The Coordinate Plane
When you drop a pin on a map, share your location, or call out a square in a game of
battleship, you are naming a spot with a pair of numbers: how far across, and how far up. That
simple trick — pinning down any point with two numbers — is the whole idea behind the
coordinate plane.
To draw a function
we first need a place to draw it. That place is the coordinate plane: a flat
grid built from two number lines that cross at right angles.
- The horizontal line is the x-axis — it runs left to right.
- The vertical line is the y-axis — it runs down to up.
They meet at one special point called the origin, the point
(0, 0). Think of the origin as "home base": every other point is
described by how far you have travelled away from it. To the right and
up the numbers are positive; to the left and
down they are negative.
Every point has an address
Just as a street address has a house number and a street name, every point on the plane has
an address made of two numbers, written as an ordered pair
(x, y). The word ordered is the whole secret — the
order of the two numbers matters:
- the first number, x, is how far
across you go — right if positive, left if negative;
- the second number, y, is how far
up you go — up if positive, down if negative.
The rule to remember is along the corridor, then up the stairs: always go
across first, then up. So to plot (3, 2) you
start at the origin, walk 3 steps right, then climb
2 steps up. Drag the sliders and watch the address move along the
same right-then-up path.
Pirates use exactly this trick. A treasure map is just a grid, and "X marks the spot" is
really an ordered pair: 3 paces east, 2 paces north from the old oak tree
is the point (3, 2). Get the order wrong — north first, then
east — and you will be digging in the wrong place!
Worked examples: reading and plotting
Every point is found the same way — start at the origin, go across, then up (or down):
- (4, 1): walk 4 right, then 1 up.
It sits low and far to the right.
- (0, 3): walk 0 across (stay put), then
3 up. A first number of 0 means the point sits
right on the y-axis.
- (-2, -3): x is negative, so go
2 left; y is negative, so go
3 down. This lands in the bottom-left.
Reading a point backwards works too: if a dot is 5 to the right of
the origin and 4 below it, its address is
(5, -4).
The order is everything — x always comes first.
- (3, 2) means 3 across, then 2 up — never
"3 up, 2 across".
- (3, 2) and (2, 3) are
different points. Swapping the numbers swaps the address, so they land
in different places. Across first, always.
See it: read the mystery point
Here is a fresh point each time. Trace the green arrow across the x-axis,
then the orange arrow up (or down) the y-axis, and check you can read off its
ordered pair. Press Refresh for a brand-new point to read.
The four quadrants
The two axes slice the plane into four regions called quadrants, numbered
with Roman numerals starting from the top-right and going counter-clockwise. The
sign of x and y tells you which one a
point lives in:
\text{I: } (+,+) \quad \text{II: } (-,+) \quad \text{III: } (-,-) \quad \text{IV: } (+,-)
The shaded diagram below labels each quadrant with its sign pattern.
In the game Battleship the whole board is a coordinate grid. When you call
out a square — "across to 4, up to 3" — you are naming the ordered pair
(4, 3) and firing at it. Real navigators do the same with the
longitude (how far across) and latitude (how far up) of a ship at sea.
The plane is everywhere once you learn to spot it.
See it in action
Khan Academy gives a friendly tour of the coordinate plane here: