Plotting Points On a Rectangular Coordinate System
Introduction
This graph illustrates the annual vehicle sales of gas motorcycles, gas cars, and electric vehicles from 1994 to 2010. It is a line graph with x- and y-axes, one of the most common types of graphs. (credit: Steve Jurvetson, Flickr)
Graphs are found in all areas of our lives—from commercials showing you which cell phone carrier provides the best coverage, to bank statements and news articles, to the boardroom of major corporations. In this tutorial, we will study the rectangular coordinate system, which is the basis for most consumer graphs. We will look at linear graphs, slopes of lines, equations of lines, and linear inequalities.
Plotting Points On a Rectangular Coordinate System
Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy-plane or the ‘coordinate plane’.
The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The x-axis and the y-axis together form the rectangular coordinate system. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See the figure below.
‘Quadrant’ has the root ‘quad,’ which means ‘four.’
In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point.
Ordered Pair
An ordered pair, \(\left(x,y\right)\), gives the coordinates of a point in a rectangular coordinate system.
The first number is the x-coordinate.
The second number is the y-coordinate.
The phrase ‘ordered pair’ means the order is important. What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is \(\left(0,0\right)\). The point \(\left(0,0\right)\) has a special name. It is called the origin.
The Origin
The point \(\left(0,0\right)\) is called the origin. It is the point where the x-axis and y-axis intersect.
We use the coordinates to locate a point on the xy-plane. Let’s plot the point \(\left(1,3\right)\) as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through \(x=1\). Then, locate 3 on the y-axis and sketch a horizontal line through \(y=3\). Now, find the point where these two lines meet—that is the point with coordinates \(\left(1,3\right)\).
Notice that the vertical line through \(x=1\) and the horizontal line through \(y=3\) are not part of the graph. We just used them to help us locate the point \(\left(1,3\right)\).
Example
Plot each point in the rectangular coordinate system and identify the quadrant in which the point is located:
\(\left(-5,4\right)\) \(\left(-3,-4\right)\) \(\left(2,-3\right)\) \(\left(-2,3\right)\) \(\left(3,\frac{5}{2}\right)\).
Solution
The first number of the coordinate pair is the x-coordinate, and the second number is the y-coordinate.
- Since \(x=-5\), the point is to the left of the y-axis. Also, since \(y=4\), the point is above the x-axis. The point \(\left(-5,4\right)\) is in Quadrant II.
- Since \(x=-3\), the point is to the left of the y-axis. Also, since \(y=-4\), the point is below the x-axis. The point \(\left(-3,-4\right)\) is in Quadrant III.
- Since \(x=2\), the point is to the right of the y-axis. Since \(y=-3\), the point is below the x-axis. The point \(\left(2,-3\right)\) is in Quadrant lV.
- Since \(x=-2\), the point is to the left of the y-axis. Since \(y=3\), the point is above the x-axis. The point \(\left(-2,3\right)\) is in Quadrant II.
- Since \(x=3\), the point is to the right of the y-axis. Since \(y=\frac{5}{2}\), the point is above the x-axis. (It may be helpful to write \(\frac{5}{2}\) as a mixed number or decimal.) The point \(\left(3,\frac{5}{2}\right)\) is in Quadrant I.
How do the signs affect the location of the points? You may have noticed some patterns as you graphed the points in the previous example.
For the point in the figure above in Quadrant IV, what do you notice about the signs of the coordinates? What about the signs of the coordinates of points in the third quadrant? The second quadrant? The first quadrant?
Can you tell just by looking at the coordinates in which quadrant the point \(\left(-2,5\right)\) is located? In which quadrant is \(\left(2,-5\right)\) located?
Quadrants
We can summarize sign patterns of the quadrants in this way.
\(\begin{array}{cccccccccc}\hfill \text{Quadrant I}\hfill & & & \hfill \text{Quadrant II}\hfill & & & \hfill \text{Quadrant III}\hfill & & & \hfill \text{Quadrant IV}\hfill \\ \hfill \left(x,y\right)\hfill & & & \hfill \left(x,y\right)\hfill & & & \hfill \left(x,y\right)\hfill & & & \hfill \left(x,y\right)\hfill \\ \hfill \left(+,+\right)\hfill & & & \hfill \left(\text{−},+\right)\hfill & & & \hfill \left(\text{−},\text{−}\right)\hfill & & & \hfill \left(+,\text{−}\right)\hfill \end{array}\)
What if one coordinate is zero as shown in the figure below? Where is the point \(\left(0,4\right)\) located? Where is the point \(\left(-2,0\right)\) located?
The point \(\left(0,4\right)\) is on the y-axis and the point \(\left(-2,0\right)\) is on the x-axis.
Points on the Axes
Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates \(\left(a,0\right)\).
Points with an x-coordinate equal to 0 are on the y-axis, and have coordinates \(\left(0,b\right)\).
Example
Plot each point:
\(\left(0,5\right)\) \(\left(4,0\right)\) \(\left(-3,0\right)\) \(\left(0,0\right)\) \(\left(0,-1\right)\).
Solution
- Since \(x=0\), the point whose coordinates are \(\left(0,5\right)\) is on the y-axis.
- Since \(y=0\), the point whose coordinates are \(\left(4,0\right)\) is on the x-axis.
- Since \(y=0\), the point whose coordinates are \(\left(-3,0\right)\) is on the x-axis.
- Since \(x=0\) and \(y=0\), the point whose coordinates are \(\left(0,0\right)\) is the origin.
- Since \(x=0\), the point whose coordinates are \(\left(0,-1\right)\) is on the y-axis.
In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the x-coordinate of a point on a graph, read the number on the x-axis directly above or below the point. To identify the y-coordinate of a point, read the number on the y-axis directly to the left or right of the point. Remember, when you write the ordered pair use the correct order, \(\left(x,y\right)\).
Example
Name the ordered pair of each point shown in the rectangular coordinate system.
Solution
Point A is above \(-3\) on the x-axis, so the x-coordinate of the point is \(-3\).
- The point is to the left of 3 on the y-axis, so the y-coordinate of the point is 3.
- The coordinates of the point are \(\left(-3,3\right)\).
Point B is below \(-1\) on the x-axis, so the x-coordinate of the point is \(-1\).
- The point is to the left of \(-3\) on the y-axis, so the y-coordinate of the point is \(-3\).
- The coordinates of the point are \(\left(-1,-3\right)\).
Point C is above 2 on the x-axis, so the x-coordinate of the point is 2.
- The point is to the right of 4 on the y-axis, so the y-coordinate of the point is 4.
- The coordinates of the point are \(\left(2,4\right)\).
Point D is below 4 on the x-axis, so the x-coordinate of the point is 4.
- The point is to the right of \(-4\) on the y-axis, so the y-coordinate of the point is \(-4.\)
- The coordinates of the point are \(\left(4,-4\right)\).
Point E is on the y-axis at \(y=-2\). The coordinates of point E are \(\left(0,-2\right).\)
Point F is on the x-axis at \(x=3\). The coordinates of point F are \(\left(3,0\right).\)
This lesson is part of:
Graphs and Equations