Positive & negative slope

- [Voiceover] Slope is defined as your change in the vertical direction, and I could use the Greek letter delta, this little triangle here is the Greek letter delta, it means change in. Change in the vertical direction divided by change in the horizontal direction. That is the standard definition of slope and it's a reasonable way for measuring how steep something is. So for example, if we're looking at the xy plane here, our change in the vertical direction is gonna be a change in the y variable divided by change in horizontal direction, is gonna be a change in the x variable. So let's see why that is a good definition for slope. Well I could draw something with a slope of one. A slope of one might look something like... so a slope of one, as x increases by one, y increases by one, so a slope of one... is going to look like this. Notice, however much my change in x is, so for example here, my change in x is positive two, I'm gonna have the same change in y. My change in y is going to be plus two. So my change in y divided by change in x is two divided by two is one. So for this line I have slope is equal to one. But what would a slope of two look like? Well, a slope of two should be steeper and we can draw that. Let me start at a different point, so if I start over here a slope of two would look like... for every one that I increase in the x direction I'm gonna increase two in the y direction, so it's going to look like... that. This line right over here, you see it. If my change in x is equal to one, my change in y is two. So change in y over change in x is gonna be two over one, the slope here is two. And now, hopefully, you're appreciating why this definition of slope is a good one. The higher the slope, the steeper it is, the faster it increases, the faster we increase in the vertical direction as we increase in the horizontal direction. Now what would a negative slope be? So let's just think about what a line with a negative slope would mean. A negative slope would mean, well we could take an example. If we have our change in y over change in x was equal to a negative one. That means that if we have a change in x of one, then in order to get negative one here, that means that our change in y would have to be equal to negative one. So a line with a negative one slope would look like... would look like this. Notice, as x increases by a certain amount, so our delta x here is one, y decreases by that same amount instead of increasing. So now this is what we consider a downward sloping line. So change in y is equal to negative one. So our change in y over our change in x is equal to negative one over one which is equal to negative one. So the slope of this line is negative one. Now if you had a slope with negative two, it would decrease even faster. So a line with a slope of negative two could look something like this. So as x increases by one, y would decrease by two. So it would look something like... it would look like that. Notice, as our x increases by a certain amount, our y decreases by twice as much. So this right over here has a slope of negative two. So hopefully this gives you a little bit more intuition for what slope represents and how the number that we use to represent slope, how you can use that to visualize how steep a line is. A very high positive slope, as x increases, y is going to increase fairly dramatically. If you have a negative slope... as x increases, your y is actually going to decrease. And then the higher the slope, the steeper, the more you increase as x increases, and the more negative the slope, the more you decrease as x increases.