Introduction to the coordinate plane

You're probably familiar with the notion of a number line where we can take a number and associate it with a point on the number line. So for example, the number 2, I would go, I would start at 0, and I'd go 1, 2 to the right, and I would end up right over there.

What we're going to do in this video is think about how do we take two numbers and associate them with a point on a plane like this. So for example, you might have the two numbers 3, 5. How does that—or how can these two numbers be represented as a point on the plane? Or how can these tell you where a point on the plane is?

So let's first get a little bit of terminology out of the way. So what we have here, this is often known as our coordinate plane. These two numbers you could view as our coordinates. Let me write this down: these are coordinates. These black lines are known as the axes; each one is an axis. The one that goes left-right here, this is known as the x-axis. It's typically known as the x-axis. In the future, it might be called other things. And the one that goes up-down in the vertical direction, this is typically known as the y-axis. As you go further in math, we might call it other things, but most of the time it's going to be called the y-axis.

So how does 3, 5—how do these coordinates specify a point on this plane? Well, the way that we typically work it through, the standard way that people will interpret these points is they'll say, "All right, this first coordinate, this is our x-coordinate." This is our x-coordinate. You can view it as how far do we move to the right along the x-axis. So what you would do is you'd say, "All right, I'm going to start right here where my axes intersect, and I'm going to go three to the right: one, two, three."

So my x-coordinate says, "All right, my point is going to be this far to the right." This far to the right, so it's going to be somewhere on this vertical line, this dotted line that I'm showing. Everything on this vertical line has an x-coordinate of 3. Now what's the y-coordinate, or another way I should say it, the second number right over here? This is the y-coordinate. The y-coordinate tells us how far do we move up.

So one way to think about it: you could start back at where the axes intersect. This point is actually called the origin. Let me write that down: that is the origin. So starting at the origin, move five up: one, two, three, four, five. So everything on this horizontal line that I'm drawing has a coordinate of five.

So what point uniquely has both an x-coordinate of three and a y-coordinate of five? Well, you can see where those two lines intersect right over here. Actually, let me do that same blue color that I wrote the coordinates in, so this point right over here has an x-coordinate of 3 and a y-coordinate of 5. That is the point 3, 5.

Now, what are the coordinates of the origin? Well, the origin is 0 to the right of the origin, and it's also 0 above the origin. So the coordinates there, the x-coordinate is 0, the y-coordinate is also 0.

Let's do a few more examples. So let's say that I wanted to plot the point 2, 5. Why don't you pause this—or let me use a different number: 2, 4. Why don't you pause this video and think about where that point would be on this coordinate plane?

All right, let's do it together. So the first number is going to be our x-coordinate. It tells us how far do we move to the right. So we move 2 to the right, and then our second number says how far do we move up. So let's first—we're going to move 2 to the right, and then we are going to move 4 up.

So you could say 1, 2 to the right, and then 1, 2, 3, 4. Right over there, this right over here is the point 2, 4. Notice its x-coordinate—how far to the right of the origin it is—that is 2, and its y-coordinate—how far above the origin is—that is 4.

Now let's go the other way around. Let's say that I were to give you—if I were to give you this point right over here, what would its coordinates be? Pause the video and try to figure that out.

All right, well, we know it's going to be two numbers, so I'll do something comma something. Now the first something, that's going to be our x-coordinate. You could think of it as what point on the x-axis are we above? You could think about how far to the right of the origin we are. And you can see that your x-coordinate right over here is—if we just drop a vertical line straight down from that point, it hits the x-axis at four. So that is four.

Another way to think about it: we're one, two, three, four to the right of the y-axis. Now how high are we? How high above the x-axis are we? Well, we're one above the x-axis, so this is going to be 4, 1. Another way to think about it: if you just take a line and you go straight to the left, you're going to hit the y-axis at the one right over here. So the coordinates here are 4, 1.

Now, just so we don't get confused—and when you first learn this, the main point of confusion is remembering that, okay, the first number is the x-coordinate, the second number is the y-coordinate. 1, 4 would be a different point. 1, 4 would be okay: one in the x-direction and then four in the y-direction. So this is 1, 4 over here.

So it's very important to realize that the standard way of interpreting these numbers is that the first one says how far do you move to the right of the origin or how far do you move along the x-axis or where are you on the x-axis, and the second number is how far do you move in the vertical direction or where are you relative to the y—or where are—or where are you in the vertical direction.