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.