Definition | Coordinate Systems | Rectangular Coordinate System

Video Transcript

You’ve seen how to add, subtract, multiply, and divide integers. You have also seen how to simplify expressions with integers. In this video we look at using integers to graph points on the Cartesian coordinate plane.

Let’s start with an example.

A construction truck goes from Building A to Building B to Building C. Describe the coordinates of these buildings.

Notice the Cartesian coordinate system. Here is the y-axis and here is the x-axis. This happens to just show Quadrant I.

This truck goes from point A here to point B here to point C here. These three points can be graphed with integer coordinates.

Here is a version of the graph with just the points. Where the x- and y-axes intersect is the origin, which has coordinates (0, 0).

Let’s assume that each tic mark along the horizontal and vertical axes represent 1 unit. Let’s look at point A. We start at the origin and count one unit along the x-axis until we are aligned with A. Now we move up 2 units to get to the point. So, point A has coordinates (1, 2).

This number is the x-coordinate and this number is the y-coordinate. Now let’s look at point B.

Start at the origin and count the number of tic marks along the x-axis to align with B. There are 1-2-3-4-5-6-7-8 units. And we move up 2 units. So point B has coordinates (8, 2).

Now let’s move on to point C. First thing, notice that it is vertically aligned with point B, which we know has an x-coordinate of 8. So C has an x-coordinate of 8. Starting here let’s count up to get to C. There are 1-2-3-4-5 tic marks. So C has coordinates (8, 5).

Let’s look at another example.

A diver takes a dive and his path is shown in the red dotted line. He passes through points A, B, and C. What are the coordinates of those points?

We can use positive and negative integers to show the coordinates of the diver at the three points. Here is a clean version to make it easier to identify the coordinates.

Recall that where the x- and y axes intersect is the origin, which has coordinates (0, 0). This coordinate system shows Quadrant I here and Quadrant II here. Quadrant I has all positive coordinates. Quadrant II has positive x and negative y coordinates.

Let’s assume that each tic mark along the horizontal and vertical axes represent 1 unit. Let’s look at point A. We start at the origin and count one unit along the x-axis until we are aligned with A. There are 1-2-3-4-5-6-7-8-9-10 units. Now we move up 1-2-3-4 units to get to the point. So, point A has coordinates (10, 4).

Now let’s look at point B. Notice that the point is on the x-axis. Any point on the x-axis has zero as a y-coordinate. So we just need t find the x-coordinate.

Start at the origin and count one unit along the x-axis until we are aligned with B, There are 1-2-3-4-5-6-7-8-9-10-11-12-13-14-15 units. So point B has coordinates (15, 0).

Now let’s move on to point C. First thing, notice that it is just one unit to the right of point B. Since point B has an x-coordinate of 15, then C has an x-coordinate of 16. Now we count down 1-2-3-4 units along the y-axis. Four units below the x-axis makes the x-coordinate -4. So point C has coordinates (16, -4).

Let’s look at a final example.

Find the coordinates of points A, B, C, and D, which are the corners of a rectangle.

Notice that this rectangle is centered at the origin. What this means is that the y-axis splits the rectangle into equal sections. The same thing happens with the x-axis: It splits the rectangle into two equal sections this way.

This means that the rectangle is symmetric about the x-axis and the y-axis. Now we can look at the coordinates for the four points to see how this symmetry is reflected in the coordinates.

Notice that positive and negative 7 and 3 are integers that make up these coordinates. Look at any collinear pair of coordinates and you’ll see that pairs of coordinates are opposites. For example, A and B have opposite x-coordinates, while D and C also have opposite x-coordinates. A-D and BC have opposite y-coordinates.