Coordinate Geometry: The Cartesian Plane

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What Are Cartesian Planes?

Cartesian Plane
Cartesian Plane. D. Russell

The Cartesian Plane is sometimes referred to as the x-y plane or the coordinate plane and is used to plot data pairs on a two-line graph. The Cartesian plane is named after the mathematician Rene Descartes who originally came up with the concept. Cartesian planes are formed by two perpendicular number lines intersect.

Points on the cartesian plane are called "ordered pairs," which become extremely important when illustrating the solution to equations with more than one data point. Simply put, though, the Cartesian plane is really just two number lines where one is vertical and the other horizontal and both form right angles with one another.

The horizontal line here is referred to the x-axis and values that come first in ordered pairs are plotted along this line while the vertical line is known as the y-axis, where the second number of ordered pairs is plotted. An easy way to remember the order of operations is that we read from left to right, so the first line is the horizontal line or the x-axis, which also comes first alphabetically.

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Quadrants and Uses of Cartesian Planes

Cartesian Plane
Cartesian Plane. D. Russell

Because Cartesian Planes are formed from two to-scale lines intersecting at right angles, the resulting image yields a grid broken into four sections known as quadrants. These four quadrants represent a full set of positive numbers on both the x- and y-axises wherein the positive directions are upward and to the right, while the negative directions are downward and to the left.

Cartesian planes are therefore used to plot the solutions to formulas with two variables present, typically represented by x and y, though other symbols can be substituted for the x- and y-axis, so long as they are properly labeled and follow the same rules as x and y in the function.

These visual tools provide students with a pinpoint using these two points that account for the solution to the equation.

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Cartesian Plane and Ordered Pairs

Ordered Pair - Locating a Point. D. Russell

The x-coordinate is always the first number in the pair and the y-coordinate is always the second number in the pair. The point illustrated on the Cartesian plane to the left shows the following ordered pair: (4, -2) wherein the point is represented by a black dot.

Therefore (x,y) = (4, -2). To identify the ordered pairs or to locate points, you start at the origin and count the units along each axis. This point shows a student who went four clicks to the right and two clicks down.

Students may also solve for a missing variable if x or y is unknown by simplifying the equation until both variables have a solution and can be plotted on a Cartesian plane. This process forms the basis for most early algebraic computations and data mapping.

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Test Your Ability to Locate Points of Ordered Pairs

Ordered Pairs
Ordered Pairs. D. Russell

Take a look at the Cartesian plane to the left and notice the four points that have been plotted on this plane. Can you identify the ordered pairs for the red, green, blue, and purple points? Take some time then check your answers with the correct responses listed below:

Red Point = (4, 2)
Green Point = (-5, +5)
Blue Point = (-3, -3)
Purple Point =(+2,-6)

These ordered pairs might remind you a bit of the game Battleship wherein players have to call out their attacks by listing ordered pairs of coordinates like G6, wherein letters lie along the horizontal x-axis and numbers form along the vertical y-axis.