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CCSS.Math: , , , ,

Which quadratic has the
lowest maximum value? So let's figure out the maximum
value for each of these-- and they're defined
in different ways-- and then see which
one is the lowest. And I'll start with the easiest. So h of x. We can just graphically look
at it, visually look at it, and say-- what's
the maximum point? And the maximum point looks
like it's right over here when x is equal to 4. And when x is equal to 4, y or
h of x is equal to negative 1. So the maximum for h of x
looks like it is negative 1. Now, what's the
maximum for g of x? And they've given us some
points here and here. Once again, we can just
eyeball it, and say-- well, what's the maximum
value they gave us? Well, 5 is the largest value. It happens when x is
equal to 0. g of 0 is 5. So the maximum value here is 5. Now, f of x. They just give us an
expression to define it. And so it's going take
a little bit of work to figure out what
the maximum value is. The easiest way to do
that for a quadratic is to complete the square. And so let's do it. So we have f of x is equal
to negative x squared plus 6x minus 1. I never like having
this negative here. So I'm going to factor it out. This is the same thing
as negative times x squared minus 6x and plus 1. And I'm going to write
the plus 1 out here because I'm fixing to
complete the square. Now, just as a review of
completing the square, we essentially want to add
and subtract the same number so that part of this
expression is a perfect square. And to figure out what number
we want to add and subtract, we look at the
coefficient on the x term. It's a negative 6. You take half of that. That's negative 3. And you square it. Negative 3 squared is 9. Now, we can't just add a 9. That would change the actual
value of the expression. We have to add a 9
and subtract a 9. And you might say-- well, why
are we adding and subtracting the same thing if
it doesn't change the value of the expression? And the whole
point is so that we can get this first
part of the expression to represent a perfect square. This x squared minus 6x
plus 9 is x minus 3 squared. So I can rewrite that
part as x minus 3 squared and then minus 9--
or negative 9-- plus 1 is negative 8. Let me do that in
a different color so we can keep track of things. So this part right over
here is negative 8. And we still have the
negative out front. And so we can rewrite
this as-- if we distribute the negative sign-- negative
x minus 3 squared plus 8. Now, let's think about
what the maximum value is. And to understand
the maximum value, we have to interpret this
negative x minus 3 squared. Well, x minus 3 squared-- before
we think about the negative-- that is always going
to be a positive value. Or it's always going
to be non-negative. But then, when we
make it negative, it's always going
to be non-positive. Think about it. If x is equal to 3, this
thing is going to be 0. And you take the negative
of that, it's going to be 0. x is anything else,
x is anything other than 3, this part
of the expression is going to be positive. But then, you have a minus sign. So you're going to subtract
that positive value from 8. So this actually
has a maximum value when this first term
right over here is 0. The only thing that this part
of the expression could do is subtract from the 8. If you want to get
a maximum value, this should be equal to 0. This equals 0 when
x is equal to 3. When x is equal to 3, this is 0. And our function hits
its maximum value of 8. So this has a max-- let
me do that in a color that you can actually read--
this has a max value of 8. So which has the
lowest maximum value? h of x.