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On the Khan Academy web app,[br]which I need to work on a

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little bit more to make it a[br]little bit faster, they have

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this one module that's called[br]the graph of the line.

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It has no directions on it, and[br]I thought I would make a little

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video here, at least to explain[br]how to do this module, and in

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the process, I think it'll help[br]people, even those of you who

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aren't using the module,[br]understand what the slope and

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the y-intercept of a line[br]is a little bit better.

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So this is a screen shot of[br]that module right here, and the

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idea is essentially to change[br]this line, and this line right

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here in orange is the line[br]specified by this equation

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right here, so right now it's[br]the equation of the

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line 1x plus 1.

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It has a slope of 1, you can[br]see that, for every amount it

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moved to the right it moves up[br]exactly 1, and has 1

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for its y-intercept.

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It intersects the y-axis[br]at exactly the point 0,1.

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Now, the goal of this exercise[br]is to change your slope and

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your y-intercept so that you go[br]through these two points, and

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this point's-- half of it's off[br]the screen, hopefully you can

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see them if you're watching[br]these in HD-- you can

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see these two points.

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Our goal is to make this line[br]go through them by essentially

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changing its equation.

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So it's a kind of a tactile way[br]of-- you know, as tactile as

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something on the computer can[br]get-- of trying to figure out

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the equation of the line that[br]goes through these two points.

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So how can we do that?

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So you can see here, when I[br]change the slope, if I make

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the slope higher, it[br]becomes more steep.

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Now the slope is 3.

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For every 1 I move to the[br]right, I have to go 3 up.

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My change in y is 3 for[br]every change in x of 1.

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Or that's my slope.

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My y-intercept is still 1.

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If I change my y-intercept, if[br]I make it go down, notice it

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just shifts the line down.

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It doesn't change its[br]inclination or its slope, it

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just shifts it down along[br]this line right there.

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So how do I make my line go[br]through those two points?

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Well it looks like, if I shift[br]it up enough-- let's shift up

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that point-- and then let's[br]say let's lower the slope.

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This looks like it has[br]a negative slope.

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So if I lower my slope, notice[br]I'm flattening out the line.

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That's a slope of 0.

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It looks like it has to be[br]even more negative than that.

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Let's see, maybe even more[br]negative than that, right?

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It has to look like a line that[br]goes bam, just down like that.

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Even more-- that looks close.

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Let me get my y-intercept[br]down to see if I can

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get closer to that.

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It still seems like my slope[br]is a little bit too high.

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That looks better.

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So let me get my y-intercept[br]down even further.

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It's now intersecting way[br]here, off the screen.

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You can't even see that.

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I just realized this is[br]copyright 2008 Khan

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Academy, it's now 2009.

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It's almost near[br]the end of 2009.

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I could just change that.

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Maybe I'll just[br]write 2010 there.

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OK.

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So y-intercept.

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Even more.

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So I lowered the y-intercept[br]but our slope is still

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not strong enough.

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The y-intercept is[br]actually off the chart.

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It's intersecting at minus 18.

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That's our current y-intercept.

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But the slope of minus 5[br]is still not enough, so

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let me lower the slope.

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So if I lower the slope, let's[br]see, if I lower the y-intercept

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a little bit more,[br]is that getting me?

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There you go.

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It got me to those points.

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So the equation of the line[br]that passes through both

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of those things is[br]minus 6x minus 22.

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Let's do another one.

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So, once again, it resets it,[br]so I just say the equation 1x

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plus 1, but it gives me these[br]two new points that I have

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to make it go through.

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And once again this is going to[br]be a negative slope, because

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for every x that I move[br]forward positive, my y

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is actually going down.

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So I'm going to have a negative[br]slope here, so let me lower

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the slope a little bit.

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It's actually doing fractions,[br]so this thing jumps

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around a little bit.

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I should probably change[br]that a little bit.

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That looks about right, so let[br]me shift the graph down a

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little bit by lowering[br]its y-intercept.

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By lowering its y-intercept,[br]can I hit those two points?

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There you go.

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This is the equation of that[br]line that goes to the points

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minus 5,1 and the[br]points 9,minus 9.

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You have a slope of minus 5/7.

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For every 7 you go to the[br]right, you go down 5.

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If you go 1, 2, 3, 4, 5,[br]6, 7, you're going to

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go down 1, 2, 3, 4, 5.

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And that, we definitely[br]see that on that line.

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And then the y-intercept is[br]minus 18 over 7, which is a

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little over 2, it's about a[br]little over-- it's what,

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a little over 2 and 1/2.

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And we see right there[br]that the y-intercept is

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a little over 2 and 1/2.

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That's the equation[br]for our line.

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Let's do another one.

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This is a fun module, because[br]there are no wrong answers.

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You can just keep messing with[br]it until you eventually get

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that line to go through both of[br]those points, but the idea is

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really give you that intuition[br]that the slope is just what the

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inclination of the line is, and[br]then the y-intercept is how far

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up and down it gets shifted.

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So this is going to be[br]a positive slope, but

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not as high as 1.

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It looks like, for every 1, 2,[br]3, 4, 5, 6, 7, 8, 9, 10, 11,

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12, for every 12 we go to the[br]right, we're going

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to go 1, 2, 3 up.

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So our slope is going to be 3[br]over 12, which is also 1 over

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4, and we can just look[br]at that visually.

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Let's lower our slope.

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That's 3/4, not low enough.

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1/2, not low enough.

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1/4, which I just figured out[br]it is, that looks right, and

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then we have to lower[br]the y-intercept.

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We're shifting it down,[br]and there we go.

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So the equation of this line,[br]its slope is 1/4, so the

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equation of the line[br]is 1/4x plus 1/4.

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So hopefully, for those of you[br]trying to do this module, that,

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1, explained how to do it, and[br]for those of you who don't even

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know what this module is, it[br]hopefully gives you a little

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intuition about what the slope[br]and the y-intercept do

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to an actual line.