Algebra: Slope and Y-intercept intuition Getting a feel for slope and y-intercept
Algebra: Slope and Y-intercept intuition
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- Good morning.
- im a nerd
- Actually I don't know what time it is for you,
- it's morning for me.
- Welcome to the presentation on slope and y-intercept.
- This presentation isn't going to teach you how to solve for
- slope and y-intercept, but hopefully it will give you
- a good intuition of what slope and y-intercept is.
- And we're going to do something a little bit different
- this time as opposed to what we normally do.
- We're not going to use the chalkboard, we're actually
- going to go on to the Khan academy website and use the
- graph of a line exercise to get a little bit of an intuition
- for what slope and y-intercept is.
- So when the application starts off, it starts with the
- equation y equals onex plus one.
- So that's the same thing as y equals x plus one.
- But we see that the slope here is one.
- If you looked at the introduction to graphing that
- I talked about, the slope is the same thing is the
- coefficient on the x term.
- And if you see here, whenever we move over
- by one, we move up by one.
- And I'm going to do another module on that slope is
- actually rise over run.
- So it's for every amount you rise, how much do you
- have to run to get that?
- And rise just means how much do you change in y, run means
- how much do you change in x.
- So here rise over run is just one, and y-intercept is where
- you intercept the y-axis.
- Now, as I change the slope and the y-intercept for this graph,
- I think it's going to make a little bit more sense to you.
- Watch what happens when the slope goes from one to three / two.
- So three / two is the same thing is one and one / two.
- So notice it got steeper.
- And if I increase the slope more it gets steeper even
- more, and y equals twox.
- If I increase any more five / two is two and one / two.
- So the more I increase the slope, I think you
- see what's happening.
- This thing jumps around.
- I should fix that.
- Let me move it back.
- And actually, the goal is to make the line go through
- those two blue points.
- That's the goal of I guess you'd call it the game.
- I don't like how this thing jumps around though.
- That was interesting, let me go back there.
- y equals zerox plus one.
- We could have rewritten this as just y equals one, because zerox
- is the same things as zero.
- And notice it's a completely flat line.
- No matter what x is y is one.
- And that makes sense because this equation would
- just be y equals one.
- Now I've been showing you what happens to the slope.
- Now notice we have a negative slope.
- The slope is now downwards sloping.
- It's downward sloping at a slope of one / two.
- Because let's say the rise in this situation is negative
- one, and the run is two.
- So that's why we get negative one over two.
- And we had just been doing slope so far and I think you
- get the idea that as we decrease slope, it's going to
- push the line further and further-- it's going to
- slope downward even more.
- I hate to use a word in its own definition, but I think you
- see that now in the picture.
- Now let's put up the y-intercept a little bit.
- And this is even more interesting.
- So y-intercept-- oh boy, how did that happen, that was
- strange --y-intercept--
- Notice, negative onex plus two, so the slope is negative one but it
- intersects the y-axis at two.
- Now if we increase y-intercept by one more it's just going
- to push this line up one.
- Let's do that.
- Oh, this is actually increasing it by increments of one / two.
- Let's do another one, I just want to see what happens
- on another graph.
- It actually depends on the actual problem.
- OK, this is interesting.
- OK, this is the same thing.
- We start at the same point.
- Let's actually try to figure out the equation of a line that
- goes through these two points.
- Well, let's see.
- It looks like the y-intercept if is going to have to
- be a little bit lower.
- I do not get why it would do that.
- It just brings the line down as we lower the y-intercept.
- And let's see I think the slope needs to be higher, because
- those two points, the line that goes through them is
- definitely steeper.
- I apologize for this thing acting up like.
- That looks like about the right slope.
- The slope is like that, and these two points are connected.
- Yeah, I think that looks like the right slope, but the
- y-intercept has to be lower.
- Almost there, I think.
- There you go!
- So the equation of this line is seven / fourx.
- So 7/4, that's the same thing as like 1.75.
- So the slope of this line slopes faster than one / one and
- you can kind of see that.
- I'll show you how to figure out all this, I just want to give
- you an intuitive sense of what sloping and y-intercept is.
- And it intersects the y-axis at negative thirteen / four.
- That's a little more than three, which you can-- negative
- three --which you can see right there.
- Let's see if we can do another one.
- And if you want, we can assign this module to you and you can
- play with it just like I'm doing right here.
- So let's see, the line that we want to get will go
- something like that.
- Looks like the current line's slope is a little too high.
- Let me lower the slope a little bit.
- That looks about right.
- seven / eight, so that means for every eight you move to the right
- you're going to move seven up.
- And I'm going to draw that better in another module.
- This module I'm kind of doing on the fly, so I apologize.
- I do every model on the fly so I guess I really
- should apologize.
- But you're not paying for this, so I shouldn't apologize.
- Oh, I distracted very easily.
- Let's see, let's move this line up.
- And you do that just by the y-intercept.
- You can see shifting the y-intercept up just shifts
- the line straight up.
- It doesn't change the inclination of the line.
- The slope changes the inclination of the line.
- There we go.
- The equation of this line is seven / eightx plus thirteen / four.
- Let's see if what I said about slope is right if we move.
- If we run eight, we should rise seven.
- So let's see.
- Run eight.
- one, two, three, four, five, six, seven, eight.
- So that gets us right there.
- And then we should rise seven.
- one, two, three, four, five, six, seven.
- Well that actually gets us those exact points.
- And we're back on the line again.
- I'm going to draw another thing like that for you so if you get
- confused don't lose heart.
- Let's do one more.
- Where's the other dot?
- I don't know.
- Let me see.
- The other dot doesn't exist.
- I gotta fix all these bugs in this thing.
- Oh there.
- It showed up.
- It showed up.
- OK, so look.
- We have to make the line go through these two points.
- It looks like the slope is negative, definitely.
- Not that negative, it's like a fractional negative slope.
- And it'll intercept the y-axis somewhere around here.
- The y-intercept is going to be like seven and something.
- seven and change.
- So first of all let's get this slope down.
- Oh boy.
- This thing is going to jump around again.
- Notice y equals zero, x plus one.
- If we increase the slope.
- This thing is doing all sorts-- I haven't seen this application
- in a while, so I must've written it when I had inferior
- programming skills, let me keep --OK, that slope
- might be right.
- Let's bring the line up higher.
- No, it still seems like my slope-- see the y-intercept,
- I'm raising the line.
- Oh good, I got it exactly right.
- And I was right.
- The slope is negative, because you can see it slopes downward.
- But it's not sloping downward that fast.
- And that make sense, that the slope is negative one / three.
- And that makes sense because if we run three, one, two, three, we rise
- negative one, we rise negative one.
- Right there.
- So that's why the slope is negative one / three.
- And then the y-intercept is twenty-two / three.
- Well that's seven and one / three.
- And right there, we intercept the y-axis one / three of the
- way between seven and eight.
- Well I think that should at least give you a little bit of
- an intuition on what slope and y-intercept are and you can
- have this module assigned for you, so you could play
- with it yourself.
- And I'm going to do some more models where you actually
- calculate slope and y-intercept and hopefully give you even
- though further intuition on what they are.
- So I hope you have fun playing around with this stuff.
- I remember I was very excited when I first learned
- this stuff, because it's very visual.
- So, have fun.
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