The Cartesian Plane

One number can locate a point on a line, but locating a point on a flat surface needs two. The Cartesian plane uses a horizontal x-axis and a vertical y-axis crossing at the origin, which is the point (0, 0). A pair of coordinates like (3, 2) means go across 3, then up 2. The order is fixed and it matters: (3, 2) and (2, 3) are different points. The memory hook most people use is that you walk along the hall before you climb the stairs — across first, then up.

The axes extend in both directions, so coordinates can be negative. A negative x means move left instead of right; a negative y means move down instead of up. The axes divide the plane into four quadrants, numbered anticlockwise starting from the top right. The first quadrant has both coordinates positive; the second has negative x and positive y; the third has both negative; the fourth has positive x and negative y. Notice that (−2, 2) and (2, −2) land in different quadrants, which is another reminder that the order of the pair is never optional.

Reading a point off a graph reverses the plotting process. Drop straight down from the point to the x-axis to read the first coordinate, then go straight across to the y-axis for the second. So the point marked here is (4, 3). The one thing to check before you read anything is the scale, because the squares are not always worth 1. If each square represents 2, then a point three squares along the x-axis is at x = 6, not x = 3. Look at the numbers written on the axes, not at the number of squares.

When you plot several points, patterns appear that are worth noticing. Here A(1, 4) and B(5, 4) share the same y-coordinate, so the line joining them is horizontal. B(5, 4) and C(5, 1) share the same x-coordinate, so that line is vertical. Recognising this saves work later: any two points with matching y-values lie on a horizontal line, and matching x-values give a vertical one. Practically, mark each point with a small dot or a neat cross rather than a big blob, because in graph work a millimetre of sloppiness turns into a wrong gradient.

Coordinates let you do geometry with arithmetic instead of a ruler. For a horizontal line from P(1, 3) to Q(5, 3), the length is just the difference in x-values: 5 − 1 = 4 units. For a vertical line you subtract the y-values instead. The midpoint is found by averaging each coordinate separately — x is (1 + 5) ÷ 2 = 3 and y is (3 + 3) ÷ 2 = 3, giving (3, 3). This is the beginning of coordinate geometry, and the reason it matters is that once a shape is described by coordinates you can calculate its properties exactly, with no measuring and no drawing error.