The Coordinate Plane

The coordinate plane is two number lines crossed at zero. The horizontal one is the x-axis, the vertical one is the y-axis, and where they meet is the origin, (0, 0).

A point is named by an ordered pair (x, y). The order is fixed: x tells you how far across, y tells you how far up or down. (3, 5) and (5, 3) are different points.

The four regions are the quadrants, numbered anticlockwise from the top right. Quadrant I is (+, +), II is (−, +), III is (−, −), IV is (+, −).

Points on an axis belong to no quadrant. (0, 4) sits on the y-axis.

The Cartesian plane establishes a bijection between ordered pairs of reals and points in the plane, which is what allows geometric objects to be described by equations and equations to be drawn.

Coordinates are signed displacements from the origin along two perpendicular axes. Because the pair is ordered, (a, b) and (b, a) denote different points unless a = b; the diagonal line y = x is exactly the set where swapping changes nothing, which is why reflecting across it is the geometric picture of finding an inverse function later.

Two formulas come straight from this setup and are used constantly. The distance between points is the Pythagorean theorem applied to the horizontal and vertical gaps, d = √((x₂−x₁)² + (y₂−y₁)²), and the midpoint is the componentwise average, ((x₁+x₂)/2, (y₁+y₂)/2).