Understanding the Four Quadrant Coordinate Plane

A coordinate plane is formed by two intersecting number lines: the horizontal $x$-axis and the vertical $y$-axis. They cross at the center, a point called the origin $(0,0)$. By extending these number lines past zero into negative numbers, the plane is divided into four sections called quadrants.

The Four Quadrants

An ordered pair $(x, y)$ tells you exactly where a point is located. The $x$-coordinate tells you how far to move left or right, and the $y$-coordinate tells you how far to move up or down.

The four quadrants are numbered using Roman numerals, starting from the top right and going counter-clockwise:

• Quadrant I: $(+, +)$ — Move right and up.
• Quadrant II: $(-, +)$ — Move left and up.
• Quadrant III: $(-, -)$ — Move left and down.
• Quadrant IV: $(+, -)$ — Move right and down.

Example: Plot $(-3, 4)$ and identify its quadrant. Because the $x$-coordinate is negative (move left $3$) and the $y$-coordinate is positive (move up $4$), this point lands in Quadrant II.

Finding Distance Between Two Points

If two points share the same $x$-coordinate or the same $y$-coordinate, you can find the distance between them by looking at the coordinates that are different. You simply find the absolute difference between those numbers.

Example: Find the distance between $(2, -3)$ and $(2, 5)$. Both points have an $x$-coordinate of $2$, so they lie on the same vertical line. We just need to find the distance between the $y$-coordinates, $-3$ and $5$. $\text{Distance} = |5 - (-3)| = |5 + 3| = 8 \text{ units}$

Drawing Polygons on the Coordinate Plane

You can draw shapes by plotting ordered pairs as vertices (corners) and connecting them with line segments. Once the shape is drawn, you can find its side lengths, perimeter, or area using the distance method above.

Example: Plot vertices $(-2,1)$, $(3,1)$, $(3,-2)$, and $(-2,-2)$ to form a rectangle, then find its perimeter.

1. Top length: Distance between $(-2,1)$ and $(3,1)$. The $y$-coordinates are the same, so find the distance between $x$-coordinates: $|3 - (-2)| = 5$ units.
2. Side width: Distance between $(3,1)$ and $(3,-2)$. The $x$-coordinates are the same, so find the distance between $y$-coordinates: $|1 - (-2)| = 3$ units.
3. Perimeter: Add all the sides together. $\text{Perimeter} = 2 \times (\text{Length} + \text{Width}) = 2 \times (5 + 3) = 16 \text{ units}$