The Coordinate Plane
Learning Objective(s)
· Plot ordered pairs on a coordinate plane.
· Given an ordered pair, determine its quadrant.
Introduction
The coordinate plane was developed centuries ago and refined by the French mathematician René Descartes. In his honor, the system is sometimes called the Cartesian coordinate system. 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.
Getting to Know 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 AF are listed along the top, and the numbers 16 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. (They are perpendicular to each other.)
The horizontal axis in the coordinate plane is called the xaxis. The vertical axis is called the yaxis. The point at which the two axes intersect is called the origin. The origin is at 0 on the xaxis and 0 on the yaxis.
The intersecting x and yaxes divide the coordinate plane into four 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 counter clockwise.
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 xaxis (the first value of the ordered pair) and along the yaxis (the second value of the ordered pair).
In an ordered pair, such as (x, y), the first value is called the xcoordinate and the second value is the ycoordinate. Note that the xcoordinate is listed before the ycoordinate. Since the origin has an xcoordinate of 0 and a ycoordinate of 0, its ordered pair is written (0, 0).
Consider the point below.
To identify the location of this point, start at the origin (0, 0) and move right along the xaxis until you are under the point. Look at the label on the xaxis. The 4 indicates that, from the origin, you have traveled four units to the right along the xaxis. This is the xcoordinate, the first number in the ordered pair.
From 4 on the xaxis move up to the point and notice the number with which it aligns on the yaxis. The 3 indicates that, after leaving the xaxis, you traveled 3 units up in the vertical direction, the direction of the yaxis. This number is the ycoordinate, the second number in the ordered pair. With an xcoordinate of 4 and a ycoordinate of 3, you have the ordered pair (4, 3).
Let’s look at another example.
Example  
Problem  Describe the point shown as an ordered pair.  

 
 (5, y)  Begin at the origin and move along the xaxis. This is the xcoordinate and is written first in the ordered pair. 
 (5, 2)  Move from 5 up to the ordered pair and read the number on the yaxis. This is the ycoordinate and is written second in the ordered pair. 
Answer  The point shown as an ordered pair is (5, 2). 
Now that you know how to use the x and yaxes, 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).
Example  
Problem  Plot the point (1, 3).  

 
 The xcoordinate is 1 because it comes first in the ordered pair. Start at the origin and move a distance of 1 unit in a positive direction (to the right) from the origin along the xaxis.  The ycoordinate is 3 because it comes second in the ordered pair. From here move directly 3 units in a positive direction (up). If you look over to the yaxis, you should be lined up with 3 on that axis. 
Answer  Draw a point at this location and label the point (1, 3). 
In the previous example, both the x and ycoordinates 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.
Example  
Problem  Plot the point (−4, −2).  

 
 The xcoordinate is −4 because it comes first in the ordered pair. Start at the origin and move 4 units in a negative direction (left) along the xaxis.  The ycoordinate is −2 because it comes second in the ordered pair. Now move 2 units in a negative direction (down). If you look over to the yaxis, you should be lined up with −2 on that axis. 
Answer  Draw a point at this location and label the point (−4, −2). 
The steps for plotting a point are summarized below.
Steps for Plotting an Ordered Pair (x, y) in the Coordinate Plane o Determine the xcoordinate. Beginning at the origin, move horizontally, the direction of the xaxis, the distance given by the xcoordinate. If the xcoordinate is positive, move to the right; if the xcoordinate is negative, move to the left. o Determine the ycoordinate. Beginning at the xcoordinate, move vertically, the direction of the yaxis, the distance given by the ycoordinate. If the ycoordinate is positive, move up; if the ycoordinate is negative, move down. o Draw a point at the ending location. Label the point with the ordered pair.

Which point represents the ordered pair (−2, −3)?

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 ycoordinates of the points within each quadrant?
Within each quadrant, the signs of the xcoordinates and ycoordinates 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  (+, +)  (5, 4)  Starting from the origin, go along the xaxis in a positive direction (right) and along the yaxis in a positive direction (up). 
II  (−, +)  (−5, 4)  Starting from the origin, go along the xaxis in a negative direction (left) and along the yaxis in a positive direction (up). 
III  (−, −)  (−5, −4)  Starting from the origin, go along the xaxis in a negative direction (left) and along the yaxis in a negative direction (down). 
IV  (+, −)  (5, −4)  Starting from the origin, go along the xaxis in a positive direction (right) and along the yaxis in a 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’s another 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.
Example  
Problem  In which quadrant is the point (−7, 10) located?  
 (−7, 10)  Look at the signs of the x and ycoordinates. For this ordered pair, the signs are (−, +). 
 Points with the pattern  Using the table or grid above, locate the pattern (−, +). 
Answer  The point (−7, 10) is in Quadrant II. 
Example  
Problem  In which quadrant is the point (−10, −5) located?  
 (−10, −5)  Look at the signs of the x and ycoordinates. For this ordered pair, the signs are (−, −). 
 Points with the pattern  Using the table or grid above, locate the pattern (−, −). 
Answer  The point (−10, −5) is in Quadrant III. 
What happens if an ordered pair has an x or ycoordinate of zero? The example below shows the graph of the ordered pair (0, 4).
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 xcoordinate is 0, the point is located on the yaxis. Similarly, any point that has a ycoordinate of 0 will be located on the xaxis.
Which of the descriptions below best describes the location of the point (8, 0)?
A) Quadrant I
B) It is on the xaxis
C) It is on the yaxis
D) The coordinate plane

Summary
The coordinate plane is a system 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 lines creates the origin, which is the point (0, 0). 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 concepts.