The Coordinate Plane

Two number lines

Draw a number line across the page, and a second number line straight up the page, so that they cross at right angles at the 0 of each. The line across is called the x-axis, and the line going up is called the y-axis. The point where they cross is called the origin.

The flat surface that the two axes lie in is called the coordinate plane. Every point on it can be named by two numbers, one read from each axis.

The x-axis runs across and the y-axis runs up. They cross at the origin, where both are 0.

A point’s address

The pair of numbers (4, 2) gives the address of a point. Start at the origin, go 4 across to the right, then go 2 up. The point you reach is (4, 2).

The two numbers are called the coordinates of the point. The first, 4, is the x-coordinate: how far across. The second, 2, is the y-coordinate: how far up. Count the steps from the origin, which is 0, not 1.

The origin itself is (0, 0): no steps across and no steps up.

The point (4, 2) is 4 across from the origin and 2 up.

Across first, then up

The order of the two numbers matters. (2, 4) means 2 across and then 4 up, which is a different point from (4, 2).

The x-coordinate is always written first and the y-coordinate second. To read the coordinates of a point, go straight down from it to the x-axis and read the first number there, then go straight across from it to the y-axis and read the second.

(4, 2) and (2, 4) use the same two numbers in a different order, and they are different points.

Negative coordinates

Each axis is a whole number line, so it carries on past 0. To the left of the origin the x-axis has negative numbers, and below the origin the y-axis has negative numbers.

So a negative x-coordinate means go left instead of right. (−3, 2) is 3 to the left of the origin and 2 up.

A negative y-coordinate means go down instead of up. (2, −3) is 2 to the right of the origin and 3 down.

The axes divide the plane into four parts, called quadrants. In the top right quadrant both coordinates are positive, like (4, 2). In the top left, x is negative and y is positive, like (−3, 2). In the bottom left both are negative, like (−2, −4), and in the bottom right x is positive and y is negative, like (2, −3).

(−3, 2) is left of the y-axis, (2, −3) is below the x-axis, and (−2, −4) is both left and below.

Distance from the origin

Going across and then up makes a right angle, so the path to a point and the straight line back to the origin form a right triangle. The point (3, 4) is 3 across and 4 up, and by Pythagoras’ theorem its distance from the origin is $\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$√(3² + 4²) = √(9 + 16) = √25 = 5.

The usual mistakes

Reading up first. For the point 4 across and 2 up, (2, 4) swaps the order. The x-coordinate, across, is always first: (4, 2).

Counting the origin as 1. The origin is 0 on both axes, so the first step to the right lands on 1.

Dropping a minus sign. (−3, 2) and (3, 2) are on opposite sides of the y-axis. Left of the origin, the x-coordinate is negative; below it, the y-coordinate is negative.

Points on the axes

A point on the x-axis is not up or down at all, so its y-coordinate is 0: (10, 0) is on the x-axis, 10 to the right of the origin. A point on the y-axis is not across at all, so its x-coordinate is 0: (0, 4) is on the y-axis, 4 up.

An equation in x and y, such as 2x + 5y = 20, draws a straight line through every point whose coordinates make it true. To find where the line meets the y-axis, put x = 0 and solve: 5y = 20, so y = 4, and the point is (0, 4). To find where it meets the x-axis, put y = 0: 2x = 20, so x = 10, and the point is (10, 0). Check (10, 0): 2 × 10 + 5 × 0 = 20.