Conic Sections: Hyperbolas 3

Let's see if we can tackle
a slightly more difficult

hyperbola graphing problem.

Let's add the hyperbola.

Make this up on the fly x minus
1 squared over 16 minus y plus

1 squared over 4 is equal to 1.

So the first thing to recognize
is that this is a hyperbola and

we'll in a few videos, do a
bunch of problems where the

first point is just to identify
what type of conic section we

have and then the second
step is actually graph

the conic section.

I already told you that we're
going to be doing a hyperbola

problem, so you know
it's a hyperbola.

But the way to recognize that
is that you have this minus of

the y squared term and then
we actually have it shifted.

The classic or the standard
non-shifted form of a hyperbola

or a hyperbola centered at 0
would look something like this.

Especially if it has the same
asymptotes just shifted, but

centered at 0 it would look
like this: x squared over 16

minus y squared over
4 is equal to 1.

And the difference between this
hyperbola and this hyperbola

the center of this hyperbola is
at the point x is equal to

1 y is equal to minus 1.

And the way to think about it
is x equals 1 makes this whole

term 0, and so that's
why it's the center.

And y equal to minus 1
makes this whole term 0.

And on here, of course,
the center is the origin.

Center is 0, 0.

So the easy way to graph this
is to really graph this one,

but you shift it so you use the
center being 1 minus 1 instead

of the center being 0, 0.

So let's do that.

So let's figure out the slope
of the two asymptotes here and

then we can shift those two
slopes so that it's appropriate

for this hyperbola right here.

So if we go with this one,
let's just solve for y.

That's what I always
like to do whenever I'm

graphing a hyperbola.

So we get minus y
squared over 4.

Subtracting x squared over
16 from both sides minus

x squared over 16 plus 1.

I'm working on this hyperbola
right here, not this one,

and then I'm going to
just shift it later.

And then let's say multiply
both sides by minus 4 and you

get y squared is equal to-- see
the minus cancels out with that

and then 4 over 16 is x squared
over 4 minus 4 and so y is

equal to plus or minus square
root of x squared

over 4 minus 4.

And to figure out the
asymptotes you just have to

think about well what happens
as x approaches positive

or negative infinity.

As x gets really positive
or x gets really negative.

And we've done this a
bunch of times already.

I think this is important.

This is more important than
just memorizing the formula,

because it gives you an
intuition of where those

equations for the lines of the
asymptote actually come from.

Because these are what this
graph or this equation or this

function approaches as x
approaches positive or

negative infinity.

As x approaches positive or
negative infinity, what is y

approximately equal
to, in this case?

Well once again, this term
is going to dominate.

This is just a 4 right here.

You could imagine when x is
like a trillion or a negative

trillion, this is going to be
huge number and this is going

to be just like you know
you almost view it like

the ground off area.

You take the square root
of that and so this

is going to dominate.

So as you approach positive or
negative infinity, y is going

to be approximately equal to
the square root, the positive

and negative square root,
of x squared over 4.

So y would be approximately
equal to positive or

negative x over 2, or 1/2x.

Let's do that.

Let's draw our asymptotes.

And remember, these are the
asymptotes for this situation.

But now of course, we're
centered at 1 negative 1.

So I'm going to draw two lines
with these slopes, with

positive 1/2 and negative 1/2
slopes, but they're going to be

centered at this point.

I just got rid of the shift
just so I could figure out the

asymptotes but of course this
is the real thing that we're

trying to graph, so
let me do that.

This is my y-axis this is my
x-axis and the center of

this is at 1 negative 1.

So x is equal to 1 y
is equal to minus 1.

And then the slopes of the
asymptotes were positive

and negative 1/2.

So let's do the positive 1/2.

So that means for every 2 you
run over, so if you go positive

in the positive x direction
2, you move up 1.

So you go to the
right 2 and up 1.

So that's the first one.

Let me draw that asymptote.

Looks something like that, and
then we draw it from this

point to that point.

Got to have a steady hand.

And then the other asymptote
is going to have a

minus 1/2 slope.

Remember this is our
center 1 minus 1, so if

I go down 1 and over.

So when I go over 2, I go down
1, so that will be right there,

Let me draw that asymptote.

And then just to continue it
in the other direction I want

to make the lines overlap.

It's going to look
something like that.

So we've drawn our asymptotes
for this function, and now we

have to figure out if it's
going to be a vertical

hyperbola or a
horizontal hyperbola.

And the easy way to think
about it is to try and make--

and we can do it two ways.

I mean if you just look at
this equation right here.

When you're taking the positive
square root, we're always going

to be slightly below
the asymptote.

The asymptote is this thing,
but we're always going

to be slightly below it.

So that tells us that were
always going to be slightly

below the asymptote on the
positive square root, and we're

always going to be slightly
above the asymptote on the

negative square root.

Because it's going to be little
less, and it's negative.

But I'll let you
think about that.

My intuition is that it's
going to be there and there.

It's more than intuition.

I know that we're going to be
a little bit less than the

negative square root, but
I'll do it the other way.

I'll do it the way I
did in the last video.

So the other way to think
about it is what happens

when this term is 0?

For this term to be 0, x
has to be equal to 1.

And does that ever happen?

Can x be equal to 1?

If x is equal to 1
here this term is 0.

And then you have a situation
where-- and then you have a

minus y squared over 4 would
have to equal 1, or this would

have to be a negative number.

So x could not be equal to 1.

So y could be equal
to negative 1.

Let's try that out.

If y is equal to negative
1, this term right

here disappears.

So when y is equal to negative
1, you're just left with-- x

minus 1 squared over
16 is equal to 1.

I just canceled out this term,
because I'm saying what happens

when y is equal to negative 1.

You multiply both sides by 16.

Let me do it over here.

These get messy. x minus 1
squared is equal to 16.

Take the square root
of both sides.

x minus 1 is equal to
positive or negative 4.

And so if x is equal to
positive 4, if you add 1 to

that x would be equal to 5.

And then if x minus 1 would be
minus 4 and you add 1 to that

you will have x is equal to 3.

So our 2 points or our 2 points
closest to our center are the

points 5 comma negative 1
and 3 comma negative 1.

Let's plot those 2.

So 5, 1 2 3 4 5, negative
1 and 3, negative 1.

Is that right?

No, minus 3, because x
minus 1 could be minus 4.

That's what happens
when you skip steps.

If you have the minus 4
situation, then x is

equal to minus 3.

You go 1 2 3 minus 3, minus 1.

So those are both points
on this hyperbola.

And then our intuition was
correct, or it was what I said.

That-- the positive square root
is always going to be slightly

below the asymptote,
so we get our curve.

It's going to look
something like this.

It's going to get closer and
closer, and then here it's

going to get closer and
closer in that direction.

It keeps getting closer and
closer to that asymptote.

And here, it's going to keep
getting closer and closer to

the asymptote on that side
and then on that side.

And of course these
asymptotes keep going

on forever and forever.

If you want you could try
out some other points

just to confirm.

You could plot that point
there, or that point there

just to confirm that
that's the case.

The hard part really is just to
identify the asymptotes and

just to figure out do we sit on
the left and the right, or do

we sit on the top
and the bottom.

And then you're done.

You can graph your hyperbola.

See you in the next video.