Chapter 14

The Coordinate Plane

Before examining the coordinate plane, you should review the number line:

The Number Line

The number line is a ruler or measuring stick that goes as far as you want in both directions. With the number line, you can say where something is positioned with a single number. In other words, you can link a position with a number:

Position	Number	Number Line
“Two units right of 0”	2
“One and a half units left of 0”	−1.5

You use both positive and negative numbers, because you can indicate positions both left and right of 0.

You might be wondering, “The position of what?” The answer is, a point, which is just a dot. When you are dealing with the number line, a point and a number mean the same thing:

If you show me where the point is on the number line, I can tell you the number.		The point is at −2.
If you tell me the number, I can show you where the point is on the number line.		The point is at 0.

This works even if you only have partial information about your point. If told something about where the point is, you can say something about the number, and vice versa.

For instance, if told that the number is positive, then you know that the point lies somewhere to the right of 0 on the number line. Even though you don’t know the exact location of the point, you do know a range of potential values:

The number is positive. In other words, the number is greater than (>) 0.
	The open circle means 0 is not included.

This also works in reverse. If you see a range of potential positions on a number line, you can tell what that range is for the number:

Therefore, the number is less than (<) 0.

Now to make things more complicated. What if you want to be able to locate a point that’s not on a straight line, but on a page?

Begin by inserting your number line into the picture. This will help you determine how far to the right or left of 0 your point is:

The point is 2 units to the right of 0. But all three points that touch the dotted line are 2 units to the right of 0. You don’t have enough information to determine the unique location of the point.

To know the location of your point, you also need to know how far up or down the dotted line you need to go. To determine how far up or down you need to go, you’re going to need another number line. This number line, however, is going to be vertical. Using this vertical number line, you will be able to measure how far above or below 0 a point is:

The point is 1 unit above 0. Notice that this number line by itself also does not provide enough information to determine the unique location of the point.

But, if you combine the information from the two number lines, you can determine both how far left or right and how far up or down the point is:

The point is 2 units to the right of 0. AND The point is 1 unit above 0.

Now you have a unique description of the point’s position. There is only one point on the page that is BOTH 2 units to the right of 0 AND 1 unit above 0. So, on a page, you need two numbers to indicate position.

Just as with the number line, information can travel in either direction. If you know the two numbers that give the location, you can place that point on the page:

The point is 3 units to the left of 0. AND The point is 2 units below 0.

If, on the other hand, you see a point on the page, you can identify its location and determine the two numbers:

The point is 1 unit to the right of 0. AND The point is 2.5 units below 0.

Now that you have two pieces of information for each point, you need to keep straight which number is which. In other words, you need to know which number gives the left-right position and which number gives the up-down position.

To represent the difference, use some technical terms:

The x-coordinate is the left-right number:

Numbers to the right of 0 are positive. Numbers to the left of 0 are negative.

The y-coordinate is the up-down number:

Numbers above 0 are positive. Numbers below 0 are negative.

Now, when describing the location of a point, you can use the technical terms:

The x-coordinate of the point is 1 and the y-coordinate of the point is 0.

You can condense this and say that, for this point, x = 1 and y = 0. In fact, you can go even further. You can say that the point is at (1, 0). This shorthand always has the same basic layout. The first number in the parentheses is the x-coordinate, and the second number is the y-coordinate. One easy way to remember the order in this “ordered pair” is that x comes before y in the alphabet. For example:

The point is at (−3, −1). OR The point has an x-coordinate of −3 and a y-coordinate of −1.

Now you have a fully functioning coordinate plane: an x-axis and a y-axis drawn on a page. The coordinate plane allows you to determine the unique position of any point on a plane (essentially, a really big and flat sheet of paper).

And in case you were ever curious about what one dimensional and two dimensional mean, now you know. A line is one dimensional, because you only need one number to identify a point’s location. A plane is two-dimensional because you need two numbers to identify a point’s location.

Check Your Skills

1. Draw a coordinate plane and plot the following points:

   1. (3, 1)
   2. (−2, 3.5)
   3. (0, −4.5)
   4. (1, 0)

2. Which point on the coordinate plane shown here is indicated by the following coordinates?

   1. (2,−1)
   2. (−1.5,−3)
   3. (−1,2)
   4. (3, 2)
   5. (2, 3)