Slope and Y-intercept Intuition Using the "Graph of a line" module to understand how a line's graph changes when its slope or y-intercept is changed.
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- 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 + 1.
- It has a slope of one, you can see that, for every amount it
- moved to the right it moves up exactly one, and has one
- 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-- 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 three.
- For every one I move to the right, I have to go three up.
- My change in y is three for every change in x of one.
- Or that's my slope.
- My y-intercept is still one.
- 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 zero.
- 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 two thousand and eight Khan
- Academy, it's now two thousand and nine.
- It's almost near the end of two thousand and nine.
- I should change that.
- Maybe I'll just write two thousand and ten 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 eighteen.
- That's our current y-intercept.
- But the slope of minus five 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 -6x - 22.
- Let's do another one.
- So, once again, it resets it, so I just say the equation 1x
- + 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
- (-5,1) and the point (9,-9).
- You have a slope of -5/7.
- For every seven you go to the right, you go down five.
- If you go one, two, three, four, five, six, seven, you're going to
- go down one, two, three, four, five.
- And that, we definitely see that on that line.
- And then the y-intercept is minus eighteen over seven, which is a
- little over two, it's about a little over-- it's what,
- a little over two and one / two.
- And we see right there that the y-intercept is
- a little over two and a half.
- 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 one.
- It looks like, for every one, two, three, four, five, six, seven, eight, nine, ten, eleven,
- twelve, for every twelve we go to the right, we're going
- to go one, two, three up.
- So our slope is going to be three over twelve, which is also one over
- four, and we can just look at that visually.
- Let's lower our slope.
- That's three / four, not low enough.
- one / two, not low enough.
- one / four, 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 one / four, so the
- equation of the line is one / fourx plus one / four.
- So hopefully, for those of you trying to do this module, that,
- one, 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.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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