So for this problem it says y equals 1/2x squared.
And what they want us to do, is they
want us to find the vertex.
They want us to find the directrix, find the focus.
Right, those are the main two important points
that we've been talking about for our parabolas, that focus
directrix, right?
Because that's what makes up the parabolas, that equi-distance.
And then also we need to just I think
sketch a graph of the parabola.
So what we're going to work in.
So for this problem, the first thing I want to do,
and if you guys just want to write down these steps.
These will be very helpful when we're doing this.
And the one thing you guys have got to remember
is we've usually just been working with parabolas
that have been vertical, that have
a vertical axis of symmetry.
Well, now we're also going to be working
with horizontal parabolas as well.
So number one, the first thing we need to do
is determine if we're going to have
a vertical or horizontal axis of symmetry or parabola.
So, axis of symmetry.
So if I look at this, I notice I have an x squared, right?
And I have a linear y.
So I'm going to look at my forms that we wrote down.
And we say, all right, well which one of those
has the x squared?
So when I look up there I have x minus h squared equals 4p times
y minus k.
That means this problem is going to be
dealing with a vertical axis.
Right, so that means my axis of symmetry
is going to be a vertical line, meaning my parabola
is going to open up or it's going to open down.
All right, but it's going to have
a vertical axis of symmetry.
Wait, how do you know that?
How did I know that from my problem?
Yeah.
Because I have an x squared.
All right, and so we look at what's our vertical axis?
What's our standard equation for a parabola
that has a vertical axis.
All right?
So the first thing I want to do is just determine it
it's horizontal or vertical.
Why do I need to determine that?
Because there's two different formulas, right?
There's one for our vertical.
And there's one for our horizontal.
The next thing I need to do, which I just did,
is write the standard equation.
So a lot of students always want to default to one of the other.
It's important for you guys notice, first off,
and just write it down.
Hey, this is going to be a vertical parabola.
It's either going to open up or it's going to open down.
But I know it's going to be a vertical parabola.
It's going to have a vertical axis of symmetry.
All right, so now what we need to do is we write the equation.
All right, so now I know what I need to look at.
Does this problem look anything like this one?
No, it doesn't.
So what I want to do is I want to write that in the form
so it looks exactly like that, all right?
So what I'll do is I'll say y equals 1/2x squared.
So let's put the x over there on the square.
So let's swap these around.
Let's just swap these out.
So I have 1/2 x squared equals y, right?
And there is nothing always in front of there.
So let's get rid of this 1/2.
So what I'll do is multiply by the reciprocal.
So therefore, I have x squared equals 2y.
Is everybody following so far what I did?
So far.
OK, now for the purposes of this problem,
just since this is our initial problem,
I want you guys to understand that our h and k are 0.
So I could write it like this, x minus 0 squared
equals 2 times y minus 0 squared.
We are going to have problems where
our vertex, our h and our k--
OK our vertex which I have written up there,
vertex is h comma k--
we are going to have problems where our h and our k
are going to be numbers, OK?
Yes?
[INAUDIBLE] So are you going to tell us that they're numbers?
Huh?
Are you going to tell us?
Yeah, when you have a problem you'll
be able to notice when you have to get the vertex.
But for right now I'm just writing it
so it looks exactly like this problem, all right?
[INAUDIBLE]
What's different?
The square on both ends.
You put a square, see.
Yeah.
[INAUDIBLE]
No, the second one.
Right side.
Oh, sorry.
Yeah, I was just kind of getting it.
So, yeah, you have two times your y minus 0.
Thank you, [INAUDIBLE].
I appreciate it.
So now the next thing is this 2.
What does the 2 represent?
The 2 is this number equals that 4p, right?
Yes.
So what I can say that is 2 equals 4p.
So what exactly is my p?
Well, divide by 4.
p equals 1/2.
So what I have is x minus 0 squared equals 4 times
1/2 times y minus 0.
Now do I have every single component, like I said,
[INAUDIBLE] numbers that looks exactly
like my standard equation?
Do I have all the components now?
Yeah, I do.
All right?
So then all I'm going to do now is
so I wrote it in standard form.
[INAUDIBLE]
I'm sorry.
[INAUDIBLE]
This number is in the spot of this 4p, right?
So 4p equals that.
So you just write 4p equals that number that you're
multiplying by your y.
So I solved for p to get p equals 1/2.
And the reason why I want do that,
is because I want to know what p is.
And I'll explain why in a second.
So the first thing is our vertex.
Vertex, remember is your h, k.
It's 0, 0.
You guys don't need to write the x minus 0 or the y minus 0.
But I wanted to show it to you, at least
for this first problem.
So you can see that you're going to have
0 comma 0 as your vertex.
All right?
If you already have it in this form, you can just say,
oh you're not minusing anything.
So it's 0 comma 0, right?
The next thing is we need to figure out what the focus is.
So the one thing with our problem,
is now we know it's a vertical parabola with a vertex at 0, 0.
So my parabola either goes up or it opens down.
All right?
And without even doing any kind of thinking,
let's take a look at how we're going to figure out our focus.
Our p is 1/2.
And what exactly does p represent?
p represents your distance from your vertex to your focus.
And it also represents the distance from your vertex
to your directrix.
So if my p is a positive 1/2, that
means I have to go up, right?
So then could it be possibly this parabola?
Could that be my parabola?
No.
Because p is 1/2.
So it has to go up.
And be 4 for y.
Be 4 for y.
What's that?
Be 4.
4?
Be 4.
4 times 1/2.
Yes, that's just part of the formula.
Your p though, is 4 times p. p is
the distance from your vertex to your focus, OK?
And here's how we figure out your vertex--
I should probably write out the formulas for you--
your vertex is h comma k, which in this example was 0, 0, OK?
Your focus, for a focus for a vertical parabola
is h comma k plus p.
All right?
And that's for a vertical one.
Because, obviously, it's going to be dealing vertically,
which in this case is going to be 0 comma 1/2.
And then the last one, the directrix, all right,
is going to be y equals k minus p, which equals negative 1/2.
Directrix is not a point.
Our focus is coordinate point.
Out directrix is a horizontal line in this case.
Because remember, our directrix is just the distance.
It's going to be whatever it is from your focus,
that distance makes up all your points.
These distances are the same.
OK?
So from your focus, to any point on the parabola,
back down to the directrix is always going to be the same.
All right, so now I've determined
what the vertex, the focus, and the directrix is.
You guys can see my focus is at 0 comma 1/2.
My directrix is y equals negative 1/2.
And my vertex is at 0, 0.
So now I've just completed just a rough sketch
of those kind of three important points to make my parabola.
Yes?
So the directrix is going to be the same as p?
But the opposite?
it's the distance from your vertex.
p is the distance from your vertex to your focus
and to your directrix.
All right.
[INAUDIBLE]
So what we're doing when it's vertically,
if you're adding p to find focus,
then you have to subtract p to find the directrix.
And it would be the other way around if my parabola was
facing downward.
But does everybody see?
Does anybody have any questions on following the rules?
How would you make the parabola [INAUDIBLE]?
These just means distance 1 distance 2.
I didn't get this.
I just sketched the parabola.
She wants to know [INAUDIBLE].
I'm just using the d1 to explain the parabola.
I didn't us that to graph it.
The only thing I did was I just showed you.
I graphed this just based on three points.
So I didn't actually [INAUDIBLE].
Right, but what about that point that was up there on the side?
What about it?
What is that?
I was just explaining that it doesn't matter what
point I pick on the parabola.
The distance from any point on the parabola,
it doesn't matter where I pick it,
it's going to be that distance to the focus
is the same distance to that to the directrix.
So it doesn't matter where it is on the parabola.
That's all I was doing in explaining.
OK?
Does anybody have any other questions on that?
No?
OK.