www.youtube.com/.../watch?v=b71r4X3Imaw

So at this point, we know that all quadratic
functions can be put in standard form

and the standard form will
help us to find the vertex.

But it turns out the vertex has, I guess,
more than just a graphical significance.

So in the case where 'a' is positive (pos.),

we're going to get a parabola
that opens upward.

And you'll notice that the
vertex occurs at a low point;

so in this case, the vertex actually
winds up giving us a minimum.

It's actually the lowest value
on the entire graph.

So we used the word
"local minimum" previously.

This is not only a local minimum

but it's also called the
absolute minimum.

It's the lowest point on the entire graph.

Contrast that where the case —

with the case where 'a' is negative (neg.).

So in that case, we're going to get
a parabola that opens downward.

And in this case, the vertex
winds up representing a maximum.

And not only is it a LOCAL maximum

but it's actually the ABSOLUTE maximum.

It's the biggest value
of y on the entire graph.

So it turns out, if a function
represents something

and it's a quadratic function,

finding the vertex is more important
than just allowing us to graph it;

but it will also tell us the maximum
or minimum value of that function.

And that's really useful if we're
trying to optimize something.

If we're trying to find the
best way to do something

or the most efficient way to do a process,

we're a lot of times trying to
maximize or minimize something.

So if we're a business, we might
want to try to minimize cost

or we might want to maximize the
amount of stuff we can fit into a box.

So a lot of maximum and
minimum problems are solved

just by finding the vertex of a parabola.

And the good news is,

it turns out mathematicians got tired
doing this over and over and over again,

and they were able to
find a formula for a vertex.

And I'm going to go through
how to find that next.

So let's suppose we have
a function in this form

and we want to try to put it in standard form.

The whole purpose is
to try to find the vertex.

So I'm going to go through the
same steps that I did with numbers.

It's going to be a little bit more abstract

because we have values here,

but the steps that we do will be very similar.

So the first thing I'm going to do
is get this constant out of the way.

By the way, this is another process

that I'm not going to expect you
to be able to do from scratch,

but I want to show you
where this comes from,

and that way, when we use it down the road,

it won't just be this magical formula
that came up, came up out of nowhere.

So we're going to have ax squared plus bx.

Now the next step is going to
be to complete the square,

but before we do that, we need to make sure

that this coefficient is out of the way.

So I'm going to leave the
left side of the equation alone

but I'm actually going to
factor 'a' out of this equation.

And factoring is just like
distribution in reverse.

So it's kind of like dividing.

We're going to have x squared (x^2),

and what this is going to turn into is,

it's not just going to be b times x anymore.

Because we're factoring an 'a' out,

it's actually going to become
b over 'a' (b divided by 'a').

And if we were to redistribute the 'a,'

you would see that we would get ax squared

and then the 'a' would cancel
here and just leave us with bx.

So that's what's going on there.

Next, we're going to complete the square.

And I've got x squared and
then I've got b over 'a' times x,

and I need to figure out
what I'm going to add here.

Well, just as a reminder,

the way to complete the square
is to cut this coefficient in half.

So as a side note, if I want
to do 1/2 of b over 'a,'

I'm going to end up multiplying
across and I'll get b over 2a.

And we're going to take that number
and then we're going to square it.

So if we square b over 2a,

what we're going to wind up getting
is b squared, and then 2 squared is 4,

so b squared over 4a squared.

And that's what we're going
to end up getting right here.

And it turns out, anything we add
to the right side of the equation,

we have to add the same
thing to the left side.

But the sneaky thing here is,

there's actually an 'a'
in front of this whole thing.

So it's actually 'a' times
b squared over 4a squared.

And it turns out, I'm actually not going
to care that much about what we get here.

What I really care about is what's
going on inside the parentheses.

So the whole purpose of
completing the square

was to hopefully get something
that factors into a perfect square.

And exactly half of this was b over 2a,

so that means that what we really have here

is x plus b over 2a in parentheses, squared.

And if we look at this other stuff,
we've got f of x, minus c.

And if we multiply these together,
one of the a's is going to cancel,

so we're going to get b squared over just 4a.

Putting this in standard form
just means getting f of x by itself,

so f of x equals 'a' times parentheses,
x plus b over 2a, squared.

And then if I add this to both sides,
that's going to become plus c.

And if I subtract this, I'm going
to get minus b squared over 4a.

And again, I don't really care about this part.

What I really care about is:

How far to the left or to the right
has this graph been shifted?

And if we look at this,

this is x plus something
inside the parentheses

and that means that the vertex has actually
been shifted b over 2a units to the left.

So the x-coordinate of the vertex

is going to be at— to the left,
so it's going to be neg. b over 2a.

And it turns out, this is the formula
that would give you the y part

(so that would give the
k value for your vertex),

but it turns out there's
another way to find that.

So if you know that x is neg. b over 2a,

you can just plug that into the function

and that will automatically tell
you what your vertex is.

So we actually have a
formula for the vertex now,

and it turns out, for any quadratic,
this is how we'll find it.

The vertex always, always, always
occurs when x is neg. b over 2a.

And then we can plug that
x value into the function

and that will tell us what y is.

So using that, we can find
maximum and minimum values.

Alright, so based on what we just determined,

we know that the maximum or minimum
value of a quadratic function

always occurs at the vertex.

And the vertex always occurs
when x is equal to neg. b over 2a.

So based on that, we actually don't have
to rewrite this in standard form to find

the maximum or minimum of this function.

So let's start with this:

Is this function right here going to have
a maximum, or is it going to have a minimum?

And how do we know?

Well, it's all based on which way
the parabola is going to open.

And if you look at this,
you'll notice that 'a' is positive

and that means that the parabola
is going to open upwards.

And if it opens upward, that means that
the vertex is going to be at a low point.

So this graph has a "min."
It's going to have a minimum value.

The vertex is going to be
the low point on this graph.

Now if we want to know what the vertex is,

we can use this formula.

So the x value of the vertex is
going to be neg. b divided by 2a,

and in this example, b is –6;

'a' is the coefficient of x squared, so that's 3;

and we don't really need c,
but just for good measure

c is equal to 1.

So according to this formula,
it's going to be neg. b

or negative –6 [–(–6)]

divided by 2 times 'a,' so 2 times 3.

And that's going to end up giving us
pos. 6 divided by pos. 6, which is 1.

So that's the x-coordinate of our minimum,

and we know it's a minimum
because of the way the graph opens,

but if we want to know the
actual value of the function,

then we need to plug 1 into this formula.

So if we want to know what y is,
that's just going to be f of 1,

so it's going to be 3 times 1 squared,

minus 6 times 1, plus 1.

And now it's just arithmetic.

So 1 squared is 1, 1 times 3 is 3,
6 times 1 is 6, and let's see...

3 minus 6 is –3, and –3 plus 1 is –2.

So we actually were able
to figure out the vertex

without putting this in standard form.

As an ordered pair,

the vertex would be the point (1, –2),

so that's the x and the y
coordinates of the vertex.

Because the graph opens upward,
we know that it has a minimum,

so that means that this value right here

is going to be our minimum value.

So that would be our answer.

The minimum value of this function is –2,

and it occurs when x is equal to 1.

Okay, next up, let's look
at this quadratic function:

–2x squared minus 8x plus 11.

In this case, you'll notice
that 'a' is a negative number,

and that means that if we were to draw the graph,

it would open in the downward direction.

But that means that the
vertex occurs at a high point,

so this graph is going to have a maximum.

And if we can figure out what the vertex is,

that will tell us the maximum
value of this function.

So according to this formula,

the vertex is when x is equal to neg. b over 2a.

And if we list out our values
for this function, 'a' is –2;

b is the coefficient of x (that would be –8);

and c is equal to 11.

So if we do neg. b over 2a,

we're going to end up with negative –8 [–(–8)]

and that's going to be divided
by 2 times 'a' (which is –2).

And that's going to wind up
giving us positive 8 over –4,

and that total is going to be –2.

So that's the x-coordinate of our vertex.

That means, if we were drawing the graph,

it was shifted two units to the left.

But if I want to know what the y-coordinate is,

I can just take that number (–2)
and plug it into the function.

So if we do y = f of –2,

we're going to get –2 times –2 squared,
minus 8 times –2, plus 11.

And we have to be really careful
with signs when we're doing this

and the order of operations.

So –2 squared (–2 times –2)
is going to be 4.

But we're taking –2 TIMES 4,
so let's be careful about that.

Here, we're taking –8 times –2,

and that's going to become positive 16.

And then we're adding 11.

So –2 times 4 is –8 and...

Oh, my mistake —

–8 times –2 is POSITIVE 16.

So a negative times negative is a positive,

so that should be a plus.

So then we're going to have
plus 16 and then plus 11.

So –8 plus 16 is positive 8,

and positive 8 plus 11 is 19.

So that is going to be our maximum value.

So the maximum is 19.

And if we wrote that as an ordered pair,

the vertex would be the point
(–2, 19) (so the x and the y)

but because the graph opens downward,

we know that this point is going to be a MAXIMUM

and that's going to be our maximum value. [END]