What it is
A coordinate plane is made of two number lines that cross at a right angle: the horizontal x-axis and the vertical y-axis. They meet at the origin, the point (0, 0). The numbers along the axes are evenly spaced, positive to the right and upwards, negative to the left and downwards.
A point is given by a coordinate pair (x, y). The first number, the x-coordinate, says how far to the right (positive) or left (negative) of the origin the point lies. The second, the y-coordinate, says how far up (positive) or down (negative).
- (3, −2) lies three steps right and two steps down.
- (−2, 3) lies two steps left and three steps up. It is a different point from (3, −2): the order matters.
- (4, 0) lies on the x-axis, because y = 0.
- (0, −3) lies on the y-axis, because x = 0.
- (0, 0) is the origin.
The axes divide the plane into four quadrants, numbered anticlockwise from the top right:
| Quadrant | x | y | Example |
|---|---|---|---|
| I | positive | positive | (2, 3) |
| II | negative | positive | (−2, 3) |
| III | negative | negative | (−2, −3) |
| IV | positive | negative | (2, −3) |
A point on an axis, such as (4, 0), lies in no quadrant.
The marks on an axis are evenly spaced, and each step has a value, the scale of that axis. The axes may have different scales. For the points (10, 500) and (20, 800) it makes sense to let each step on the x-axis be 5 and each step on the y-axis 100, so that both points fit.
There are other ways to give a point. In space three coordinates are needed, (x, y, z), and polar coordinates give a point in the plane by its distance from the origin and an angle.
Why it matters
The coordinate plane joins geometry and algebra: a point becomes two numbers and a shape becomes an equation.
- The graph of a function is all the points (x, f(x)). For f(x) = 2x + 1, the points (0, 1), (1, 3) and (2, 5) lie on a straight line, described by the equation of a straight line.
- The distance between two points, for example (1, 2) and (4, 6), is √(3² + 4²) = 5, and the midpoint between them is (2.5, 4).
- The circle with centre at the origin and radius 5 consists of the points where x² + y² = 25, for example (3, 4); see the equation of a circle.
- A vector is written in component form as two numbers, and polar coordinates are converted to x and y with the coordinate converter.
Plot the points (1, 2) and (4, 6) in the graphing calculator and read them off the grid.
Common misconceptions
- “(3, 5) and (5, 3) are the same point.” The order decides. The first number is x: (3, 5) lies three steps right and five up, (5, 3) five right and three up.
- “Points with x = 0 lie on the x-axis.” It is the other way round. Points on the x-axis have y = 0, and points on the y-axis have x = 0. (0, 3) lies on the y-axis.
- “The quadrants are numbered clockwise.” They are numbered anticlockwise: I top right, II top left, III bottom left and IV bottom right.
- “A coordinate plane has only positive numbers.” The axes continue both ways, and the negative numbers lie to the left and downwards. A picture with only the first quadrant is a cut-out.
- “Both axes must have the same scale.” They may have different scales, but then a line with gradient 1 does not look like 45°, and a circle does not look round.
Common questions
How do I plot a point?
Start at the origin. Move x steps along the x-axis, right if x is positive and left if it is negative. Then move
y steps up or down. For (−2, 3) that is two steps left and three steps up.
Why is it called the Cartesian coordinate system?
After the French mathematician and philosopher René Descartes, who used coordinates in the 1600s to join
geometry and algebra.
What is the difference between a coordinate system and a graph?
The coordinate system is the axes and the grid. The graph is what is drawn in it, for example the line
y = 2x + 1.
How do I give a point in space?
With three coordinates (x, y, z), where z is measured along a third axis perpendicular to the other two. The point
(2, 3, 4) lies 2 steps along the x-axis, 3 along the y-axis and 4 along the z-axis.