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We're asked to first find the focus and directrix
of the parabola by moving the orange point
and line to their correct positions,
then use that information to find
the equation of the parabola.
So just as a little bit of review,
a parabola is a conic section that
is formed from all of the points that
are equidistant to some point that we call
the focus and some line that we called the directrix.
And what we have right over here-- what it does
is, as we move along the parabola,
as we move our cursor along the parabola
right over here, it looks at the distance to where
the current dot is and then it tries
to plot that same distance to some theoretical directrix.
And that's where that line drops down.
And you can clearly see right now
that this isn't working out, that this is not equidistant.
So the first thing that I like to do when I try to figure out
where the focus is is that the parabola
is going to be symmetric around the focus,
or at least around the vertical line that the focus is on,
I should say.
And so the focus is going to sit right above the vertex.
So that's the first thing to do.
And so let me keep manipulating it.
So if I look below down there-- and it's
hard to do that because when I move the cursor then
that green thing disappears-- but if you look
at that vertical green line, that shouldn't be moving--
the bottom of it shouldn't be moving up and down.
It should be equidistant to a flat line.
So let me keep playing with it.
It looks like that might be an improvement.
Let me see.
So this actually looks pretty good.
Notice the bottom-- oops-- notice
the bottom of the vertical green line
is not moving in the vertical direction.
It's just moving in the horizontal direction.
So if I move my directrix up there, that should work out.
So let me move it up there.
And there you have it.
Notice now my parabola-- these green lines
show the distance to the focus and the distance
to the directrix, and notice that they are always the same.
So it looks like I got the right focus and the right directrix
now.
So I got the right focus and the right directrix.
And the focus is 1/4 comma negative 3/8.
And the directrix is y equals negative 5/8.
So now let's try to use that information
to actually find the equation of the parabola in this form
right over here.
And this is going to involve a little bit of algebra for us.
So actually let me copy and paste this.
So I'm going to copy and paste this
so that we know what we need, and let
me open my little scratch pad up.
All right.
So this is our information, and we
need to figure out the equation of the parabola.
So just as a little bit of review, we have the point.
So let's say this is our point.
This is our focus-- is the point 1/4 comma negative 3/8.
And then we have the directrix at y equals negative 5/8.
So the directrix looks something like this.
That is y is equal to negative 5/8.
And then we have our parabola.
And it looks something like this.
It's equidistant to the 2, so our parabola
looks something like that.
And the parabola, the x's and y's that satisfy this equation
are all of the x's and y's that are
equidistant to these two things.
So let's call that x comma y.
So let's find an equation for all of the x's and y's that
are equidistant to this point and to this line
right over here.
So, first, let's think about the distance between this focus
and this x, y right over here.
Well, that distance just comes straight out of the distance
formula, which comes straight out of the Pythagorean theorem.
It's going to be x minus 1/4 squared
plus y minus negative 3/8 squared.
So that's going to be y plus 3/8 squared.
So this is our distance squared.
We could take the square root if we-- Why don't we just do that?
So let me just take the square root.
So this is the distance between any point on the parabola x, y
and the focus.
And the distance between that same point and the directrix--
well, that's just going to be a straight, vertical line.
So that's just going to be our change in y.
So this is going to be equal to y minus negative 5/8,
which is the same thing as y plus 5/8.
And just to make sure that this is always positive,
let's square it and then take the square root.
Now, to simplify this, we could square
both sides of this equation.
And we get x minus 1/4 squared plus y--
and actually let me expand this out.
This will be useful for us in a second,
so let me expand this out.
So this is going to be y squared plus-- let's see,
2 times 3/8 is 3/4-- so plus 3/4y plus 9/64 is
equal to this thing squared, which is just y-- well, let me
actually expand that out as well.
So that's y squared plus 2 times 5/8 is going to be 5/4.
5/4y plus 25 over 64.
Now, let me see if I can simplify it.
I have a y squared on both sides,
so I can cancel those terms out.
So y squared, y squared.
Let me subtract 3/4y from both sides.
And let me subtract 9/64 from both sides.
Minus 9/64.
Minus 9/64.
Actually, we had a little more space.
So that's going to be gone and that's going to be gone.
Our left hand side is just going to simplify
to x minus 1/4 squared.
And that's going to be equal to-- let's see.
5/4 minus 3/4 is 2/4, which is the same thing as 1/2y.
And then 25/64 minus 9/64 is 16/64,
which is just the same thing as 1/4.
So it's going to be plus 1/4 there.
Now, we're almost got where we need to go.
We have this x minus 1/4 squared.
We just need to get a coefficient out here and write
this in terms of y minus something.
So what we can do right over here
is we can factor out a 1/2.
So let's do that.
So this is going to be x minus 1/4 squared is
equal to-- factoring out a 1/2, it's 1/2 times y plus 1/2.
And now, in order to put it in the form that they care about,
they just have y minus something right over here
and then something times x minus something squared.
So we need to get this 1/2 or get this coefficient
on the other side.
The easiest way I know how to do that
is to multiply both sides by 2.
So you multiply both sides by 2.
That cancels.
And if we really want to put it in the form
that they care about, we can swap the sides.
Instead of writing y plus 1/2, we
can write that as y minus negative 1/2.
That's the same thing as y plus 1/2--
I'm going to do it in the same form
that they want-- is equal to 2 times x minus 1/4 squared.
So let me see.
Did I do that right?
Yep.
y minus negative 1/2 is the same thing as y plus 1/2
is equal to 2 times x minus 1/4 squared.
So let me see if we got that actually right.
So let's see.
It's negative 1/2 here, positive 2, and then 1/4.
So we have it is negative 1/2, positive 2, and then 1/4.
Let's see if we got it right.
And we did.