Slope of a line: negative slope
Find the slope of the line pictured on the graph. So the slope of a line is defined to be rise over run. Or you could also view it as change in y over change in x. And let me show you what that means. So let's start at some arbitrary point on this line, and they highlight some of these points. So let's start at one of these points right over here. So if we wanted to start one of these points-- and let's say we want to change our x in the positive direction. So we want to go to the right. So let's say we want to go from this point to this point over here. How much do we have to move in x? So if we want to move in x, we have to go from this point to this point. We're going from negative 3 to 0. So our change in x-- and this triangle, that's delta. That means "change in." Our change in x is equal to 3. So what was our change in y when our change in x is equal to 3? Well, when we moved from this point to this point, our x-value changed by 3, but what happened to our y-value? Well, our y-value went down. It went from positive 3 to positive 2. Our y-value went down by 1. So our change in y is equal to negative 1. So we rose negative 1. We actually went down. So our rise is negative 1 when our run-- when our change in x-- is 3. So change in y over change in x is negative 1 over 3, or we could say that our slope is negative 1/3. Let me scroll over a little bit. It is negative 1/3. And I want to show you that we can do this with any two points on the line. We could even go further than 3 in the x-direction. So let's go the other way. Let's start at this point right over here and then move backwards to this point over here, just to show you that we'll still get the same result. So to go from this point to that point, what is our change in x? So our change in x is this right over here. Our change in x is that distance right over there. We started at 3, and we went to negative 3. We went back 6. Over here, our change in x is equal to negative 6. We're starting at this point now. So over here our change in x is negative 6. And then when our change in x is negative 6, when we start at this point and we move 6 back, what is our change of y to get to that point? Well, our y-value went from 1. That was our y-value at this point. And then when we go back to this point, our y-value is 3. So what did we do? We moved up by 2. Our change in y is equal to 2. Slope is change in y over change in x, or rise over run. Change in y is just rise. Change in x is just run, how much you're moving in the horizontal direction. So rise over run in this example right over here is going to be 2 over negative 6, which is the same thing as negative 1/3. And you could verify it for yourself. Take any of these two points, start at one of these two points, and figure out what is the run to get to the next point, and then what is the rise to get the next point. And for any line, the slope won't change. Let me do it again. Over here, we had to move in the positive 3 direction, so that is our run. So this right here is positive 3. That's our run. But what's our rise? Well, we actually went down, so we have a negative rise. Our rise is negative 1. So we have negative 1 as our rise. We went down. And our run was positive 3. So our slope here is negative 1/3.