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Lesson 3: Slope

# Intro to slope

Slope tells us how steep a line is. It's like measuring how quickly a hill goes up or down. We find the slope by seeing how much we go up or down (vertical change) for each step to the right (horizontal change). If a line goes up 2 steps for every 1 step to the right, its slope is 2.

## Want to join the conversation?

• Without doing any calculations how do you know that one slope has a higher slope than another line?
• You can know if one slope has a higher slope without calculations because the higher the slope the steeper the line. Great question!
• I don't get slope at all; can somebody explain it to me?
• it's really simple, dude.
basically it's just the rise over the run, which means its the amount that goes up, divided by the amount going sideways. think of it this way. when your rising, your going up, so your going up on your graph, but when your running, your going sideways (usually) meaning across your graph. if the line is steeper, you will get a larger slope. if it is not as steep your slope will be smaller
• How do you find slope of a straight horizontal line? my brother said it would be one, but im not sure... Thanks in advance!
• 0 always
• Why can't it be 'increase in horizontal/ increase in vertical' or 'run over rise'?
• It kinda makes no sense as we are measuring the amount of steepness i.e higher the number --> steeper the slope. In your question it is the opposite
• So slope-intercept form is y=mx+b and Standard Form is Ax+Bx=C. What I don't get is how to create the Standard Form right from a graph, can someone explain to me how it's done?
• how do I caculate a slope
• Without doing any calculations how do you know that one slope has a higher slope than another line?
• You just imagine both of them in on place
or look which one has greater angle from same x line
• why it is change in y / change in x, not the other way?
• I think its the convection we have chosen for it
• what if the line is straight??
• If a line is straight horizontally then the slope would be 0 but if the line is straight vertically the slope would be undefined
(1 vote)
• Slope is also y2 - y1 / x2 - x1, right?
• Yes, that is the slope formula, though it would be better to put these in parentheses and add the m to get m=(y2-y1)/(x2-x1). On a graph, you can count rise over run, but you are still counting the difference between y values (change in y) divided by difference between x values (change in x).

## Video transcript

- [Voiceover] 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.