Graph axis of symmetry vertex and max and min, domain and range

>> Ladies and gentlemen, I got

a lot of questions off of this.

What's even more disturbing
though is I got a lot of

non questions that people
that got Hs or even Ts.

Ladies and gentlemen,
if you have a question,

please write that question

on your homeworkers
because otherwise,

I don't know what the
answer to the question.

I don't know where your
misunderstanding is or not.

If you have a misunderstanding,

please make sure you
guys write that down,

or if you have a question on

what to do with the problem.

The question asked,
they give us a

function f(x) = -x^2+6x-15.

What they ask us to
do for this problem,

is define the maximum
and the minimum point.

Then also determine the
domain and the range.

If you didn't write
this down, Alex,

you're probably going
to want to write

this down on a sheet of paper,

so you have it either
in your notes or not.

The first thing we look
at this range on is,

remember, we got to talk about

what do quadrats produce?

We wrote down our
notes that quadrats

produce a graph that we
call a parabola. Thank you.

Our graph is either going to

open up or it's
going to open down.

It's going to look
something like that.

Then what we talked about was

the maximum point
was right there,

or it has a minimum point.

It's either one or the other.

What you need to do is determine

what is that maximum point.

To do that, if you
guys remember,

that point is what

we call the what Sean.
Do you remember?

>> Maximum point.

>> Maximum point, which we call,

it started with a V.

>> Vertex.

>> Vertex. How did
we find the vertex?

Well, what we remembered was

the vertex went through
the axis of symmetry.

The first thing I want to do is

determine what is the
axis of symmetry.

The axis of symmetry,

is if you guys remember
the formula, -b/2a.

Now, what was b and a?

Remember all quadratic equations

come in the form ax^2+bx+c,

where a, b and c are

real numbers where a
cannot equal zero.

In this formula, I say
the axis asymmetry or

the x coordinate of my vertex
is going to equal x=b,

which in this case is a -6/2*-1,

where b is -6 and a is -1.

Anybody having any
questions up to this point?

No. Therefore, I figure this

out and I get the x
coordinate equals three.

That means my x coordinate

is equal to three and my vertex.

It's three, what?
Ladies and gentlemen,

in a function, if
you know one value,

if you know the input
or the x value,

how do you find the
y value? Yes, Sean.

>> Plug in.

>> You plug it in
for your x value.

We know the x, we need to

figure out the f(x)
or the output value.

What I do then is I take f(3)

and I get -3^2+6*3-15.

This becomes a -9+18-15,

which ends up giving
me f(3) = -6.

If I was going to plot
this ladies and gentlemen,

I don't really know exactly
what this graph looks like.

But I know right now
that I have a point,

my vertex is at (3,-6), 1,

2, 3, 4, 5, 6.

My points right there. Now
I need to do is determine,

is it a maximum or a minimum?

What is that point?

Is that the maximum or is that

the minimum of the graph?

Remember, there was a rule,

there's a test we looked at.

That test said, when
a was less than zero,

that means you had
a maximum point.

When a was greater than zero,

that means your vertex
was your minimum point.

We look at this
and we write down,

we say, what was my a?

My a is -1, correct?

Is a greater than zero
or less than? Less than.

Therefore, that means
that's the maximum point.

My graph is going to look
something like this.

We're not going to work
on sketching right now.

We'll do table values later.

My maximum point
looks like that.

We could say, that's my vertex,

which is a maximum point.

That's what you guys should have

wrote for your maximum point.

That is my vertex, which is

a maximum point
because the graph

opens down because of this
rule that we worked on.

Then the next thing
that got students

very confused was the
domain and range.

Remember, the domain
is the set of

all x values that are
part of your graph.

Ladies and gentlemen, look
at these two parabolas.

Think about any parabola you've

ever graphed or looked at.

These parabolas are going to

infinitely just keep on
getting wider and wider.

As they go up, they're
going to keep on expanding.

As they go down, they just
keep on getting wider.

Eventually, they are
going to encompass

every single x value that

we have in the number
system, every single one.

Therefore, the domain for

a quadratic without
any constraints

is going to be all real numbers.

Your domain is going to go

from negative
infinity to infinity.

Pick any value to be x.
Emma, give me a value for x.

>> Two.

>> Two. Is two a value?

Yes, it's right there.
Give me another one.

>> Eighteen.

>> Eighteen, probably
be over here.

But when you go down,
will this graph

eventually have a
coordinate at 18?

Yes you can do -300
way over there.

But it's still eventually
all the way down there,

eventually it's going
to have a point.

Because these graphs
keep on expanding.

However, the range is
going to be the y values.

You look at where
are the coordinate

points for all the y values,

do I have a coordinate point?

Well, all these
negative numbers, yeah.

But what about the positives?

Nope. When does it stop?

Well, the highest y value that I

have a coordinate
point was my maximum,

and my maximum was at -6.

My range is my lowest value,

which was negative infinity

to my highest
value, which is -6.

I don't have a y
coordinate higher than -6.

What about when y = 0?

Is that a point on this graph?

No. The highest y
value is when y = -6.

You go lowest to your max.

Does anybody have any
questions on what I just did?

No, okay [BACKGROUND].