 So, we'll introduce this notion of graphing points as a prelude to other types of graphing. And so, here's a very common problem in life, which is we want to locate something. We want to figure out where something is. And so, the idea in mathematical terms, well, suppose I have a point that's located someplace in space. So, here's my point A, someplace out here, and I want to be able to identify where this point is located. Well, I could say, here it is. It's over here. Here it is. Here it is. But that doesn't really do a lot of good if I'm trying to describe this location as somebody else. And so, how do we specify where that point's located? And so, what we can do is we need to locate a specific point, which we call the origin. You can think about this as this is where you're starting from. And we also need to identify a specific line through the origin, which we'll call the principal direction. And our idea at this point is I'm going to describe the location of this point as a set of directions for how to get from here to there. Now, I'm from Boston, so if this were in New England, I might have the opportunity of saying, well, you can't get there from here, but this is mathematics. So, we can actually find those directions. And so, what I might do is I might say something like the following. So, start at your origin. And what you're going to do is you're going to walk along the principal direction some distance. So, I'll take a couple of steps. That's one, two, and three. And at this point, so I take three steps out along our principal axis and then I'm going to make a right, sorry, left turn. Get right and left next up all the time. And then I'm going to walk towards that point A. And I'll walk up and up and up and up and there I am. And so, now I can describe my directions for getting there. What I've done is I've got one, two, three steps to the right and one, two, three, four steps up. And now, I can specify this number in a couple of different ways. So, what I might do is I might actually just write it out. The point A is located three to the right for up. And that implies I'm going to take three steps to the right, four steps up. And that's going to get me right to the point. The other thing I can do is write this in what's called coordinate form. It's a set of directions. So, I do have to specify how far over horizontally and how far up vertically. So, I need to specify two numbers, two ordnance, which are the coordinates. And that's going to be three, four. Where I specify the horizontal distance first, the vertical distance second, and I throw the whole thing in a set of parentheses. Now, it's important to remember that when we do that, this is a single point. These are directions for getting there. The ether number by itself doesn't really tell you anything. This says go three over, one, two, three. If I go over three, I'm not at the point. This says go up four. If I go up four from the origin, I'm not at the point. This only makes sense as a set of directions if I have both. Also, when we write this in coordinate form, the first coordinate always specifies the left-right, the horizontal amount, and the second always specifies the up-down, the vertical amount. Well, maybe I have multiple points I want to describe. So, again, I have my origin, I have my principal line, and I have a whole bunch of points I want to get to. Maybe I'm going on a shopping trip. I need to go here, then here, then here, then here before heading back to home. If I have multiple points, it's convenient to set down what's called a coordinate grid that marks out all of the distances. So, I throw the grid down, and it looks really like a street map. And I could specify all of my points with reference to the grid. Now, on this grid, if I'm going right or up, if I'm going to the right or if I'm going up, I'm going to reference those distances by positive numbers. While if I'm going left or down, I'm going to call those by negative numbers. So, if I want to get from A to O, if I want to get to A starting at O, again, we always start at the origin, let's see what I have to do. Well, I have to travel one, two spaces over, one space up. I've gone right, two, up, one, and because that's right and up, they're both positive. So, that's right, two, up, one, and my coordinates are going to be two, one. Now, if I want to get to B, well, here's the important thing to remember. I'm always starting at the origin, and I'm going to get to the point I'm interested in. So, what do I have to do? Well, I have to go one, two, three, four. I have to go left, four spaces, up, one. So, I have to travel left, four, one, up. And so, because I'm traveling left, that's a negative amount. So, this four units, I'm going to write this as negative for one unit up. And so, I could write this as negative for one. So, that negative says I'm going left, four units, and the one positive says I'm going up one unit. So, I'm going back for up one. From C to get to C, I have to go back one, two, three, and then down one, two. I have to go left three, down two. And so, because I'm going left and down, both of these will be written using negative numbers. So, my coordinates negative three, negative two. And finally, to get to D from O, I have to travel one, two, three, four, five. I have to travel five to the right and two down. So, that's going to be five right, two down. Travel right is positive, travel down is negative. So, this is coordinates five, negative two. And my coordinates will be that. What if I actually want to plot a point? So, given the coordinates of a point, how do I find it? Well, again, the coordinates tell you how to get there based on where the origin is located and where the principal direction is. So, for example, let's say I want to get to the point four, negative two. Well, I have the origin, I have my principal direction, and if I want to plot this point, this says four. It's positive, so I'm going to go four units to the right. This is negative two. Negative says I'm going down to units. Notice that I only had to plot the origin and the principal direction. You may have been taught to plot the x and y axes. You don't actually need both axes if you're just plotting a couple of points. It's convenient, but we don't need them. So, the directions here say go four units to the right, two units down. So, I'll take a walk. One, two, three, four, and I'm going to go one, two units down, and that's where the point F is located. As before, if I am finding many points, it's helpful to set down a coordinate grid, but not absolutely necessary to do so. So, here I have a bunch of points. A being point one, two. B being two, one. C being negative two, negative one. So, let's take a look at that. That point A, I'll read this. First coordinate tells you left, right. Second coordinate tells you up, down. Both of them are positive, so I'm going right, one, up, two. And that's always from the origin. So, I'll take a step from the origin, right, one, up, one, two, and that'll put me right there. And there's my point A. My point B, two, one. Right, two, up, one. So, I'll be at the origin. I'll take a step out, one, two, and up, one. And I'm going to be right there. There's my point B. C, negative two, negative one. So, this is negative, so I'm going left, two units. Negative one says I'm going down, one unit. So, here I am. I'm going to go back, one, two, and down, one unit. I should put my point C right there, and there it is.