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>> Hi, this is Julie Harland
and I'm YourMathGal.
Please visit my website
at yourmathgal.com
where you could search for any
of my videos organized
by topic.
[ Pause ]
This is Part 5 of Equations
and Graphs of Ellipses
and we're going
to do the following problem
on this video.
We're going
to take this equation
of the ellipse
which in general form
and we're going to write it
in standard form.
Then we'll state the center,
we'll graph it,
graph the ellipse,
then we'll state the vertices
and then we'll check
that at least one
of these vertices satisfies
the original equation below.
So, first of all,
remember what the standard
form of an ellipse looks like.
Notice, we have something
in the form X minus H squared
over A squared,
H squared over A squared
and Y minus K squared
over B squared equals 1.
H, K would be the center
and the general form,
it just looks
like a polynomial equal to 0,
so this is in general form.
So, here we are.
Here is the problem
and the task is to write it
in standard form.
So, we're going to have
to employ completing the
square which was a method we
also used to put circles
in standard form.
So, what I'm going
to do is I'm going
to group the X squared terms
together and the X
term together.
So, I'm going
to put the 4X squared and--
oops, the minus 16X.
4X squared minus 16X,
and then I'm going
to put the Y term--
Y squared and Y term together.
And I'm going to go ahead
and write this negative 13
on the other side.
You know, it won't be a
problem completing the square
with this Y squared plus 2Y
because I basically have a 1
in front of the Y squared,
but I have a problem over here
with the X terms
because I have a 4X squared
and we want the coefficient
to be 1.
So, if you have a coefficient
of X that is not 1,
you need to factor it out.
So, I have to factor
out of this first parenthesis
of 4 which is X squared
minus 4X.
Now, if you're not sure what
it is to factor out,
for instance,
if 4 didn't go evenly into 16,
you're basically just dividing
that number by 4.
So, by the way,
if that was a 9X
and you're factoring out 4,
you would write 9/4.
Okay, that's the trick.
But this one was easy enough
to do, so we are going
to add something
in this parenthesis
to make it a prefect square.
All right, and now,
we didn't have any problem
with this one.
We didn't have
to factor anything out,
but we're also going to have
to add something to make
that a perfect square.
And on this side,
we have negative 13
and we see, we're going
to have to add 2 numbers.
Whatever we put
on the blank here,
we've got to be careful.
The first blank,
whatever I put
on that blank here,
I'm going to have
to multiply it by 4 by the way
because I'm really not just
adding the number
on the blank,
I'm adding to that left side
of the equation 4 times
that number.
All right, so let's go down
and complete this square
inside the parenthesis.
So, inside the parenthesis,
we're going to make
that a prefect square.
But I have that 4
out in the front, okay?
So, X squared minus 4X plus
something, I think, well,
what-- this--
would this be a perfect square
if it'd be half
of that coefficient,
X minus 2?
Which means I'm adding a 4,
right?
2 squared was--
makes that a prefect square.
So, this is basically the same
thing as that, right?
4 times X squared minus 4X
plus 4 is the same thing
as 4 times X minus 2 squared.
Okay, for the second one,
it's a little bit easier
because there's not a
coefficient in front
of the parenthesis.
And so, we have Y plus half
of 2 is just 1,
so that's simpler.
1 squared is what will go
on the parenthesis here.
The trick is what I really
added to the left side
of the equation.
You didn't really just add 4,
you added 4 times 4.
So, you have to add 16
for that little 4 right here
and we also added a 1,
tricky algebra.
So, what do I have here
on the right-hand side?
Negative 13 plus 16 plus 1
or negative 13 plus 17 is 4.
So, we're getting closer
to the standard form
of this ellipse.
The only thing I have left
to do here is divide both
sides by 4
so that the equation is equal
to 1.
So, let's divide both sides
by 4 and when I did that,
it also happened to take care
of this coefficient up here.
But if not, you'd have to deal
with that at the next step.
So, this is an X minus 2
squared over what?
Just a 1, so I'm going
to write this
as 1 squared 'cause to see it
in standard form, sometimes,
it's easier
to write it that way.
Of course,
this is the same thing
as just X minus 2 squared,
correct?
It's fine,
and we have Y minus 1 squared
over 4 or I could write
that as 2 squared.
So, remember,
it's up to you whether you
want to write it
over 1 squared or write over,
you know, and you want
to write this over 1 and this
over 4, or you want
to write 1 squared
and 2 squared.
But this is in standard form
and now, I can see what the
center is.
The center is what?
Okay, what's your H
and your K?
So, what's an equation equal
to 0, 2?
That's the X-coordinate
of the center,
and the Y-coordinate is
negative 1.
So, now, we're ready
to graph it.
Okay, so here was our
original equation.
If we did everything correct,
this would be our standard
form of the equation.
And so, then we picked
out what the center is,
so 2 negative 1.
So, I know
where the center is,
it's at 2,
negative 1 about right here,
and it's up to you if you want
to dot it.
I like to sort of dot it,
it's easier for me
to see the center.
So, I like to put a dot here.
It's up to you.
And so, how would I draw the
ellipse from here?
Well, this one tells you,
you're going to go
from that center,
one to the right and one
to the left,
so in the X direction,
one in each direction,
all right?
And in the Y direction,
you're going to go two
in each direction
from the center,
so you go up here
and down here.
And then that gives us a--
well, you know, we're--
we are only estimating, right?
What that ellipse looks like.
We're sketching it,
so those 4 points guide us.
Now, it also--
I also asked for--
to write the vertices,
so let's state the vertices.
[ Pause ]
All right, so we've 4 of them,
what's this one?
This is one is 1, negative 1,
so we've got 1, negative 1.
And let's say this is one is
2, 1, this one over here is 3,
negative 1, and the one
down here is 2, negative 3,
so we've got those 4 vertices.
And the question is, now,
we want to check our work
and that's the reason we want
to put one of the vertices
in because we've said
that this ellipse is going
through all 4 of these points,
so we want
to make sure we didn't make a
mistake some place.
So, that's the last thing
we're going to do,
is we're going to check
at least one
of those vertices.
All right,
so here are our 4 vertices.
In other words,
we say that all 4
of these points satisfy
this equation.
We-- make sure
that when you always check
your work,
you're always checking the
original equation
because you might have made a
mistake on step 1.
So, let's just check any
of these, all right?
You could put the video
on pause and try
that on your own.
How about we just check this
first ordered pair, 2,
negative 1?
So, I'm going to put in 2
for X and 1 for Y. So,
we're just going to check 2,
1 into this equation.
So, we have 4 times 2 squared
plus, and for Y,
I'm putting a 1,
so that's a 1 squared.
And for the X,
I'm putting a 2;
and for the Y,
I'm putting a 1.
And the big question is
that does that make
that a true statement?
So, we have 4 times 4 plus 1
minus 32 plus 2 plus 13,
is that equal to 0?
So, we have 16 plus 1 minus
32, and let's see,
2 and 13 is 15.
And, yes, that does equal 0,
doesn't it?
Hopefully, you could see
that at this point,
it's equal to 0
because 16 plus 1 plus 15
is 32.
And so, yes, it checks, 2, 1.
Let's just try one more
ordered pair.
How about 3, negative 1?
So, we're just going
to check again
in the original equation
up here.
So, I'm putting 4 times--
I'm putting in 3 for X,
so this time,
it's 4 times 3 squared,
all right, plus Y squared.
Now, be careful here,
Y is negative 1,
so it's negative 1 squared.
You have the quantity there.
Then we have minus 16,
so 16 times--
I'm plugging in 3 for X
and I'm putting in negative 1
for Y, right?
2Y, 2 times negative 1 plus
13, and we want to know does
that equal to 0 as well?
So, we've got 4 times 9 plus--
remember, this is negative 1
times negative 1 which 1,
minus 16 times 3 is 48.
2 times negative 1 is negative
2 plus 13, is that equal 0?
So, we have 36 plus 1 minus
48, let's see,
how about doing this together?
Negative 2 plus 13 is--
or actually,
I think it's even simpler.
Negative 48 minus 2,
that's the negative 50 right
there then I have plus 13,
does that equal to 0?
And, yes, because 36 and 1
and 13 add up to 50,
so 50 minus 50 is
definitely 0.
So, it looks
like we did everything right,
it all checked out.
I know a lot of people don't
like checking their work, but,
you know, that's the only way
you know whether you got the
right answer.
So, we've done everything we
were asked to do.
We wrote the equation that was
in general form here
in standard form,
we stated the center,
we graphed it,
and we stated the 4 vertices.
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.