Use completing the square to write the quadratic equation y
is equal to negative 3x squared, plus 24x, minus 27 in
vertex form, and then identify the vertex.
So we'll see what vertex form is, but we essentially
complete the square, and we generate the function, or we
algebraically manipulate it so it's in the form y is equal to
A times x minus B squared, plus C.
We want to get the equation into this form right here.
This is vertex form right there.
And once you have it in vertex form, you'll see that you can
identify the x value of the vertex as what value will make
this expression equal to 0.
So in this case it would be B.
And the y value of the vertex, if this is equal to 0, then
the y value is just going to be C.
And we're going to see that.
We're going to understand why that is the vertex, why this
vertex form is useful.
So let's try to manipulate this equation to get
it into that form.
So if we just rewrite it, the first thing that immediately
jumps out at me, at least, is that all of these numbers are
divisible by negative 3.
And I just always find it easier to manipulate an
equation if I have a 1 coefficient out in front of
the x squared.
So let's just factor out a negative 3
right from the get-go.
So we can rewrite this as y is equal to negative 3 times x
squared, minus 8x-- 24 divided by negative 3 is
negative 8-- plus 9.
Negative 27 divided by negative 3 is positive 9.
Let me actually write the positive 9 out here.
You're going to see in a second why I'm doing that.
Now, we want to be able to express part of this
expression as a perfect square.
That's what vertex form does for us.
We want to be able to express part of this expression as a
perfect square.
Now how can we do that?
Well, we have an x squared minus 8x.
So if we had a positive 16 here-- because, well, just
think about it this way, if we had negative 8, you divide it
by 2, you get negative 4.
You square that, it's positive 16.
So if you had a positive 16 here, this would
be a perfect square.
This would be x minus 4 squared.
But you can't just willy-nilly add a 16 there, you would
either have to add a similar amount to the other side, and
you would have to scale it by the negative 3 and all of
that, or, you can just subtract a 16 right here.
I haven't changed the expression.
I'm adding a 16, subtracting a 16.
I've added a 0.
I haven't it changed it.
But what it allows me to do is express this part of the
equation as a perfect square.
That right there is x minus 4 squared.
And if you're confused, how did I know it was 16?
Just think, I took negative 8, I divided by 2, I
got negative 4.
And I squared negative 4.
This is negative 4 squared right there.
And then I have to subtract that same amount so I don't
change the equation.
So that part is x minus 4 squared.
And then we still have this negative 3 hanging out there.
And then we have negative 16 plus 9, which is negative 7.
So we're almost there.
We have y equal to negative 3 times this whole thing, not
quite there.
To get it there, we just multiply negative 3.
We distribute the negative 3 on to both of these terms. So
we get y is equal to negative 3 times x, minus 4 squared.
And negative 3 times negative 7 is positive 21.
So we have it in our vertex form, we're done with that.
And if you want to think about what the vertex is, I told you
how to do it.
You say, well, what's the x value that makes
this equal to 0?
Well, in order for this term to be 0, x minus 4 has to be
equal to 0.
x minus 4 has to be equal to 0, or add 4 to both sides. x
has to be equal to 4.
And if x is equal to 4, this is 0, this whole thing becomes
0, then y is equal to 21.
So the vertex of this parabola-- I'll just do a
quick graph right here-- the vertex of this parabola occurs
at the point 4, 21.
So I'll draw it like this.
Occurs at the point.
If this is the point 4, if this right here is the-- so
this is the y-axis, that's the x-axis-- so this is
the point 4, 21.
Now, that's either going to be the minimum or the maximum
point in our parabola, and to think about whether it's the
minimum or maximum point, think about what happens.
Let's explore this equation a little bit.
This thing, this x minus 4 squared is always greater than
or equal to 0.
Right?
At worst it could be 0, but you're taking a square, so
it's going to be a non-negative number.
But when you take a non-negative number, and then
you multiply it by negative 3, that guarantees that this
whole thing is going to be less than or equal to 0.
So the best, the highest, value that this function can
attain, is when this expression right
here is equal to 0.
And this expression is equal to 0 when x is equal
to 4 and y is 21.
So this is the highest value that the function can attain.
It can only go down from there.
Because if you shift the x around 4, then this expression
right here will become, well, it'll become non-zero.
When you square it, it'll become positive.
When you multiply it by negative 3,
it'll become negative.
So you're going to take a negative number plus 21, it'll
be less than 21, so your parabola is going
to look like this.
Your parabola is going to look like that.
And that's why vertex form is useful.
You break it up into the part of the equation that changes
in value, and say, well, what's its
maximum value attained?
That's the vertex.
That happens when x is equal to 4.
And you know its y value.
And because you have a negative coefficient out here
that's a negative 3, you know that it's going to be a
downward opening graph.
If that was a positive 3, then this thing would be, at
minimum, 0 and it would be an upward opening graph.