is:

y = 3x + 4

---

When x = 0 Is Not in the Table

Consider:

Calculate slope:

m = (15 − 9)/(4 − 2)

m = 6/2

m = 3

Now use:

y = 3x + b

Substitute:

(2, 9)

9 = 3(2) + b

9 = 6 + b

b = 3

Therefore:

y = 3x + 3

---

Connecting Tables, Graphs, and Equations

Consider:

y = 2x + 5

Equation:

m = 2

b = 5

Table:

Graph:

• crosses the y-axis at (0, 5)
• rises 2 for every movement of 1 to the right

All three representations show the same relationship.

---

Slope-Intercept Form and Direct Variation

Direct variation has the form:

y = kx

Slope-intercept form is:

y = mx + b

For direct variation:

b = 0

Therefore:

y = mx

and the graph passes through:

(0, 0)

For example:

y = 5x

has:

m = 5

b = 0

It is both linear and directly proportional.

---

Comparing Two Lines

Consider:

A: y = 2x + 8

B: y = 5x + 2

Line A:

m = 2

b = 8

Line B:

m = 5

b = 2

Therefore:

• A has the greater y-intercept
• B has the greater slope
• A starts higher
• B increases more quickly

Slope-intercept form makes these comparisons particularly easy.

---

Same Slope, Different Intercepts

Consider:

y = 3x + 2

and:

y = 3x − 6

Both have:

m = 3

Their y-intercepts are different.

Therefore, their graphs are parallel.

They increase at exactly the same rate but begin at different vertical positions.

---

Same Intercept, Different Slopes

Consider:

y = 2x + 4

and:

y = −3x + 4

Both cross the y-axis at:

(0, 4)

But their slopes differ.

The first line increases.

The second line decreases.

Therefore, the lines move away from their shared intercept in different directions.

---

Interpreting Slope in Context

Suppose:

C = 4d + 7

where:

C = taxi cost in dollars

d = distance in kilometres

The slope is:

4 dollars/km

This means the taxi cost increases by $4 for every kilometre travelled.

---

Interpreting the Intercept in Context

For:

C = 4d + 7

the intercept is:

7

At:

d = 0

C = 7

Therefore, the taxi charges a:

$7 starting fee

The intercept represents the initial value before any distance is travelled.

---

Real-World Example: Hourly Pay

Suppose:

P = 20h + 50

where:

P = total pay

h = hours worked

Slope:

m = 20 dollars/hour

Interpretation:

The worker earns an additional $20 for each hour worked.

Intercept:

b = 50 dollars

Interpretation:

The worker begins with a $50 fixed payment or bonus.

---

Real-World Example: Water Tank

Suppose:

V = −15t + 300

where:

V = water volume in litres

t = time in minutes

Slope:

−15 L/min

This means the tank loses 15 L every minute.

Intercept:

300 L

This means the tank contains 300 L at:

t = 0

---

Real-World Example: Temperature

Suppose:

T = −2t + 24

where:

T = temperature in °C

t = time in hours

Slope:

−2°C/hour

The temperature decreases by 2°C each hour.

Intercept:

24°C

The initial temperature is 24°C.

---

Real-World Example: Savings

Suppose:

S = 35w + 120

where:

S = savings in dollars

w = number of weeks

Slope:

$35/week

The person saves an additional $35 each week.

Intercept:

$120

The person already had $120 at week zero.

---

Why Units Matter

Suppose an equation has:

m = 5

Without context, this tells us only the mathematical rate.

Depending on the variables, it might mean:

• $5 per item
• 5 km per hour
• 5 L per minute
• 5°C per hour
• 5 people per year

The units of slope come from:

y-units/x-units

The intercept uses the same units as y.

---

When an Intercept May Not Be Realistic

Suppose:

P = 25a − 100

models profit P based on the number of items a sold.

Mathematically:

when a = 0:

P = −100

The intercept represents:

−$100

This could represent a fixed initial cost.

Sometimes the intercept is mathematically meaningful even when it is negative.

In other situations, x = 0 might fall outside the realistic domain, so the intercept may have little practical meaning.

Always interpret the equation within its context.

---

Rearranging into Slope-Intercept Form

Not every linear equation is initially written as:

y = mx + b

Consider:

2x + y = 7

Solve for y.

Subtract 2x:

y = −2x + 7

Now we can identify:

m = −2

b = 7

---

Another Rearranging Example

Consider:

3x + 2y = 10

Subtract 3x:

2y = −3x + 10

Divide everything by 2:

y = −(3/2)x + 5

Therefore:

m = −3/2

b = 5

---

Watch the Signs When Rearranging

Consider:

4x − y = 6

Subtract 4x:

−y = −4x + 6

Now divide everything by −1:

y = 4x − 6

Therefore:

m = 4

b = −6

A common mistake is forgetting that dividing by −1 changes the signs of every term.

---

Determining Whether an Equation Is in Slope-Intercept Form

Consider:

y = 5x − 3

Yes. It is already in:

y = mx + b

Consider:

2x + y = 8

No. It must first be rearranged.

Consider:

x = 4

No. This is a vertical line and cannot be expressed in slope-intercept form.

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Finding an Unknown Value Using the Equation

Suppose:

y = 3x + 7

Find y when x = 6.

Substitute:

y = 3(6) + 7

y = 18 + 7

y = 25

Therefore:

(6, 25)

lies on the line.

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Finding x from a Known y-Value

Suppose:

y = 4x + 5

and:

y = 29

Substitute:

29 = 4x + 5

Subtract 5:

24 = 4x

Divide by 4:

x = 6

Therefore:

(6, 29)

lies on the line.

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Verifying a Point

Does:

(4, 13)

lie on:

y = 3x + 1?

Substitute:

13 = 3(4) + 1

13 = 12 + 1

13 = 13

Yes.

Therefore, the point lies on the line.

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Detecting an Incorrect Equation

Suppose a graph appears to have:

slope = 2

y-intercept = −3

A student writes:

y = −3x + 2

The student has switched the slope and intercept.

The correct equation is:

y = 2x − 3

Always remember:

y = mx + b

m = slope

b = y-intercept

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From Words to Slope-Intercept Form

Suppose:

"A gym charges a $30 registration fee and $12 per visit."

Let:

C = total cost

v = number of visits

Starting value:

30

Rate:

12 per visit

Therefore:

C = 12v + 30

This has exactly the same structure as:

y = mx + b

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From Slope-Intercept Form to Words

Consider:

D = 75t + 20

Suppose D is distance in kilometres and t is time in hours.

A possible interpretation is:

"An object begins 20 km from the reference point and moves away at 75 km/h."

The slope gives the rate.

The intercept gives the initial position.

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Comparing Representations

Slope and intercept can be identified from different representations.

Being able to move between these representations is one of the main advantages of understanding slope-intercept form.

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A Reliable Strategy for Graphing y = mx + b

1. Make sure the equation is written as y = mx + b.
2. Identify m.
3. Identify b.
4. Plot the y-intercept (0, b).
5. Write the slope as a fraction.
6. Use rise/run to locate another point.
7. Repeat the slope pattern if needed.
8. Check the coordinates.
9. Draw a straight line through the points.
10. Extend the line appropriately.

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A Reliable Strategy for Finding an Equation from a Graph

1. Locate the y-intercept.
2. Record b.
3. Choose another clear point on the line.
4. Calculate rise and run.
5. Find m.
6. Write y = mx + b.
7. Substitute m and b.
8. Simplify the equation.
9. Test another point from the graph.
10. Check that the slope direction matches the graph.

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Common Mistakes

Switching m and b

In:

y = 4x + 7

m = 4

b = 7

not the other way around.

Giving b instead of the y-intercept point

If b = 5, the y-intercept is:

(0, 5)

Missing an implied slope

In:

y = x + 3

m = 1

In:

y = −x + 3

m = −1

Forgetting that a whole-number slope has denominator 1

m = 4 means:

4/1

Moving the wrong direction for a negative slope

A negative slope can be shown by moving right and down.

Using the x-intercept instead of the y-intercept

The y-intercept occurs where:

x = 0

Reading slope only from visual steepness

Use rise/run or coordinates whenever possible.

Forgetting to solve for y first

In:

2x + y = 8

the slope is not 2.

Rearrange:

y = −2x + 8

so the slope is −2.

Forgetting to change all signs when dividing by a negative

Be especially careful when isolating y.

Trying to write a vertical line in slope-intercept form

Vertical lines have undefined slope and cannot be written as y = mx + b.

Ignoring context

Slope and intercept should be interpreted using appropriate quantities and units.

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Key Terms

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

Slope — rate of change represented by m.

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

Rate of change — change in y compared with change in x.

Rise — vertical change between points.

Run — horizontal change between points.

Coefficient — number multiplying a variable.

Constant term — term without a variable; b in slope-intercept form.

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

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

Zero slope — slope of a horizontal line.

Undefined slope — slope of a vertical line.

Linear equation — equation whose graph forms a straight line.

Initial value — starting value of the dependent variable, often represented by the y-intercept.

Parallel lines — non-intersecting lines with equal slopes.

Direct variation — proportional relationship of the form y = kx, equivalent to slope-intercept form with b = 0.

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Key Takeaways

• Slope-intercept form is y = mx + b.
• The value m represents the slope.
• The value b represents the y-intercept.
• The y-intercept occurs at (0, b).
• Slope represents the rate of change.
• Positive slopes rise from left to right.
• Negative slopes fall from left to right.
• Zero slope produces a horizontal line.
• Vertical lines have undefined slope and cannot be written in slope-intercept form.
• Whole-number slopes can be written over 1 to show rise/run.
• Fractional slopes directly show a rise/run pattern.
• To graph from slope-intercept form, plot the y-intercept first and then use the slope.
• A line can be written in slope-intercept form when its slope and y-intercept are known.
• If two points are known, calculate the slope and then determine b.
• Tables can be used to determine both slope and intercept.
• Equations may need to be rearranged to isolate y before m and b can be identified.
• Lines with the same slope and different intercepts are parallel.
• Direct variation is the special case in which b = 0.
• In real situations, slope usually represents a rate such as dollars/hour, km/h, or L/min.
• The y-intercept often represents a starting amount, initial condition, or fixed charge.
• The intercept should only be interpreted within the realistic domain of the situation.
• Tables, graphs, equations, and verbal descriptions can all communicate the same slope and intercept.

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Check Your Understanding

1. Identify the slope and y-intercept of:

y = 5x + 8

2. Identify the slope and y-intercept of:

y = −3x − 4

3. Identify m and b in:

y = (2/5)x + 7

Then describe how you could use them to graph the line.

4. Determine the slope and y-intercept of:

y = −x + 6

5. Write an equation in slope-intercept form with:

m = 4

and:

b = −7

6. Write an equation for a line with slope −3/2 and y-intercept (0, 5).
7. Describe how you would graph:

y = 2x − 3

using only its slope and y-intercept.

8. A line crosses the y-axis at (0, 4) and also passes through (3, 13). Calculate its slope and write its equation in slope-intercept form.
9. Determine the equation of the line passing through (2, 5) and (6, 17).
10. Write this equation in slope-intercept form and identify m and b:

2x + y = 9

11. Write this equation in slope-intercept form:

3x + 2y = 12

Then identify its slope and y-intercept.

12. A taxi company's cost is represented by:

C = 2.75d + 6

where d is distance in kilometres. Explain the meaning of both 2.75 and 6, including appropriate units.

13. A tank's water volume is represented by:

V = −18t + 450

where V is measured in litres and t in minutes. Identify and interpret the slope and intercept. What does the negative slope tell you?

14. Two lines are represented by:

A: y = 4x + 9

B: y = 4x − 3

Compare their slopes and y-intercepts. Explain what their graphs would look like relative to each other.

15. A student looks at a graph and determines that its y-intercept is −2. The graph also contains the points (3, 4) and (6, 10). Determine the slope, write the equation in slope-intercept form, verify that all the information is consistent, and explain how you would graph the line using only m and b.