Good morning.
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 1x plus 1.
So that's the same thing as y equals x plus 1.
But we see that the slope here is 1.
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 1, we move up by 1.
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 1, 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 1 to 3/2.
So 3/2 is the same thing is 1 and 1/2.
So notice it got steeper.
And if I increase the slope more it gets steeper even
more, and y equals 2x.
If I increase any more 5/2 is 2 and 1/2.
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 0x plus 1.
We could have rewritten this as just y equals 1, because 0x
is the same things as 0.
And notice it's a completely flat line.
No matter what x is y is 1.
And that makes sense because this equation would
just be y equals 1.
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 1/2.
Because let's say the rise in this situation is negative
1, and the run is 2.
So that's why we get negative 1 over 2.
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 1x plus 2, so the slope is negative 1 but it
intersects the y-axis at 2.
Now if we increase y-intercept by 1 more it's just going
to push this line up 1.
Let's do that.
See.
Oh, this is actually increasing it by increments of 1/2.
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 7/4x.
So 7/4, that's the same thing as like 1.75.
So the slope of this line slopes faster than 1/1 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 13/4.
That's a little more than 3, which you can-- negative
3 --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.
7/8, so that means for every 8 you move to the right
you're going to move 7 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 7/8x plus 13/4.
Let's see if what I said about slope is right if we move.
If we run 8, we should rise 7.
So let's see.
Run 8.
1, 2, 3, 4, 5, 6, 7, 8.
So that gets us right there.
And then we should rise 7.
1, 2, 3, 4, 5, 6, 7.
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.
OK.
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.
Good.
It showed up.
Excellent.
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 7 and something.
7 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 0, x plus 1.
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 1/3.
And that makes sense because if we run 3, 1, 2, 3, we rise
negative 1, we rise negative 1.
Right there.
So that's why the slope is negative 1/3.
And then the y-intercept is 22/3.
Well that's 7 and 1/3.
And right there, we intercept the y-axis 1/3 of the
way between 7 and 8.
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.