The Coordinate Plane

A map for numbers

Imagine trying to tell a friend exactly where a treasure is buried in a field. "Over there" is useless. But "go 3 steps east and 5 steps north from the big tree" works perfectly. The coordinate plane is the maths version of that idea: a grid that lets us pin down the exact location of any point using just two numbers.

It is one of the most useful inventions in all of mathematics. It connects numbers to pictures, lets us draw graphs, and is the foundation of everything from video-game screens to GPS. It is sometimes called the Cartesian plane, after Rene Descartes, who developed it in the 1600s.

The two axes

The coordinate plane is built from two number lines that cross at right angles. If number lines are new to you, the Introduction to Integers lesson is the perfect warm-up, because the plane uses positive and negative numbers in both directions.

• The x-axis is the horizontal line (left–right).
• The y-axis is the vertical line (up–down).
• The point where they cross is the origin, written (0, 0).

                y
                |
        Q II    |    Q I
                |
  --------------+--------------  x
                | (origin 0,0)
        Q III   |    Q IV
                |

On the x-axis, right is positive and left is negative. On the y-axis, up is positive and down is negative.

Ordered pairs: (x, y)

Every point is named by an ordered pair of numbers inside brackets: (x, y).

• The first number is the x-coordinate — how far across (right is +, left is −).
• The second number is the y-coordinate — how far up or down (up is +, down is −).

The word ordered is the key. (3, 5) and (5, 3) are different points, because the order tells you which direction each number controls. A simple memory trick: "across the hall, then up the stairs" — you always go across first, then up or down.

Plotting a point

To plot a point, always start at the origin and follow the two numbers in order.

Worked example 1 — plot (4, 2):

1. Start at the origin (0, 0).
2. The x-value is 4 → move 4 units right.
3. The y-value is 2 → move 2 units up.
4. Mark the point. That spot is (4, 2).

Worked example 2 — plot (−3, 1):

1. Start at the origin.
2. x = −3 → move 3 units left.
3. y = 1 → move 1 unit up.
4. Mark it. The point lands in the top-left region.

Worked example 3 — plot (2, −3):

1. Start at the origin.
2. x = 2 → 2 units right.
3. y = −3 → 3 units down.
4. Mark it in the bottom-right region.

Reading a point

Reading is just plotting in reverse. To find the coordinates of a marked point, ask two questions:

1. How far across is it from the origin? (left = negative) → that is x.
2. How far up or down is it? (down = negative) → that is y.

Write your two answers as (x, y) and you have named the point.

The four quadrants

The two axes chop the plane into four regions called quadrants. They are numbered anticlockwise starting from the top-right, using Roman numerals. Each quadrant has its own sign pattern, which is a fast way to check your plotting.

Quadrant	Position	x sign	y sign	Example
I	top-right	+	+	(4, 2)
II	top-left	−	+	(−3, 1)
III	bottom-left	−	−	(−2, −5)
IV	bottom-right	+	−	(2, −3)

Points on the axes are special. If a point sits on the x-axis, its y-coordinate is 0, like (5, 0). If it sits on the y-axis, its x-coordinate is 0, like (0, 4). Points on an axis belong to no quadrant.

Why graphs come alive here

Once you can plot points, you can plot whole patterns. Take the rule y = x + 1 and make a small table:

x	y = x + 1	point
0	1	(0, 1)
1	2	(1, 2)
2	3	(2, 3)
3	4	(3, 4)

Plot those four points and you will see they form a straight line. This is how algebra and geometry meet: an equation like the ones in Solving Linear Equations turns into a picture you can see. Plotting points is the first step toward graphing every kind of function.

Practice activity

Draw a grid from −5 to 5 on both axes, then:

1. Plot these points and label each: A(3, 4), B(−2, 3), C(−4, −1), D(1, −4).
2. State which quadrant each of A, B, C, D is in.
3. Plot E(0, −3) and F(4, 0). Which axis does each one sit on?
4. Make a table for y = 2x using x = 0, 1, 2, 3, plot the points, and check they line up straight.

Answers: 2) A → I, B → II, C → III, D → IV. 3) E is on the y-axis, F is on the x-axis. 4) Points (0,0), (1,2), (2,4), (3,6) lie on one straight line.

Why this matters

The coordinate plane turns abstract numbers into pictures you can see and measure. It powers graphs in science, maps and navigation, computer screens (every pixel has coordinates), and the entire field of graphing functions. The whole system rests on one tidy idea: two numbers, in order — across, then up or down. Get that habit firm, always start from the origin, and you will read and draw points with total confidence.