The Coordinate Plane
Why a coordinate system is needed, what the axes and origin are, and how to plot a point from its (x, y) pair including negative coordinates.
--
Imagine describing exactly where a ball is sitting on a football pitch to someone who cannot see it. "Near the middle" will not do. Mathematics needs precision, and precision needs a fixed reference that everyone agrees on.
Concept
Why a system is needed
Geography solved this problem long ago with north, south, east and west, measured from agreed reference lines. Mathematics does the same thing with two axes and an origin.
Once the system is agreed, a position becomes a pair of numbers — and anyone, anywhere, will place it in exactly the same spot.
Concept
The parts of the plane
Two perpendicular lines meet at the origin. The horizontal one is the x-axis and the vertical one is the y-axis. The origin is written %20%7D)
and every position is measured from it.
Concept
Reading a coordinate pair
A point is written as two numbers in brackets, and the order is fixed:
The first number is the x-coordinate — how far across.
The second number is the y-coordinate — how far up.
Always start at the origin, move along the x-axis first, then move parallel to the y-axis. Reversing the order lands you somewhere else entirely: and
are different points.
Concept
What the signs mean
A positive x moves right; a negative x moves left.
A positive y moves up; a negative y moves down.
The signs carry direction while the digits carry distance — the same idea you already met on the number line, now applied in two directions at once.
Example
The ball on the pitch
The ball sits at . Where exactly is that?
Begin at the origin, the agreed corner of the pitch.
Move 20 units along the x-axis.
Then move 5 units up the y-axis.
⟹ 20 across and 5 up from the origin
Two numbers have replaced a vague description, and there is now no room for disagreement.
Example
Plotting a negative coordinate
How do you plot ?
Start at the origin .
The x-coordinate is 
, so move 3 units left.
The y-coordinate is 
, so move 4 units up.
⟹ 3 left and 4 up from the origin
The procedure never changes. Only the direction of each step depends on the sign.
Note
Mistakes to avoid
Reading the pair in the wrong order — x always comes first.
Moving up before moving across, and losing track of the position.
Ignoring a minus sign and moving right when you should move left.
Measuring from somewhere other than the origin.
Treating %20%7D)
and %20%7D)
as the same point.
Summary
- The coordinate plane gives every point an exact, agreed address.
- Two perpendicular axes meet at the origin, written (0, 0).
- A point is written (x, y): across first, then up.
- Negative x moves left and negative y moves down.
- Order matters — (3, 2) and (2, 3) are different points.