Contenuto principale

## Caratteristiche e forme delle funzioni quadratiche

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# Forms & features of quadratic functions

## Trascrizione del video

I have a function here defined
as x squared minus 5x plus 6. And what I want
us to think about is what other forms we can
write this function in if we, say, wanted to find the
0s of this function. If we wanted to figure out where
does this function intersect the x-axis, what form
would we put this in? And then another form
for maybe finding out what's the minimum
value of this. We see that we have a positive
coefficient on the x squared term. This is going to be an
upward-opening parabola. But what's the
minimum point of this? Or even better, what's the
vertex of this parabola right over here? So if the function looks
something like this, we could use one
form of the function to figure out where does
it intersect the x-axis. So where does it
intersect the x-axis? And maybe we can manipulate
it to get another form to figure out what's
the minimum point. What's this point right
over here for this function? I don't even know if the
function looks like this. So I encourage you
to pause this video and try to manipulate this
into those two different forms. So let's work on it. So in order to find the
roots, the easiest thing I can think of doing
is trying to factor this quadratic
expression which is being used to define
this function. So we could think
about, well, let's think of two numbers whose
product is positive 6 and whose sum is negative 5. So since their
product is positive, we know that they
have the same sign. And if they have the same sign
but we get to a negative value, that means they both
must be negative. So let's see-- negative 2
times negative 3 is positive 6. Negative 2 plus negative
3 is negative 5. So we could rewrite f of x. And so let me write it this way. We could write f of
x as being equal to x minus 2 times x minus 3. Now, how does this help
us find the zeroes? Well, in what situations is
this right-hand expression, is this expression on the right
hand going to be equal to 0? Well, it's the product
of these two expressions. If either one of these is equal
to 0, 0 times anything is 0. 0 times anything else is 0. So this whole thing is going to
be 0 if x minus 2 is equal to 0 or x minus 3 is equal to 0. Add 2 to both sides
of this equation. You get x is equal to
2 or x is equal to 3. So those are the two
zeroes for this function, I guess you could say. And we could already
think about it a little bit in
terms of graphing it. So let's try to
graph this thing. So this is x equals 1. This is x equals 2. This is x equals 3
right over there. So that's our x-axis. That, you could say, is our
y is equal to f of x axis. And we're seeing that we
intersect both here and here. When x is equal to 2,
this f of x is equal to 0. When x is equal to 3,
f of x is equal to 0. And you could substitute
either of these values into the original expression. And you'll see it's
going to get you to 0 because that is
the same thing as that. Now, what about the vertex? What form could we write
this original thing in order to pick out the vertex? Well, we're already
a little familiar with completing the square. And when you complete the
square with this expression, that seems to be a pretty good
way of thinking about what the minimum value
of this function is. So let's just do
that right over here. So I'm just going to rewrite it. So we get f of x is equal
to x squared minus 5x. And I'm just going to throw
the plus 6 right over here. And I'm giving myself
some real estate because what I need to do, what
I want to think about doing, is adding and subtracting
the same value. So I'm going to add it here, and
I'm going to subtract it there. And I can do that because
then I've just added 0. I haven't changed the value
of this right-hand side. But I want to do that so that
this part that I've underlined in this magenta color, so that
this part right over here, is a perfect square. And we've done
this multiple times when we've completed the square. I encourage you to
watch those videos if you need a little
bit of a review on it. But the general idea
is this is going to be a perfect
square if we take this coefficient
right over here. We take negative 5. We take 1/2 of that, which is
negative 5/2, and we square it. So we could write this
as plus negative-- what's negative 5/2 squared? So I could write this--
negative 5/2 squared. Well, if we square
a negative number, it's just going
to be a positive. So it's going to be the
same thing as 5/2 squared. 5 squared is 25. 2 squared is 4. So this is going
to be plus 25/4. Now, once again, if we want
this equality to be true, we either have to add the
same thing to both sides. Or if we're just
operating on one side, if we added it to that
side, we could just subtract it from that side. And we haven't changed the
total value on that side. So we added 25/4, and
we subtracted 25/4. So what is this part
right over here? What does this become,
the part that I've underlined in magenta? Well, this is going to
be-- the whole reason why we engineered it in
this way is so that this could be x minus 5/2 squared. And I encourage
you to verify this. And we go into more
detail about why taking 1/2 the coefficient here
and then squaring it, adding it there and then subtracting
there, why that works. We do that in the completing
the square videos. But these two things,
you can verify that they are equivalent. So that's that part. And now we just have to
simplify 6 minus 25/4. So 6 could be rewritten as 24/4. 24/4 minus 25/4 is negative 1/4,
so minus 1/4, just like that. So we've rewritten
our original function as f of x is equal to x
minus 5/2 squared minus 1/4. Now, why is this
form interesting? Well, one way to think
about it is this part is always going to
be non-negative. The minimum value of this part
in magenta is going to be 0. Why? Because we're
squaring this thing. If you're taking
something like this-- and we're just dealing
with real numbers-- and you're squaring
it, you're not going to be able to
get a negative value. At the minimum value,
this is going to be 0. And then it obviously could
be positive values, as well. So if we want to think about
when does this thing hit its minimum value-- well,
it hits its minimum value when you're squaring 0. And when are you squaring 0? Well, you're squaring 0 when
x minus 5/2 is equal to 0, or when x is equal
to 5/2 if you just want to add 5/2 to both
sides of that equation. So this thing hits its minimum
value when x is equal to 5/2. And then what is y, or what is
f of x, when x is equal to 5/2? f of 5/2-- and once again, you
could use any of those forms to evaluate 5/2. But it's really
easy in this form. When x is equal to 5/2, this
term right over here becomes 0. 0 squared, 0. You're just left
with negative 1/4. So another way to think
about it is our vertex is at the point x equals
5/2, y equals negative 1/4. So x equals 5/2. That's the same
thing as 2 and 1/2. So x equals 5/2. And y is equal to negative 1/4. So if that is negative 1, 1/4
would be something like that. So that right over
there is the vertex. That is the point--
let me make it clear-- that's the point 5/2
comma negative 1/4. And what's cool is we've
just used this form to figure out the minimum
point, to figure out the vertex in this case. And then we can use the
roots as two other points to get a rough sketch of what
this parabola will actually look like. So the interesting-- or I guess
the takeaway from this video is just to realize that we can
rewrite this in different forms depending on what we're
trying to understand about this function.