# Slope and y-intercept intuition

## Video transcript

On the Khan Academy web app, which I need to work on a little bit more to make it a little bit faster, they have this one module that's called the graph of the line. It has no directions on it, and I thought I would make a little video here, at least to explain how to do this module, and in the process, I think it'll help people, even those of you who aren't using the module, understand what the slope and the y-intercept of a line is a little bit better. So this is a screen shot of that module right here, and the idea is essentially to change this line, and this line right here in orange is the line specified by this equation right here, so right now it's the equation of the line 1x plus 1. It has a slope of 1, you can see that, for every amount it moved to the right it moves up exactly 1, and has 1 for its y-intercept. It intersects the y-axis at exactly the point 0,1. Now, the goal of this exercise is to change your slope and your y-intercept so that you go through these two points, and this point's-- half of it's off the screen, hopefully you can see them if you're watching these in HD-- you can see these two points. Our goal is to make this line go through them by essentially changing its equation. So it's a kind of a tactile way of-- you know, as tactile as something on the computer can get-- of trying to figure out the equation of the line that goes through these two points. So how can we do that? So you can see here, when I change the slope, if I make the slope higher, it becomes more steep. Now the slope is 3. For every 1 I move to the right, I have to go 3 up. My change in y is 3 for every change in x of 1. Or that's my slope. My y-intercept is still 1. If I change my y-intercept, if I make it go down, notice it just shifts the line down. It doesn't change its inclination or its slope, it just shifts it down along this line right there. So how do I make my line go through those two points? Well it looks like, if I shift it up enough-- let's shift up that point-- and then let's say let's lower the slope. This looks like it has a negative slope. So if I lower my slope, notice I'm flattening out the line. That's a slope of 0. It looks like it has to be even more negative than that. Let's see, maybe even more negative than that, right? It has to look like a line that goes bam, just down like that. Even more-- that looks close. Let me get my y-intercept down to see if I can get closer to that. It still seems like my slope is a little bit too high. That looks better. So let me get my y-intercept down even further. It's now intersecting way here, off the screen. You can't even see that. I just realized this is copyright 2008 Khan Academy, it's now 2009. It's almost near the end of 2009. I could just change that. Maybe I'll just write 2010 there. OK. So y-intercept. Even more. So I lowered the y-intercept but our slope is still not strong enough. The y-intercept is actually off the chart. It's intersecting at minus 18. That's our current y-intercept. But the slope of minus 5 is still not enough, so let me lower the slope. So if I lower the slope, let's see, if I lower the y-intercept a little bit more, is that getting me? There you go. It got me to those points. So the equation of the line that passes through both of those things is minus 6x minus 22. Let's do another one. So, once again, it resets it, so I just say the equation 1x plus 1, but it gives me these two new points that I have to make it go through. And once again this is going to be a negative slope, because for every x that I move forward positive, my y is actually going down. So I'm going to have a negative slope here, so let me lower the slope a little bit. It's actually doing fractions, so this thing jumps around a little bit. I should probably change that a little bit. That looks about right, so let me shift the graph down a little bit by lowering its y-intercept. By lowering its y-intercept, can I hit those two points? There you go. This is the equation of that line that goes to the points minus 5,1 and the points 9,minus 9. You have a slope of minus 5/7. For every 7 you go to the right, you go down 5. If you go 1, 2, 3, 4, 5, 6, 7, you're going to go down 1, 2, 3, 4, 5. And that, we definitely see that on that line. And then the y-intercept is minus 18 over 7, which is a little over 2, it's about a little over-- it's what, a little over 2 and 1/2. And we see right there that the y-intercept is a little over 2 and 1/2. That's the equation for our line. Let's do another one. This is a fun module, because there are no wrong answers. You can just keep messing with it until you eventually get that line to go through both of those points, but the idea is really give you that intuition that the slope is just what the inclination of the line is, and then the y-intercept is how far up and down it gets shifted. So this is going to be a positive slope, but not as high as 1. It looks like, for every 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, for every 12 we go to the right, we're going to go 1, 2, 3 up. So our slope is going to be 3 over 12, which is also 1 over 4, and we can just look at that visually. Let's lower our slope. That's 3/4, not low enough. 1/2, not low enough. 1/4, which I just figured out it is, that looks right, and then we have to lower the y-intercept. We're shifting it down, and there we go. So the equation of this line, its slope is 1/4, so the equation of the line is 1/4x plus 1/4. So hopefully, for those of you trying to do this module, that, 1, explained how to do it, and for those of you who don't even know what this module is, it hopefully gives you a little intuition about what the slope and the y-intercept do to an actual line.