The Coordinate Plane
Two number lines crossed at right angles, and suddenly every pair of numbers has a place. The two things people get wrong are the order of the pair and which quadrant is which — both worth practising rather than reading.
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Before the coordinate plane, a line was something you drew and a circle was something you measured. After it, both are equations you can calculate with. The whole of graphing — and most of what you will do in algebra afterwards — rests on one small idea: a pair of numbers names a place.
Click anywhere on the grid below. The dashed lines show how the pair is built.
Click anywhere. The dashed lines show how the pair is read: across the x-axis first, then up the y-axis.
Across first, then up
The two axes each do one job. The x-axis runs left to right and measures how far across; the y-axis runs bottom to top and measures how far up. Where they cross is the origin, which is (0, 0) — no distance in either direction.
To name a point you give two numbers in a fixed order: across, then up. So (3, 2) means three to the right and two up. Nothing about the point itself says which number is which, so the order carries that information — which is why these are called ordered pairs, and why (3, 2) and (2, 3) are different places.
If you keep mixing them up, one trick tends to stick: x comes before y in the alphabet, so it comes first in the pair.
Negatives just reverse the direction. A negative x-coordinate means go left instead of right; a negative y means go down instead of up. Click around the bottom-left of the grid above and watch both signs turn negative together.
The four quadrants
The two axes cut the plane into four regions, and they are numbered from the top right, anticlockwise:
| quadrant | x | y | example |
|---|---|---|---|
| I | positive | positive | (3, 2) |
| II | negative | positive | (-4, 1) |
| III | negative | negative | (-2, -3) |
| IV | positive | negative | (4, -2) |
The anticlockwise order looks like a strange choice until you meet angles, which are also measured anticlockwise from the positive x-axis. The two conventions agree on purpose.
There is no fifth region for the axes themselves. A point like (0, 3) sits on the y-axis, and a point like (-4, 0) on the x-axis; neither is in a quadrant, because the axes are the borders. The widget above says so explicitly when you click on one, because it is the case people most often guess at.
Plotting a whole set of points
Once single points are automatic, a list of them starts to say something. Here are four points plotted on the same axes:
They are not scattered — they line up. That is the moment the coordinate plane starts earning its keep: a pattern in a list of numbers becomes a shape you can see, and a shape you can see can usually be written as an equation. In this case every y is half its x, so the line is y = 1/2x.
Try one
The grid is above if you need it, but try to place this one in your head first.
The point is (-3, 5) — and the question was deliberately about the second number, because the habit of reading a pair in order is the thing worth building.
Common questions
- What is the coordinate plane?
- It is two number lines crossed at right angles: a horizontal one called the x-axis and a vertical one called the y-axis. They meet at a point called the origin, which is (0, 0). Every point on the flat surface can then be named by a pair of numbers — how far across, and how far up.
- Which number comes first in an ordered pair?
- The x-coordinate, always. The pair (3, 2) means go 3 across and 2 up, not the other way round. Alphabetical order is a reliable way to remember it: x comes before y in the alphabet and it comes first in the pair. The word 'ordered' in 'ordered pair' is doing real work — (3, 2) and (2, 3) are two different points.
- How are the four quadrants numbered?
- They start at the top right and go anticlockwise: quadrant I is top right where both coordinates are positive, II is top left (x negative, y positive), III is bottom left (both negative), and IV is bottom right (x positive, y negative). Anticlockwise looks arbitrary but it matches the direction angles are measured in, which is why it is worth learning that way round.
- Which quadrant is a point on an axis in?
- None of them. The axes are the boundaries between quadrants, not part of any one, so a point like (0, 3) or (-4, 0) is described as being on the y-axis or on the x-axis rather than in a quadrant. The origin is on both.
- Why is it called the Cartesian plane?
- After René Descartes, who published the idea of naming points by pairs of numbers in 1637. It is one of the most useful ideas in mathematics: it makes geometry into algebra. A line stops being a drawing and becomes an equation, which is what lets you calculate with it.
Now do it yourself, properly
Reading someone else's working is not the same as being able to do it. The Coddy math course puts you on a board that checks every move you make, so you find out where you actually stand.
