The coordinate plane was developed centuries ago (in 1637, to be exact) and refined by the French mathematician René Descartes. In his honor, the system is sometimes called the Cartesian coordinate system. In this chapter, we will study how the coordinate plane can be used to plot points and graph lines. This system allows us to describe algebraic relationships in a visual sense, and also helps us create and interpret algebraic concepts.

The components of the coordinate plane

You have likely used a coordinate plane before. For example, have you ever used a gridded overlay to map the position of an object? (This is often done with road maps, too.)

This “map” uses a horizontal and vertical grid to convey information about an object’s location. Notice that the letters A–F are listed along the top, and the numbers 1–6 are listed along the left edge. The general location of any item on this map can be found by using the letter and number of its grid square. For example, you can find the item that exists at square “4F” by moving your finger along the horizontal to letter F and then straight down so you are in line with the 4. You’ll find a blue disc is at this location on the map.

The coordinate plane has similar elements to the grid shown above. It consists of a horizontal axis and a vertical axis, number lines that intersect at right angles (perpendicular to each other),

Typically, the horizontal axis in the coordinate plane is called the x-axis and the vertical axis is called the y-axis. The point at which the two axes intersect is called the origin. The origin is at 0 on the x-axis and 0 on the y-axis.

Locations on the coordinate plane are described as ordered pairs. An ordered pair tells you the location of a point by relating the point’s location along the x-axis (the first value of the ordered pair) and along the y-axis (the second value of the ordered pair).

In an ordered pair, written (x, y), the first value is called the x-coordinate and the second value is the y-coordinate. Since the origin has an x-coordinate of 0 and a y-coordinate of 0, its ordered pair is written (0, 0).

Describe the point shown as an ordered pair

Consider the point below.

To identify the location of this point, start at the origin (0, 0) and move right along the x-axis until you are under the point. Look at the label on the x-axis. The 4 indicates that, from the origin, you have traveled four units to the right along the x-axis. This is the x-coordinate, the first number in the ordered pair.

From 4 on the x-axis move up to the point and notice the number with which it aligns on the y-axis. The 3 indicates that, after leaving the x-axis, you traveled 3 units up in the vertical direction, the direction of the y-axis. This number is the y-coordinate, the second number in the ordered pair. With an x-coordinate of 4 and a y-coordinate of 3, you have the ordered pair (4, 3).

Let’s look at another example.

Plotting points in the coordinate plane

Now that you know how to use the x- and y-axes, you can plot an ordered pair as well. Just remember, both processes start at the origin—the beginning! The example that follows shows how to graph the ordered pair (1,3).

In the previous example, both the x- and y-coordinates were positive. When one (or both) of the coordinates of an ordered pair is negative, you will need to move in the negative direction along one or both axes. Consider the example below in which both coordinates are negative.

The steps for plotting a point are summarized below.

The next example include fractional coordinates.

Sometimes the scales on the x-axis and y-axis won’t match. It’s always important to pay attention to the scales so that you plot points correctly, as seen in the next example:

 

Determine the quadrant an ordered pair is located in

The intersecting x- and y-axes of the coordinate plane divide it into four distinct sections. These four sections are called quadrants. Quadrants are named using the Roman numerals I, II, III, and IV, beginning with the top right quadrant and moving counterclockwise.

Ordered pairs within any particular quadrant share certain characteristics. Look at each quadrant in the graph below. What do you notice about the signs of the x- and y-coordinates of the points within each quadrant?

Within each quadrant, the signs of the x-coordinates and y-coordinates of each ordered pair are the same. They also follow a pattern, which is outlined in the table below.

Quadrant	General Form of Point in this Quadrant	Example	Description
I	[latex](+,+)[/latex]	[latex](5,4)[/latex]	Starting from the origin, go along the x-axis in the positive direction (right) and along the y-axis in the positive direction (up).
II	[latex](−,+)[/latex]	[latex](−5,4)[/latex]	Starting from the origin, go along the x-axis in the negative direction (left) and along the y-axis in the positive direction (up).
III	[latex](−,−)[/latex]	[latex](−5,−4)[/latex]	Starting from the origin, go along the x-axis in the negative direction (left) and along the y-axis in the negative direction (down).
IV	[latex](+,−)[/latex]	[latex](5,−4)[/latex]	Starting from the origin, go along the x-axis in the positive direction (right) and along the y-axis in the negative direction (down).

Once you know about the quadrants in the coordinate plane, you can determine the quadrant of an ordered pair without even graphing it by looking at the chart above. Here is a more visual way to think about it.

The example below details how to determine the quadrant location of a point just by thinking about the signs of its coordinates. Thinking about the quadrant location before plotting a point can help you prevent a mistake. It is also useful knowledge for checking that you have plotted a point correctly.

What happens if an ordered pair has an x– or y-coordinate of zero? The example below shows the graph of the ordered pair [latex](0,4)[/latex].

A point located on one of the axes is not considered to be in a quadrant. It is simply on one of the axes. Whenever the x-coordinate is 0, the point is located on the y-axis. Similarly, any point that has a y-coordinate of 0 will be located on the x-axis.

Summary

The coordinate plane is a system that can be used for graphing and describing points and lines. The coordinate plane is comprised of a horizontal (x-) axis and a vertical (y-) axis. The intersection of these number lines creates the origin, which is the point [latex](0,0)[/latex]. The coordinate plane is split into four quadrants. Together, these features of the coordinate system allow for the graphical representation and communication about points, lines, and other algebraic objects.