Slope and Y-intercept Intuition

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.