A pair of perpendicular number lines turns numerical position into geometry. Their intersection is the origin, and a point in the plane is described by an ordered pair

$$(x,y),$$

where $x$ gives the horizontal position and $y$ the vertical position.

Axes and quadrants

The horizontal line is the $x$-axis and the vertical line is the $y$-axis. Their signs divide the plane into four quadrants:

          y
          ↑
     II   |   I
          |
──────────+──────────→ x
          |
    III   |   IV
          ↓

For example, $(3,2)$ lies three units to the right of the origin and two units above it, while $(-3,2)$ lies in the second quadrant.

Changes in coordinates

If two points are

$$A=(x_1,y_1),\qquad B=(x_2,y_2),$$

then their horizontal and vertical changes are

$$\Delta x=x_2-x_1,\qquad \Delta y=y_2-y_1.$$

These differences describe how far and in which direction one moves along each axis.

Reading graphs

A graph can represent a relationship between two quantities by placing one quantity on each axis. A point $(x,y)$ then records a pair of corresponding values. Reading a graph means connecting positions, coordinates and changes in the plotted quantities.