Contenuto principale

## Caratteristiche e forme delle funzioni quadratiche

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# Vertex & axis of symmetry of a parabola

## Trascrizione del video

We need to find the vertex
and the axis of symmetry of this graph. The whole point of doing this
problem is so that you understand what the vertex
and axis of symmetry is. And just as a bit of a
refresher, if a parabola looks like this, the vertex is the
lowest point here, so this minimum point here, for an
upward opening parabola. If the parabola opens downward
like this, the vertex is the topmost point right like that. It's the maximum point. And the axis of symmetry is
the line that you could reflect the parabola around,
and it's symmetric. So that's the axis
of symmetry. That is a reflection of the
left-hand side along that axis of symmetry. Same thing if it's a
downward-opening parabola. And the general way of telling
the difference between an upward-opening and a
downward-opening parabola is that this will have a positive
coefficient on the x squared term, and this will have
a negative coefficient. And we'll see that in a little
bit more detail. So let's just work on this. Now, in order to figure out the
vertex, there's a quick and dirty formula, but I'm not
going to do the formula here because the formula really tells
you nothing about how you got it. But I'll show you how to apply
the formula at the end of this video, if you see this on a math
test and just want to do it really quickly. But we're going to do it the
slow, intuitive way first. So let's think about how we can
find either the maximum or the minimum point of
this parabola. So the best way I can think
of doing it is to complete the square. And it might seem like a very
foreign concept right now, but let's just do it one
step at a time. So I can rewrite this as y is
equal to-- well, I can factor out a negative 2. It's equal to negative 2 times
x squared minus 4x minus 4. And I'm going to put the
minus 4 out here. And this is where I'm going
to complete the square. Now, what I want to do is
express the stuff in the parentheses as a sum of a
perfect square and then some number over here. And I have x squared minus 4x. If I wanted this to be a perfect
square, it would be a perfect square if I had a
positive 4 over here. If I had a positive 4 over
there, then this would be a perfect square. It would be x minus 2 squared. And I got the 4, because I said,
well, I want whatever half of this number is,
so half of negative 4 is negative 2. Let me square it. That'll give me a positive
4 right there. But I can't just add
a 4 willy-nilly to one side of an equation. I either have to add it to the
other side or I would have to then just subtract it. So here I haven't changed
equation. I added 4 and then
I subtracted 4. I just added zero to this little
expression here, so it didn't change it. But what it does allow me to do
is express this part right here as a perfect square. x
squared minus 4x plus 4 is x minus 2 squared. It is x minus 2 squared. And then you have this
negative 2 out front multiplying everything, and
then you have a negative 4 minus negative 4, minus
8, just like that. So you have y is equal to
negative 2 times this entire thing, and now we can multiply
out the negative 2 again. So we can distribute it. Y is equal to negative 2 times
x minus 2 squared. And then negative 2 times
negative 8 is plus 16. Now, all I did is algebraically arrange this equation. But what this allows us to do
is think about what the maximum or minimum point
of this equation is. So let's just explore
this a little bit. This quantity right here, x
minus 2 squared, if you're squaring anything, this
is always going to be a positive quantity. That right there is
always positive. But it's being multiplied
by a negative number. So if you look at the larger
context, if you look at the always positive multiplied by
the negative 2, that's going to be always negative. And the more positive that this
number becomes when you multiply it by a negative, the
more negative this entire expression becomes. So if you think about it,
this is going to be a downward-opening parabola. We have a negative coefficient
out here. And the maximum point on this
downward-opening parabola is when this expression right here
is as small as possible. If this gets any larger, it's
just multiplied by a negative number, and then you subtract
it from 16. So if this expression right here
is 0, then we have our maximum y value, which is 16. So how do we get x is
equal to 0 here? Well, the way to get x minus
2 equal to 0-- so let's just do it. x minus 2 is equal to 0,
so that happens when x is equal to 2. So when x is equal to 2,
this expression is 0. 0 times a negative number,
it's all 0, and then y is equal to 16. This is our vertex, this
is our maximum point. We just reasoned through it,
just looking at the algebra, that the highest value this
can take on is 16. As x moves away from 2 in
the positive or negative direction, this quantity right
here, it might be negative or positive, but when you square
it, it's going to be positive. And when you multiply it by
negative 2, it's going to become negative and it's going
to subtract from 16. So our vertex right here
is x is equal to 2. Actually, let's say each
of these units are 2. So this is 2, 4, 6,
8, 10, 12, 14, 16. So my vertex is here. That is the absolute maximum
point for this parabola. And its axis of symmetry is
going to be along the line x is equal to 2, along the
vertical line x is equal to 2. That is going to be its
axis of symmetry. And now if we're just curious
for a couple of other points, just because we want to plot
this thing, we could say, well, what happens when
x is equal to 0? That's an easy one. When x is equal to 0,
y is equal to 8. So when x is equal to 0, we have
1, 2, 3, 4-- oh, well, these are 2. 2, 4, 6, 8. It's right there. This is an axis of symmetry. So when x is equal to 3, y is
also going to be equal to 8. So this parabola is a really
steep and narrow one that looks something like this, where
this right here is the maximum point. Now I told you this is the slow
and intuitive way to do the problem. If you wanted a quick and dirty
way to figure out a vertex, there is a formula
that you can derive it actually, doing this exact same
process we just did, but the formula for the vertex, or
the x-value of the vertex, or the axis of symmetry, is x is
equal to negative b over 2a. So if we just apply this-- but,
you know, this is just kind of mindless application
of a formula. I wanted to show you the
intuition why this formula even exists. But if you just mindlessly
apply this, you'll get-- what's b here? So x is equal to negative--
b here is 8. 8 over 2 times a. a right here is a negative 2. 2 times negative 2. So what is that going
to be equal to? It is negative 8 over negative
4, which is equal to 2, which is the exact same thing we
got by reasoning it out. And when x is equal to
2, y is equal to 16. Same exact result there. That's the point 2 comma 16.