The Four-Quadrant Plane

The coordinate plane is a flat surface formed by two intersecting number lines. The horizontal line is the x-axis, and the vertical line is the y-axis. They meet at a special center point called the origin, represented by the coordinates $(0, 0)$. Think of the x-axis as a floor and the y-axis as an elevator. To find any location, you must first walk along the floor (left or right) and then take the elevator (up or down). This location is written as an ordered pair $(x, y)$, where the first number tells you your horizontal position and the second tells you your vertical position.

When the two axes cross, they divide the plane into four regions called quadrants. We number them $I, II, III,$ and $IV$ in a counter-clockwise direction, starting from the top-right.

1. Quadrant I: Both $x$ and $y$ are positive $(+, +)$. 2. Quadrant II: $x$ is negative, $y$ is positive $(-, +)$. 3. Quadrant III: Both $x$ and $y$ are negative $(-, -)$. 4. Quadrant IV: $x$ is positive, $y$ is negative $(+, -)$.

A great trick to remember the order is to draw a large letter 'C' on the plane. Start in Quadrant I, curve through II and III, and end in IV!

Sometimes we need to know the distance between two points. If two points share the same $x$-coordinate or the same $y$-coordinate, they lie on a straight vertical or horizontal line. To find the distance, we look at the coordinates that are different. If the points are in the same quadrant, we subtract their absolute values. If they are in different quadrants (across an axis), we add their absolute values. Distance is always positive because you can't walk a 'negative' number of steps!