So we've talked about two different formulas, right?
We talked about a formula when we
have a vertical parabola-- it either opens up or opens down.
But then we also introduced a formula for horizontally,
right?
OK.
So right now they're saying find the standard form
of the equation if we have a vertex at 2,1
and a focus at 2,4.
So right now, I need to determine, well,
which formula am I going to be using with the focus?
Am I going to use the vertical or the horizontal?
So the best thing I would tell you guys to do first
is to plot the points and see which one would make sense.
So let's plot the two points here.
So if I have a vertex at 2,1 and then I have a focus at 2,4.
One, two, three, four.
Now remember-- the definition of a parabola
is the set of points that are equal
distance from a focus and your directrix.
So remember, our directrix has to be the opposite direction
from our focus.
So if our focus is up, our directrix--
so this distance is one, two, three--
sorry.
One, two, three, four.
That's right there.
So there's our focus.
And then this one's going to be up three.
So now-- one, two, three.
So we don't really have to write our directrix.
But now you guys can see-- here's our directrix.
Here's our focus.
What way is the parabola going to do?
Is it going to open up, down, left, right?
Which way is our parabola going to open up?
Anybody?
Bueller?
Right angles?
Yeah, but which way?
Is it going to open up?
Is it going to open down?
Left?
Right?
Up.
Right.
Because it always is going to open so it contains your focus.
Because remember-- these distances--
and I know my parabola is not the best.
But you remember the distance from the focus to the directrix
has to be equal for your parabola.
So you're parabola is always going to contain inside of it
its focus.
Is it always going to have to contain
a number, like [INAUDIBLE]?
Remember when it's a vertical--
right.
When you have a vertical, it means
it opens up or opens down.
Yes, your focus and-- your focus and your vertex
are going to rely-- have the same x-coordinate.
And if it was horizontal, they'd have the same y-coordinate.
So now we now we know we're dealing
with a parabola that's going to be vertical,
so we can write the equation.
So we have x minus h squared equals 4p times x-- sorry--
times y minus k.
All right.
So they said find the standard form.
So we know that we have our vertex, which
is our h and our k, so we can plug that in.
All right.
That was our h and our k.
That's from what we've done before.
But now we need to look at-- what is our p?
Remember, the p is your distance from your vertex to your focus.
And it's also the distance from your vertex to your directrix.
But we're just going to be looking primarily and seeing
what is this absolute value?
Obviously it's positive to the focus
and negative to the directrix, but the absolute value of that
is going to be three.
Well, I'm sorry.
Your p-- we're just going to be looking at the positive.
Since this opens up, we're going to want
to make sure it's positive.
So we know that this is going to be 3,
because, obviously, if our--
if our vertex opened down, the p would be negative, right?
So we're just going to look at from this distance, when
it's going from your vertex to your focus, which would be 3.
So to have x minus 2 squared equals 4 times 3 y minus 1.
All right?
So this is going back to our standard form,
but could we write this back into a standard equation--
standard form that we've done before?
Yeah.
So we have like this.
So now you could just do x minus 4 x plus 4 equals 12y minus 12.
[INAUDIBLE]
Yes it is.
All right.
Then what you could do is you can
add the 12 to the other side.
So you have x squared minus 4x plus 16 equals 12y.
Then you want to get the y by itself, right?
So we divide by 12.
And rather than having each one of these terms divide by 12,
we could just say it's going to be y equals 1/12 times x
squared minus 4x plus 16.
Or you could have every term simplified-- every term
by dividing by 12 and leave him as fractions.
So you have 1/12 x squared minus 1/3 x plus 4/3.
You could do it that way as well.