 In this video, we're gonna talk about slope. Slope is the rate of change for a line, a linear graph, we could say it. It's measuring how steep a line is. So imagine we have a linear graph that looks something like the following. Then when one talks about slope, we're interested in the following basic idea. We have, first of all, the quantity to which we call the rise. How far up does the graph go? So it raises, we call that the rise, and then we measure the horizontal displacement, which is commonly referred to as the run. The idea here is unless you're Spider-Man, you run horizontally. So running is a horizontal change, rising is a vertical change. And so if there were two points on this line, call it X1, Y1 for the first point, and then for the second one, call it X2, Y2, the rise is gonna be the difference of the Y coordinates. As you go from the lower Y coordinate to the higher one, we could find that rise by taking the difference, Y2 minus Y1. We're gonna abbreviate this as delta Y. The triangle there is the Greek letter delta, delta is for difference here. Run is gonna be similar. If you wanna go from this point to this point, the distance there is the difference of the X coordinates, in which case we're gonna get X2 minus X1, which is what we call delta X for short. So again, a difference of the X is there. So for a line, the slope is constant. The rate in which it raises is the same. It doesn't matter whether we take like a small little section or a big section, the fact it's a line, so that the proportions between rise and run will always be the same. And so if you take any two points on a line, Y1, X1, Y1, X2, Y2, it doesn't matter. The slope can always be computed by the following formula, M, we typically use the letter M to write slope of a generic line. M equals Y2 minus Y1 over X2 minus X1. And in short, slope here is just rise over run. Or in other words, it's this delta Y over delta X. That's how we calculate the slope of any line. So as a very simple example of that, let's take a line which passes through the points negative two, negative three and negative five comma one. So to find the slope of the line, we're gonna take M equals M2 minus M, sorry, Y2 minus Y1 over X2 minus X1, just the formula we have from the previous slide. Now we have to make a decision. Who's gonna be the first point? Who's gonna be the second point? And it turns out it doesn't matter who the first point is or who the second point is. You could specify any one. Only thing that matters, you have to be consistent. If you're gonna choose negative two to be X1, then negative three has to be Y1. On the other hand, if you want negative five to be X1, then one has to be Y1. It doesn't matter as long as we're consistent in that regard. So for the sake of argument, I'm just gonna put, I should say for the sake of example, we'll take the first point to be the first point just as they're listed, right? And so we're gonna take one minus a negative three, the difference of the Ys, the rise. Then we're gonna take negative five minus negative two, the difference of the Xs, the rise over one. And so you get one minus a negative three, it's a double negative, so you get one plus three. Same thing in the bottom, you get negative five plus two in that situation. One plus three is four. Negative five plus two is a negative three. And so we can write this as negative four over three. This is the slope. And so I usually like to think of the rise as being positive or negative, and I always keep the run to be positive. It doesn't really matter though. You just don't wanna put a negative in both situations. Cause this fraction, you could write as four over negative three like this one. You could talk about negative four over three or you could say negative four thirds. All of those are the same rational number. Just make sure you don't do a double negative because negative four over negative three is actually four thirds, which is a different quantity. Let's take a more geometric approach to something like this. Let's say that the line is given by the illustration you see right here. And we can very quickly see that the run of the function would be right here. The rise would be right here for a line. If we wanna find the slope of this thing, M equals delta Y over delta X. The calculation's exact same. It just comes down to identifying points on the graph. If at all possible, I like to choose integer points on the graph. That is points whose X and Y corner are both integers. You can often see that when you look at the lattice. So you look at this first point right here. You're gonna get one, two, three for the X coordinate. And you're gonna get one, two, three, four for the Y coordinate. So three comma four is a point on the graph. The other one you get, you can get something like this, one, two, three, four. So you get the X coordinate of four and then you get the Y coordinate of one, two, like so. And that's one way. Then we could just calculate the formula like we did before, four minus two over three minus four. You get the slope from there. But when you have the picture in front of you, it turns out these coordinates are not necessary whatsoever. All we have to do is just kind of measure just the rise and run specifically, right? So we go one, two, go down. Since we went down, I would say that's negative two. So we get negative two over and then our run is just one. So negative two over one, which typically would be abbreviated just as negative two is the slope of the slope. We don't actually need the coordinates, although you can use the coordinates. We just wanna pick two definite points on the line and then you count off the rise, you count off the run. The ratio would give us the slope of the line.