Video Transcript: Slope

Intro to slope 

As we start to graph lines, we might notice that they're differences between lines. For example, this pink or this magenta line here, it looks steeper than this blue line. And what we'll see is this  notion of steepness, how steep a line is, how quickly does it increase or how quickly does it  decrease, is a really useful idea in mathematics. So ideally, we'd be able to assign a number to  each of these lines or to any lines that describes how steep it is, how quickly does it increase or  decrease? So what's a reasonable way to do that? What's a reasonable way to assign a number  to these lines that describe their steepness? Well one way to think about it, could say well, how  much does a line increase in the vertical direction for a given increase in the horizontal  direction? So let's write this down. So let's say if we an increase increase, in vertical, in vertical,  for a given increase in horizontal for a given increase a given increase in horizontal. So, how can this give us a value? Well let's look at that magenta line again. Now let's just start at an  arbituary point in that magenta line. But I'll start at a point where it's going to be easy for me to figure out what point we're at. So if we were to start right here, and if I were to increase in the  horizontal direction by one. So I move one to the right. To get back on the line, how much do I  have to increase in the vertical direction? Well I have to increase in the vertical direction by two. By two. So at least for this magenta line, it looks like our increase in vertical is two, whenever  we have an increase in one in the horizontal direction. Let's see, does that still work if I were to  start here, instead of increasing the horizontal direction by one, if I were increase in the  horizontal direction... So let's increase by three. So now, I've gone plus three in the horizontal  direction, then to get back on the line, how much do I have to increase in the vertical direction?  I have to increase by one, two, three, four, five, six I have to increase by six. So plus six. So when I increase by three in the horizontal direction, I increase by six in the vertical. We were just  saying, hey, let's just measure how much to we increase in vertical for a given increase in the  horizontal? Well two over one is just two and that's the same thing as six over three. So no  matter where I start on this line, no matter where I start on this line, if I take and if I increase in  the horizontal direction by a given amount, I'm going to increase twice as much twice as much  in the vertical direction. Twice as much in the vertical direction. So this notion of this increase in vertical divided by increase in horizontal, this is what mathematicians use to describe the  steepness of lines. And this is called the slope. So this is called the slope of a line. And you're  probably familiar with the notion of the word slope being used for a ski slope, and that's  because a ski slope has a certain inclination. It could have a steep slope or a shallow slope. So  slope is a measure for how steep something is. And the convention is, is we measure the  increase in vertical for a given in increase in horizontal. So six two over one is equal to six over  three is equal to two, this is equal to the slope of this magenta line. So let me write this down.  So this slope right over here, the slope of that line, is going to be equal to two. And one way to  interpret that, for whatever amount you increase in the horizontal direction, you're going to  increase twice as much in the vertical direction. Now what about this blue line here? What  would be the slope of the blue line? Well, let me rewrite another way that you'll typically see  the definition of slope. And this is just the convention that mathematicians have defined for  slope but it's a valuable one. What is are is our change in vertical for a given change in  horizontal? And I'll introduce a new notation for you. So, change in vertical, and in this  coordinate, the vertical is our Y coordinate. divided by our change in horizontal. And X is our 

horizontal coordinate in this coordinate plane right over here. So wait, you said change in but  then you drew this triangle. Well this is the Greek letter delta. This is the Greek letter delta. And  it's a math symbol used to represent change in. So that's delta, delta. And it literally means,  change in Y, change in Y, divided by change in X, change in X. So if we want to find the slope of  the blue line, we just have to say, well how much does Y change for a given change in X? So, the slope of the blue line. So let's see, let me do it this way. Let's just start at some point here. And  let's say my X changes by two so my delta X is equal to positive two. What's my delta Y going to  be? What's going to be my change in Y? Well, if I go by the right by two, to get back on the line,  I'll have to increase my Y by two. So my change in Y is also going to be plus two. So the slope of  this blue line, the slope of the blue line, which is change in Y over change in X. We just saw that  when our change in X is positive two, our change in Y is also positive two. So our slope is two  divided by two, which is equal to one. Which tells us however much we increase in X, we're  going to increase the same amount in Y. We see that, we increase one in X, we increase one in Y. Increase one in X, increase one in Y. >From any point on the line, that's going to be true. You  increase three in X, you're going to increase three in Y. It's actually true the other way. If you  decrease one in X, you're going to decrease one in Y. If you decrease two in X, you're going to  decrease two in Y. And that makes sense from the math of it as well Because if you're change in  X is negative two, that's what we did right over here, our change is X is negative two, we went  two back, then your change in Y is going to be negative two as well. Your change in Y is going to  be negative two, and negative two divided by negative two, is positive one, which is your slope  again. 

Positive & negative slope 

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. 

Worked example: slope from graph 

Find the slope of the line in the graph. And just as a bit of a review, slope is just telling us how  steep a line is. And the best way to view it, slope is equal to change in y over change in x. And  for a line, this will always be constant. And sometimes you might see it written like this: you  might see this triangle, that's a capital delta, that means change in, change in y over change in  x. That's just a fancy way of saying change in y over change in x. So let's see what this change in y is for any change in x. So let's start at some point that seems pretty reasonable to read from  this table right here, from this graph. So let's see, we're starting here-- let me do it in a more  vibrant color-- so let's say we start at that point right there. And we want to go to another point that's pretty straightforward to read, so we can move to that point right there. We could  literally pick any two points on this line. I'm just picking ones that are nice integer coordinates,  so it's easy to read. So what is the change in y and what is the change in x? So first let's look at  the change in x. So if we go from there to there, what is the change in x? My change in x is equal to what? Well, I can just count it out. I went 1 steps, 2 steps, 3 steps. My change in x is 3. And  you could even see it from the x values. If I go from negative 3 to 0, I went up by 3. So my  change in x is 3. So let me write this, change in x, delta x is equal to 3. And what's my change in  y? Well, my change in y, I'm going from negative 3 up to negative 1, or you could just say 1, 2. So my change in y, is equal to positive 2. So let me write that down. Change in y is equal to 2. So  what is my change in y for a change in x? Well, when my change in x was 3, my change in y is 2.  So this is my slope. And one thing I want to do, I want to show you that I could have really  picked any two points here. Let's say I didn't pick-- let me clear this out-- let's say I didn't pick  those two points, let me pick some other points, and I'll even go in a different direction. I want  to show you that you're going to get the same answer. Let's say I've used this as my starting  point, and I want to go all the way over there. Well, let's think about the change in y first. So the

change in y, I'm going down by how many units? 1, 2, 3, 4 units, so my change in y, in this  example, is negative 4. I went from 1 to negative 3, that's negative 4. That's my change in y.  Change in y is equal to negative 4. Now what is my change in x? Well I'm going from this point,  or from this x value, all the way-- let me do that in a different color-- all the way back like this.  So I'm going to the left, so it's going to be a negative change in x, and I went 1, 2, 3, 4, 5, 6 units  back. So my change in x is equal to negative 6. And you can even see I started it at x is equal to  3, and I went all the way to x is equal to negative 3. That's a change of negative 6. I went 6 to  the left, or a change of negative 6. So what is my change in y over change in x? My change in y  over change in x is equal to negative 4 over negative 6. The negatives cancel out and what's 4  over 6? Well, that's just 2 over 3. So it's the same value, you just have to be consistent. If this is  my start point, I went down 4, and then I went back 6. Negative 4 over negative 6. If I viewed  this as my starting point, I could say that I went up 4, so it would be a change in y would be 4,  and then my change in x would be 6. And either way, once again, change in y over change in x is going to be 4 over 6, 2/3. So no matter which point you choose, as long as you kind of think  about it in a consistent way, you're going to get the same value for slope. 

Graphing a line given point and slope 

We are told graph a line with the slope of negative two, that contains the point four comma  negative three. And we have our little Khan Academy graphing widget right over here, where  we just have to find two points on that line, and then that will graph the line for us. So pause  this video and even if you don't have access to the widget right now, although it's all available  on Khan Academy, at least think about how you would approach this. And if you have paper  and pencil handy, I encourage you to try to graph this line on your own, before I work through it with this little widget. All right, now let's do it together. So we do know that it contains point  four comma negative three. So that's I guess you could say the easy part, we just have to find  the point x is four y is negative three. So it's from the origin four to the right, three down. But  then we have to figure out where could another point be? Because if we can figure out another  point, then we would have graphed the line. And the clue here is that they say a slope of  negative two. So one way to think about it is, we can start at the point that we know is on the  line, and a slope of negative two tells us that as x increases by one, y goes down by two. The  change in why would be negative two. And so this could be another point on that line. So I  could graph it like this is x goes up by one, as x goes from four to five, y will go, or y will change  by negative two. So why we'll go from negative three to negative five. So this will be done, we  have just graphed that line. Now another way that you could do it, because sometimes you  might not have space on the paper, or on the widget to be able to go to the right for x to  increase, is to go the other way. If you have a slope of negative two, another way to think about it is, if x goes down by one, if x goes down by one, then y goes up by two. 'Cause remember,  slope is change of y over change in x. So you could either say you have a positive change in y of  two when x has a negative one change, or you could think of it when x is a positive one change,  y has a negative two change. But either way notice, you got the same line. Notice this line is  the same thing, as if we did the first way is we had x going up by one and y going down by two,  it's the exact same line. 

Calculating Slope from tables

We are asked, what is the slope of the line that contains these points? So pause this video and  see if you can work through this on your own before we do it together. Alright, now let's do it  together, and let's just remind ourselves what slope is. Slope is equal to change in y, this is the  

Greek letter delta, look likes a triangle, but it's shorthand for change in y over change in x.  Sometimes you would see it written as y2 minus y1 over x2 minus x1 where you could kind of  view x1 y1 as the starting point and x2 y2 as the ending point. So let's just pick two xy pairs  here, and we can actually pick any two if we can assume that this is actually describing a line.  So we might as well just pick the first two. So let's say that's our starting point and that's our  finishing point. So what is our change in x here? So we're going from two to three, so our  change in x is equal to three minus two which is equal to one, and you can see that to go from  two to three you're just adding one. And what's our change in y? Our change in y is our finishing y one minus our starting y four, which is equal to negative three. And you could of, you didn't  even have to do this math, you would have been able to see to go from two to three you added  one, and to go from four to one, you have to subtract three. For there we have all the  information we need. What is change in y over change in x? Well, it's going to be, our change in  y is negative three and our change in x is one. So our slope is negative three divided by one is  negative three. Let's do another example. Here we are asked, what is the slope of the line that  contains these points? So pause this video and see if you can figure it out or pause the video  again and see if you can figure it out. Alright, so remember, slope is equal to change in y over  change in x. And we should be able to pick any two of these pairs in order to figure that out if  we assume that this is indeed a line. Well, just for variety, let's pick these middle two pairs. So  what's our change in x? To go from one to five, we added four. And what's our change in y? To  go from seven to 13, we added six. So our change in y is six when our change in x is four. And I  got the signs right, in both case it's a positive. When x increases, y increased as well. So our  slope is six fourths, and we could rewrite that if we like. Both six and four are divisible by two,  so let be divide both the numerator and the denominator by two and we get three halves, and  we're done. 

Calculating slope from tables 

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.