# Coordinate Graph: Definition & Examples

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Learn what a coordinate graph is and about its characteristics. Through definition, story, and example, we will see how to navigate a coordinate graph and how to locate and plot points on these types of graphs.

## A Fly On The Wall

Rene Descartes was a French mathematician. He lived from 1596-1650 and made many contributions to mathematics.

Legend has it that when Descartes was a child, he was often ill and spent a lot of time resting in bed. One time, while in bed, he noticed a fly on the ceiling. As the fly moved around, he noticed that the fly's position was determined by its distance from each of the corners of the walls with the ceiling of the room. After making this realization, he decided to represent the lines in which two of the walls met the ceiling with number lines, and the rest is history!

## Coordinate Graph

Descartes' observation of that fly led him to develop a coordinate graph. A coordinate graph is also sometimes called a coordinate plane, a Cartesian plane, or a Cartesian coordinate system. A coordinate graph consists of two number lines that run perpendicular to each other. These number lines are called axes. The horizontal line is called the x-axis, and the vertical line is called the y-axis. These two axes intersect where they are both equal to zero. This intersection point is called the origin.

## Ordered Pair

Now that we have seen what a coordinate graph looks like, let's consider Descartes' fly observation. Imagine there is a fly somewhere on our graph as shown in the image below.

Notice that if we start at the origin, we have to move 2 units to the left, and then we have to move 3 units up to get to the fly. This position lines up with -2 on the x-axis and 3 on the y-axis. We represent this position using an ordered pair. An ordered pair (x, y) represents a position of a point on a coordinate graph, where x is the number on the x-axis that the point lines up with, and y is the number on the y-axis that the point lines up with. The numbers x and y in the ordered pair (x, y) are called coordinates. The first number in the pair is called the x-coordinate, and the second number in the pair is called the y-coordinate.

In our fly example just described, we said that the fly lines up with -2 on the x-axis and 3 on the y-axis. Therefore the ordered pair representing the position of the fly on our graph is (-2,3).

## Plotting Ordered Pairs

When given an ordered pair (a, b), where a is the x-coordinate and b is the y-coordinate, we can plot it on a coordinate graph. To do so, we follow these steps.

1. Start at the origin, and move a units horizontally along the x-axis. If a is positive, we move a units to the right of the origin, and if a is negative, we move a units to the left of the origin.
2. From the point we just moved to on the x-axis, move vertically b units. If b is positive, we move up b units, and if b is negative, we move down b units.
3. We are now at the position that our ordered pair represents, so we draw a point, and we have plotted our ordered pair.

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