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Worked example: slope from two points

CCSS Math: HSF.IF.C.7a
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Find the slope of the line that goes through the ordered pairs 4 comma 2 and negative 3 comma 16. So just as a reminder, slope is defined as rise over run. Or, you could view that rise is just change in y and run is just change in x. The triangles here, that's the delta symbol. It literally means "change in." Or another way, and you might see this formula, and it tends to be really complicated. But just remember it's just these two things over here. Sometimes, slope will be specified with the variable m. And they'll say that m is the same thing-- and this is really the same thing as change in y. They'll write y2 minus y1 over x2 minus x1. And this notation tends to be kind of complicated, but all this means is, is you take the y-value of your endpoint and subtract from it the y-value of your starting point. That will essentially give you your change in y. And it says take the x-value of your endpoint and subtract from that the x-value of your starting point. And that'll give you change in x. So whatever of these work for you, let's actually figure out the slope of the line that goes through these two points. So we're starting at-- and actually, we could do it both ways. We could start at this point and go to that point and calculate the slope or we could start at this point and go to that point and calculate the slope. So let's do it both ways. So let's say that our starting point is the point 4 comma 2. And let's say that our endpoint is negative 3 comma 16. So what is the change in x over here? What is the change in x in this scenario? So we're going from 4 to negative 3. If something goes from 4 to negative 3, what was it's change? You have to go down 4 to get to 0, and then you have to go down another 3 to get to negative 3. So our change in x here is negative 7. Actually, let me write it this way. Our change in x is equal to negative 3 minus 4, which is equal to negative 7. If I'm going from 4 to negative 3, I went down by 7. Our change in x is negative 7. Let's do the same thing for the change in y. And notice, I implicitly use this formula over here. Our change in x was this value, our endpoint, our end x-value minus our starting x-value. Let's do the same thing for our change in y. Our change in y. If we're starting at 2 and we go to 16, that means we moved up 14. Or another way you could say it, you could take your ending y-value and subtract from that your starting y-value and you get 14. So what is the slope over here? Well, the slope is just change in y over change in x. So the slope over here is change in y over change in x, which is-- our change in y is 14. And our change in x is negative 7. And then if we want to simplify this, 14 divided by negative 7 is negative 2. Now, what I want to show you is, is that we could have done it the other way around. We could have made this the starting point and this the endpoint. And what we would have gotten is the negative values of each of these, but then they would've canceled out and we would still get negative 2. Let's try it out. So let's say that our start point was negative 3 comma 16. And let's say that our endpoint is the 4 comma 2. 4 comma 2. So in this situation, what is our change in x? Our change in x. If I start at negative 3 and I go to 4, that means I went up 7. Or if you want to just calculate that, you would do 4 minus negative 3. 4 minus negative 3. But needless to say, we just went up 7. And what is our change in y? Our change in y over here, or we could say our rise. If we start at 16 and we end at 2, that means we went down 14. Or you could just say 2 minus 16 is negative 14. We went down by 14. This was our run. So if you say rise over run, which is the same thing as change in y over change in x, our rise is negative 14 and our run here is 7. So notice, these are just the negatives of these values from when we swapped them. So once again, this is equal to negative 2. And let's just visualize this. Let me do a quick graph here just to show you what a downward slope would look like. So let me draw our two points. So this is my x-axis. That is my y-axis. So this point over here, 4 comma 2. So let me graph it. So we're going to go all the way up to 16. So let me save some space here. So we have 1, 2, 3, 4. It's 4 comma-- 1, 2. So 4 comma 2 is right over here. 4 comma 2. Then we have the point negative 3 comma 16. So let me draw that over here. So we have negative 1, 2, 3. And we have to go up 16. So this is 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16. So it goes right over here. So this is negative 3 comma 16. Negative 3 comma 16. So the line that goes between them is going to look something like this. Try my best to draw a relatively straight line. That line will keep going. So the line will keep going. So that's my best attempt. And now notice, it's downward sloping. As you increase an x-value, the line goes down. It's going from the top left to the bottom right. As x gets bigger, y gets smaller. That's what a downward-sloping line looks like. And just to visualize our change in x's and our change in y's that we dealt with here, when we started at 4 and we ended at-- or when we started at 4 comma 2 and ended at negative 3 comma 16, that was analogous to starting here and ending over there. And we said our change in x was negative 7. We had to move back. Our run we had to move in the left direction by 7. That's why it was a negative 7. And then we had to move in the y-direction. We had to move in the y-direction positive 14. So that's why our rise was positive. So it's 14 over negative 7, or negative 2. When we did it the other way, we started at this point. We started at this point, and then ended at this point. Started at negative 3, 16 and ended at that point. So in that situation, our run was positive 7. And now we have to go down in the y-direction since we switched the starting and the endpoint. And now we had to go down negative 14. Our run is now positive 7 and our rise is now negative 14. Either way, we got the same slope.