 This video is going to talk about slope and the introduction to slope. So we have this situation here where we have some x values. We have an equation that is y is 2x minus 2. And we want to figure out what's going on here with all these values. So I've already done the first two. When x is negative 1, we have negative 1 times 2 is negative 2 minus 2. We'll give us negative 4. And that means that y is equal to negative 4. And then if I let it put 0 in here, 2 times 0 is 0 minus 2 gives me negative 2. And so let's do the other two. So 2 times 1 minus 2, that tells me that y is going to be equal to 2 minus 2 would be 0. And if I have y is equal to 2 times 2 minus 2, that's really 4 minus 2. So y is going to be equal to 2. Now I have a couple of the columns here. Change in x. Well the first one, there was no change. But in the second one, I started with 0 for the second one. And I wanted to know how I had changed from negative 1. So I've subtracted the previous number. So here at 1, when x is 1, I take 1 and I subtract the previous number. And 1 minus 1, or 1 minus 0 is 1. And I'm going to take 2 minus the previous x, which is 1, and 2 minus 1 is 1. So you can see that my change in x is always 1. Now I come over here and I look at my change in y. Again, I take the newest value minus the one before it. So negative 2 minus the negative 4 would be plus 4 or positive 2. Now I'm going to take 0, which is my y value here, and subtract from that the negative 2, which gives me 2. And then for this last one, I start with the 2 that the y value for this line is. And I subtract the previous y, which was 0. And again, I get 2. So if I do the change in y, it was 2 over the change in x, which was 1. You can see every time the change in y is 2 and the change in x is 1. That's called our slope. And we often call slope M. It'll make more sense later why we call it M. But I'll just clue you in right now. Well, there's a formula for that. It's called y2 minus y1. That's exactly what we did here. This was my y2 and this was my y1. Negative 2 was my y2 and negative 4 was my y1, okay? So that's the first part of my formula down here. And then I have x2 minus x1. And that's exactly what I did here. X2 was 0 and I subtracted from that the first x that I used, which was negative 1. So y2, which is the change in y, we're subtracting to find the change in y over the change in x. Now, a couple of things you need to remember. You have to start with the same point. It's the y's over the x's. We usually do x first, but in this formula, we're going to put the y's first. And if I start with the second point, then I have to start with the second point in my x's, just like I did with my y's. I can't mix those up. They have to start with the same point. All right, let's practice. So here I have a bunch of points and I want to know what the slope of the line is. Well, I need to do. I'm going to let this one be x1 and I'm going to let this one be x2. We'll take the first two points. And I'm going to let this one, since it corresponds with x1, it'll be y1. And this one will be y2. So again, it's y2 minus y1 over x2 minus x1. Y2, if you look over there, it's 7 and y1 is 11. And x2 is the bottom one of my two points, so it's negative 1. Notice that it's negative 1, 7, negative 1, 7. And then minus my yx2, minus my x1, which is negative 2. And that gives me 7 minus 11 is going to be 4. And negative 1 minus and negative 2 or plus 2 is going to be a positive 1. So my slope is 4. Let's try two other points just to try. I'm going to keep my x1 and my y1, so I want to keep these two. But this time, let's try the last one. Let's let this be x2 and let this be y2. We're going to do the black ones. So y2 is this bottom number down here, negative 1 minus my y1, which is the top y, which is 11. And yx2 is going to be 1 minus my x1, which is at the top. So if we look at that, negative 1 minus 11 is going to be negative 12. And 1 minus a negative 2 or plus 2 would be over 3 and we get negative 4. And I had one mistake up here that we need to correct because 7 minus 11 isn't 4, it's negative 4. So negative 4 over 1 is going to be negative 4. And now we do definitely get the same thing no matter what two points you pick. You should always get the same slope if they're on the same line. So now we know how to find slope with a formula. What about if I just have a graph? Well, I could pick two points and use the slope formula, but I could also do it graphically. So let's pick a couple nice points where it goes through the graph nicely. Like this point right here is very nice and so is this one right here. Both of those go through some nice grid marks. So we're going to learn that slope is also called rise because that's the y direction over run because that's the, you run horizontally which is the x direction. So I'm going to look at this graph and I'm going to say, okay, well I went 1, 2 to get across from that other point. So my rise was 2 and then I went and it was up to get to the next point from left to right. And then I went over one to the right which is a positive one. So that means that my slope would be 2. What if I had a line that looked like this one went through here and here and here and here and here. Okay. So here's a nice point right here and here's a nice point right here. Let's try and see if we can find the slope. Well remember we're going to go from left to right. So when I go from this point down to the one that's to the right of it I actually have to go down to and remember in the extra in the y direction down is negative. So I go down to but I go to the right and that's a positive 2. So rise over run in this case is going to be negative 2 over positive 2 and the slope of that blue line should be negative 1. It's going down that's why it's a negative slope. Our black line here was going up and we had a positive slope.