Investigation 7: The Coordinate Plane

New Concept

A coordinate plane is a grid formed by two perpendicular number lines, the x-axis (horizontal) and y-axis (vertical). The numbers that tell the location of a point are the coordinates, written as an ordered pair like $(x, y)$.

What’s next

This is just the foundation. Soon, you'll use this system in worked examples to plot points, find coordinates, and analyze geometric shapes like rectangles.

Property

A coordinate plane is a grid formed by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. The point at which the number lines intersect is called the origin.

Examples

• The origin is the special starting point with the coordinates $(0, 0)$.
• A point in the upper-right section, like $(5, 3)$, has two positive coordinates.
• A point in the lower-left section, like $(-2, -4)$, has two negative coordinates.

Explanation

Think of the coordinate plane as a giant treasure map for math! The horizontal x-axis tells you how far to walk right or left, while the vertical y-axis tells you how far to climb up or down. The origin is your 'X marks the spot' starting point where every adventure begins before you follow the two crucial directions.

Property

The coordinates of a point are written as an ordered pair of numbers in parentheses, such as $(x, y)$. The first number is the x-coordinate (horizontal travel), and the second number is the y-coordinate (vertical travel).

Examples

• To graph the point $(3, -2)$, you start at the origin, move 3 units right, and then 2 units down.
• The point $(-5, 1)$ means you move 5 units to the left and 1 unit up from the origin.
• The ordered pairs $(4, 6)$ and $(6, 4)$ are different points on the coordinate plane.

Explanation

An ordered pair is like a secret code giving you two-step directions. The first number tells you to run along the x-axis hallway, and the second tells you to take the y-axis elevator. The order matters—get it wrong, and you might end up on the wrong floor! Remember this simple rule: you have to run before you can jump.

Property

By graphing the vertices of a rectangle on a coordinate plane, we can determine its side lengths and then calculate the rectangle's perimeter and area.

Examples

• For a rectangle with vertices at $(-1, 2)$, $(3, 2)$, $(3, -1)$, and $(-1, -1)$, the length is 4 units and the width is 3 units.
• The perimeter of the rectangle with those vertices is $2(4) + 2(3) = 14$ units.
• The area of the same rectangle is found by multiplying the sides: $4 \times 3 = 12$ square units.

Explanation

Become a digital architect by plotting the corners (vertices) of a shape to create a blueprint on the grid. From there, you can just count the squares to find the length and width of your design. This makes it super easy to find the distance around it (perimeter) or calculate the space it covers (area), all without needing a physical ruler!