Transcription
Welcome to the presentation on graphing lines. Let's get started. So let's say I had the equation-- let me make sure that this line doesn't show up too thick. Let's say I had the equation-- why isn't that showing up? Let's see. Oh, there you go. y is equal to 2x plus 1. So this is giving a relationship between x and y. So say x equals 1, then y would be 2 times 1 plus 1, or 3. So for every x that we can think of, we can think of a corresponding y.
So let's do that. If we said that-- put a little table here. x and y. And let's just throw out some random numbers for x. If x was, let's say, negative 1, then y would be 2 times negative 1, which is negative 2. Plus 1, which would be negative 1. If x was 0, that's easy. It'd be 2 times 0, which is 0. Plus 1, which is 1. If x was 1, y would be 2 times 1, which is 2. Plus 1, which is 3. If x was 2, then I think you get the idea here. y would be 5. And we could keep on going. Obviously, there are an infinite number of x's we could choose, and we could pick a corresponding y. So now you see we have a little table that gives the relationships between x and y.
What we can do now is actually graph those points on a coordinate axis. So let me see if I can draw this somewhat neatly. I'll use this line so I get straight lines. That's pretty good. Again, let me draw some coordinate points. So let's say that's 1, that's 2, that's 3. This is negative 1, negative 2, negative 3. So this is the x-axis. We have 1, 2, 3. Notice we could keep going. 1, 2, 3, and this is the y-axis. And this would be 1, 2, 3, and so on. This would be negative 1. I think you get the idea. So we can graph each of these points. So if we have the point x is negative 1, y is negative 1. So x, we go along the x-axis here, and we go to x is equal to negative 1. Then we go to y is equal to negative 1, so the point would be right here. Hope that makes sense to you. That's the point. I'll label it: negative 1, negative 1. It's a little messy. That says negative 1, negative 1. That point I just x'ed right there.
Let's do another one. That's this point. I'll do it in a different color this time. Let's say we had the point 0, 1. Well, x is 0, which is here. And y is 1, so that point is right there. Let's do one more. If we have the point 1, 3. Well, 1, 3, x is 1 and we have y is 3. So we have the point right there. Hope that's making sense for you. And we could keep graphing them, but I think you see here, and especially if I had drawn this a little bit neater, that these points are forming a line. Let me draw that line in. The line looks something like this. That's not a good line. Let me do it better than that. The line looks something like this. You see that? Well, that's actually a pretty bad line that I just drew. So it would be a line that goes through-- let me change tools. It'd be a line that goes through here, through here, and through here. I don't know if I'm making this clear at all. Let me make these points a little bit. You see the line will go through all of these points, but it will also go through the point 2, 5, which will be up here some place. For any x that you can think of, if you had x is equal to 10,380,000,000, the corresponding y will also be on this line. So this pink line, and it keeps going on forever, that represents every possible combination of x's and y's that will satisfy this equation. And of course, x doesn't have to just be whole numbers or integers. x could be pi-- 3.14159. In which case it would be someplace here and in which case y would be 2 pi plus 1. So every number that x could be, there's a corresponding y.
Let's do another one. So if I had the equation y is equal to-- that's an ugly y. y is equal to negative 3x plus 5. Well, I'm going to draw it quick and dirty this time. So that's the x-axis. That's the y-axis. Let's put some values here. x and y. Let's say if x is negative 1, then negative 1 times negative 3 is 3 plus y is 8. If x is 0, then y is 5. That's pretty easy. If x is 1, negative 3 times 1 is negative 3. Then y is 2. If x is 2, negative 3 times 2 is negative 6. Then y is 1. Is that right? Negative 6-- no, no. Negative 1. I knew something was wrong there. So let's graph some of these points. So when x is negative 1 and I'm just kind of approximating. When x is negative 1, y is negative 8. So that point would be someplace around here. And there's a whole module I'm graphing coordinates if you're finding the graphing a coordinate pair to be a little confusing. Oh, wait. I just made a mistake. When x is negative 1, y is 9. Not negative 8, so ignore this right here. When x is negative 1, y is positive 8. So y being up here someplace. When x is 0, y is 5. So it'd be here someplace. When x is 1, y is 2. So it's like here. When x is 2, y is negative 1. So as you can see-- and I've approximated it. If I had graphing paper or if I had a better drawn chart you could have seen it and it would have been exactly right. I think this line will do the job. That every point that satisfies this equation actually falls on this line. And something interesting here I'll point out. You notice that this line it slopes downwards. It goes from the top left to the bottom right. While the line we had drawn before had gone from the bottom left to the top right.
Is there anything about this equation that seems a little bit different than the last? I'll give you a little bit of a hint. This number-- the negative 3, or you could say that the coefficient on x-- that determines whether the line slopes upward, or the line slopes downward, and it tells you also how steep the line is. And that actually, negative 3 is the slope. And I'm going to do a whole nother module on slope. And this number here is called the y-intercept. And that actually tells you where you're going to intersect the y-axis. And it turns out here, that you intersect the axis at 0, 5.
Let's do one more real fast. y is equal to 2-- we already did 2x. y is equal to 1/2 x plus 2. So real fast. x and y. And you only need two points for a line, really. So you could just say let's say, x equals 0. That's easy. y equals 2. And if x equals 2 then y equals 3. So before when we were doing 3 and 4 points that was just to kind of show you, but you really just need two points for a line. So 0, 1 2. So that's on there. And then 1, 2, 3. So it's there. So the line is going to look something like this. So notice here, once again, we're upward sloping and that's because this 1/2 is positive. But we're not sloping-- we're not moving up as quickly as when we had y equals 2x. y equals 2x looked something like this. It was sloping up much, much, much faster. I hope I'm not confusing you. And then the y intercept of course is at 0, 2, which is right here. So if you ever want to graph a line it's really easy. You have to just try out some points and you can graph it. And now in the next module I'm going to show you a little bit more about slope and y-intercept and you won't even have to do this. But this gives you good intuitive feel, I think, what a graph of a line is. I hope you have fun.