So we want to talk about linear functions a little bit more. We already talked about graphing them. And some of this will be mostly review for some of you guys. So it just depends on your background, and how much you remember, and probably how recently you took a math class. So this is about linear functions, linear functions. We've kind of done this before, right? They look like f of x is equal to mx plus b, or can be written like that, or possibly y is equal to mx plus b. You might be used to that form, before. And we've talked about that form. And we'll talk about it again. Associated with linear functions is this quantity m, which is the idea of the slope of the line. Now slope, for those of you who remember, is the change in y divided by the change in x, which sometimes we write as a Delta y divided by Delta x. And some of you might remember the phrase rise over run. That's exactly what we're talking about here. So in this case, we measure change of y, and in two points, and let me just identified those. So I have two points, x, y, points. We name them x1, y1, and x2, y2. So those are their coordinates. And then the slope between those two points is the change in y, y2 minus y1, divided by x2 minus x1. Slope is the defining characteristic of linear functions. So linear functions, for linear functions, slope, or this quantity m, doesn't change, no matter which two points you pick. And maybe you never thought about that before, but that is not true for quadratic functions. It's not true, for example, for square root functions, the slope will change, if you go to different places in a square root function. But for a linear function, no matter where you are in the entire real line, the slope, this quantity m, the change in y over the change in x doesn't change. It stays constant. So let's do some pretty typical examples about slope. Find the slope of the line passing through the points 3, 5, and negative 7, 2. And for those of you who remember this, this is the kind of idea, you label these-- one of these is x1, y1, and one is x2, y2. And then you go ahead and plug in, to this idea, this slope. So I have 2, y2 is 2, and y1 is 5, 2 minus 5, over negative 7 minus 3. That's a negative 3 over negative 10, or a slope of positive 3/10. This means that every time I go-- when it's positive here, it means I go up 3 and to the right by 10. That's the idea of the slope. And we have talked before about specific, or special, types of lines. Let me do those again for you, just to make sure. We have horizontal lines, we talked about these in the graphing section. They are of the form y is equal to b. And horizontal lines have the same y value all across. So when you're talking about computing this quantity, these two y values will be the same number. So you'll get a zero over something, right? If your two x's aren't the same, which give you a zero slope. So these all have-- horizontal lines have m is 0. The other special type of line that we talked about before, that kind of has a little bit different look to it, are vertical lines. These are not functions. Let me make that note. If you just picture a vertical line, it doesn't pass the vertical line test. But they have equations that look like x is equal to some number c. And in that case, when you think about the slope, you get something up here, the y values might be different, if you're picturing what I'm picturing in my head. And the x values are the same, you'll get division by 0. So this means that the slope is undefined for vertical lines. Otherwise, I want to just make sure that we have a pretty, kind of intuitive understanding. If I have something-- a graph that looks like this, positive slope is moving up and to the right, something like that. And it could be less slanted, but that's kind of positive slope. And negative slope is moving down and to the right, maybe something like this. We use several different forms-- let me just make this note on here-- this is positive slope, and this is negative slope. We use several different forms of lines. And you might remember these. We use the slope intercept form, that we have seen in the graphing section, y equals mx plus b. We also use the point slope form, y minus y1 is equal to m times x minus x1. That's the point slope form. In this case m is still the slope here. And this x1, y1 is a point on the line. We also have the general form of a line, which looks something like Ax plus By equals C. You see this form more often when you're solving systems of equations, so you might remember seeing something like that. So let's do an example of finding the equation of a line. Find the equation of the line that passes through 2, 5, and 1, negative 2. So usually if we start with points, one of the more natural, depending on what they are, one of the natural equations start with is the point slope form. In order to use that form I need-- so I think I want to use the point slope form to find this equation. Because I already have a point, point, that I actually have two points, there's another one, and I just have to find the slope. And I can do that. I know how to do that. So let's find the slope, y2 minus y1, change in y, x2 minus x1, change in x. And you can label these whatever you want. I'll label this one as x1, y1, and x2, y2. It does not matter, you'll get the same answer either way. Minus y1, 5, over 1 minus 2, so I get negative 7 over, I think, negative 1. You guys get 7? So I'm going to go ahead and-- now I have a point in my slope, so I use the point slope form. So y minus y1, is m times x minus x1. y minus-- and you can use again-- either one of those points, but I will probably use this first one, minus 2. This is the equation of the line in-- this is point slope form. And you can graph the equation from this form, plot your point and then use the slope to get a few more points. Now you can actually-- you can go from this equation or from this form of the line to some of the other forms. Let's go to, let's say, slope-intercept form. In slope-intercept form I just need to solve for y. So usually I go from here, get y minus 5, and I will multiply the 7 out, and then add 5 to both sides, 7x minus 9. And there's my slope-intercept form. To get to general form, I just move both x and y pieces over to the left. So in this case, I'll move the 7x and subtract 7x from both sides. And there's my equation in general form. So those are kind of the three different forms of equations of the line, equations of linear functions that we'll use here. I think that's about all that we're going to do online. So, yeah, because we already did graphing. So just a quick review of linear functions. So let me know if you have any questions.