- [Instructor] It might not be obvious
when you look at these three equations
but they're the exact same equation.
They've just been algebraically manipulated.
They are in different forms.
This is the equation
and sometimes called standard form for a quadratic.
This is the quadratic in factored form.
Notice this has been factored right over here.
And this last form
is what we're going to focus on in this video.
This is sometimes known as vertex form
and we're not gonna focus
on how do you get from one of these other forms
to a vertex form in this video,
we'll do that in future videos, but what we're going to do
is appreciate why this is called vertex form.
Now to start, let's just remind ourselves what a vertex is.
As you might remember from other videos,
if we have a quadratic, if we're graphing y
is equal to some quadratic expression in terms of x,
the graph of that will be a parabola,
and it might be an upward opening parabola
or a downward opening parabola.
This one in particular
is going to be an upward opening parabola,
and so it might look something like this.
It might look something
like this right over here.
And for an upward opening parabola like this,
the vertex is this point right over here.
You could view it as this minimum point.
You have your x-coordinate of the vertex right over there
and you have your y-coordinate
of the vertex right over here.
Now the reason why this is called vertex form
is it's fairly straightforward to pick out the coordinates
of this vertex from this form.
How do we do that?
Well, to do that, we just have to appreciate the structure
that's in this expression.
Let me just rewrite it again.
We have y is equal to three times x plus two squared
minus 27.
The important thing to realize
is that this part of the expression
is never going to be negative.
No matter what you have here, if you square it,
you're never going to get a negative value.
And so this is never going to be negative
and we're multiplying it by a positive right over here.
This whole thing right over here
is going to be greater than or equal to zero.
So another way to think about it,
it's only going to be additive to negative 27.
So your minimum point for this curve right over here,
for your parabola, is going to happen
when this expression is equal to zero,
when you're not adding anything to negative 27.
And so, when will this equal zero?
Well, it's going to be equal to zero
when x plus two is going to be equal to zero.
So you could just say,
if you wanna find the x-coordinate of the vertex,
well, for what x value does x plus two equal zero?
And of course we can subtract two from both sides
and you get x is equal to negative two, so we know
that this x-coordinate right over here is negative two.
And then what's the y-coordinate of the vertex?
You could just say, "Hey, what is the minimum y
"that this curve takes on?"
Well, when x is equal to negative two,
this whole thing is zero and y is equal to negative 27.
Y is equal to negative 27,
so this right over here is negative 27.
And so the coordinates of the vertex here
are negative two comma negative 27.
And you are able to pick that out
just by looking at the quadratic in vertex form.
Now let's get a few more examples under our belt
so that we can really get good at picking out the vertex
when a quadratic is written in vertex form.
So let's say let's pick a scenario
where we have a downward opening parabola,
where y is equal to, let's just say negative two
times x plus five,
actually, let me make it x minus five.
X minus five squared,
and then let's say plus 10.
Well here, this is gonna be downward opening
and let's appreciate why that is.
So here, this part is still always going to be non-negative
but it's being multiplied by a negative two,
so it's actually always gonna be non-positive.
So this whole thing right over here
is going to be less than or equal to zero for all x's,
so it could only take away from the 10.
So, where do we hit a maximum point?
Well, we hit a maximum point
when x minus five is equal to zero,
when we're not taking anything away from the 10.
And so, x minus five is equal to zero.
Well, that of course is going to happen
when x is equal to five,
and that indeed is the x-coordinate for the vertex.
And what's the y-coordinate for the vertex?
Well, if x is equal to five and this thing is zero,
you're not gonna be taking anything away from the 10
and so y is going to be equal to 10.
And so the vertex here is x equals five,
and I'm just gonna eyeball it,
maybe it's right over here, x equals five.
And y is equal to 10.
If this is negative 27, this would be positive 27,
10 would be something like this.
I'm not using the same scales for the x and y-axis,
but there you have it.
So it's five comma 10
and our curve is gonna look something like this.
I don't know exactly where it intersects the x-axis
but it's going to be a downward opening parabola.
Let's do one more example
just so that we get really fluent
at identifying the vertex from vertex form.
So let's say, I'm just gonna make this up,
we have y is equal to negative pi
times x minus 2.8 squared
plus 7.1.
What is the vertex of the parabola here?
Well, the x-coordinate is going to be the x value
that makes this equal to zero, which is 2.8.
And then if this is equal to zero,
then this whole thing is going to be equal to zero
and y is going to be 7.1.
So now, you hopefully appreciate
why this is called vertex form.
It's quite straightforward to pick out the vertex
when you have something written in this way.