https:/.../Key+features.mp4

All right, we're going to look at some
key features of polynomial functions now,

and the first feature
we're going to look at

are intervals of increase and decrease.

Now, the easiest way that I know of

to look at an interval
of increase and decrease

is to think about your polynomial
function as a roller coaster.

Because that's truly
what you're looking at.

You're looking at an interval
of when the graph is increasing,

and then when it's decreasing.

So in this situation
it's increasing first,

then it hits this peak
and it starts coming down,

and then it hits this bottom level
and it starts going back up.

Now I'm not using precise vocabulary yet,
because I haven't introduced it.

But before we're done with the
discussion today,

you'll have all the vocabulary words
for all the pieces of that description

that I just gave you.

So let's get started.

There are two pieces of this graph
that are increasing:

right at the beginning, and at the end.

And in-between there, it's decreasing.

So if you think about it this way,
there's actually three parts to the graph.

So I'm drawing an imaginary dotted line
where the -- divides out the graph

into those three parts.

This is something you can do with
your students if they're struggling

as to where things end.

Like when an interval of increase ends
or when an interval of decrease begins

or something like that.

By looking at it like this, you could
even fold it for students that are --

that can't see, you know,
want to see one thing at a time.

You could fold it back so that they only
saw the one thing at a time.

You can also use the colored pencils like
I did at the beginning and the end here,

to really highlight where it's increasing
and where it's decreasing.

But -- so this is great
to show on a graph.

But you have to be able to name
the interval of increase

or the interval of decrease.

So that gets into reading
the domain values.

So the domain values are x-values.

So of course they're along
this axis right here. Okay?

Now this is a common
misconception kids have.

Because the interval I give you says that
it is from negative infinity to 24.5.

So this number right here is 24.5.
That is a domain value.

This negative infinity
is also a domain value.

It is not saying that this goes to
negative infinity on the y-axis.

It is saying that because this goes to
negative infinity on the y-axis,

it also goes to negative infinity
on the x-axis.

Does that make sense?

Okay, so we are saying that the --
whoops, I'm sorry.

I hit it the other way.

We are saying that the values on
the domain of this interval of increase

is from negative infinity way past
this picture here,

all the way to 24.5.

And this interval of increase

starts at this x-value of 75.9,
and goes all the way up

to positive infinity.

Again, domain values,
not range values. Okay?

So you might wonder where I got
the 24.5 and the 75.9.

So this graph should look
pretty familiar to you,

because you guys came up with it
in learning experience one.

So there's a lot of ways that you
could find these points.

You could actually solve
for them in the equations,

you could use a table and keep
playing with it until you --

oh it's going up, it's going up,
it's going up, it's going up --

uh-oh, it's starting to go back down.

So you can kind of play with it
and see when that point is

that it changes direction.

Or you can do what we did, which is
I put the equation into the...

Actually, I was using Desmos at the time.

So I put it into Desmos, and I traced
the function until I found those points

at which it changes direction.

So any of those ways will work,

but you need to understand
that it is the domain values

that name our intervals of increase.

Well they also name
the interval of decrease.

But our interval of decrease
has nothing to do with infinity,

so it's not as complicated in this one.

But sometimes it is, so you just have to
look at each one as you come up with them.

So this one, our decrease is
in-between our two increases.

So if we look at that, we take this
number from the first one,

because that's the point
that it starts the decrease,

and this number from the second one,
and that's where it starts the increase.

If you have graphs that are continuous,

then the intervals of increase and
decrease should also be connected

with these numbers.

Now, there's some in the future that
you're gonna find out

are not always continuous,
and that's a totally different thing.

But right now when you have
continuous graphs like this,

they will be linked one-to-one
all the way through.

So these points that we talk about
appear at the top

and down here at the bottom.

They have special names.
That's why I haven't been using them.

I say it kind of goes up there, and it
kind of goes down to the bottom here,

but the reason I haven't
used specific names yet

is because they have very
specific reasons.

I keep using that word, sorry.

But there's very specific reasons on why
they are called the local maximum

and the local minimum.

Now, a maximum value is where the graph
goes from increasing to decreasing.

Okay? So at first glance,
this appears to be a maximum value.

But because we have a part of the graph
over here that shoots off to infinity,

that actually is higher than this maximum.

So that's the reason we can't
call this a maximum.

We can only call it a local maximum.

Now some of you may be thinking,
so is infinity then the maximum?

No. This one doesn't have a maximum.

Because you cannot call infinity
a maximum.

Because we cannot graph it,

you cannot actually give me
the number of what infinity is,

so there's no value in it.

We just know it's out there
somewhere, okay?

So we don't consider it a maximum value.

Same way with the local minimum.

This is the local minimum, but actually
the minimum goes a lot lower than that

because it goes all the way
to negative infinity.

But because there is no such number
that we can actually quantify,

put on a graph, count to,

we don't consider that an actual minimum.

So we have a local maximum
and a local minimum,

which are the points where it changes
from increasing to decreasing,

and then decreasing back to increasing.

That only leaves us with
one other key feature

that I want to point out right now,

and that is our intercepts.

So this graph has three x-intercepts.

And when we were talking
about linear functions,

there was only one x-intercept.

Then we got into quadratic
or exponential,

and we had to talk about
intercepts in a different way.

And now there's three of them.

And there can even be more,
depending on the power of our polynomial.

But in this case, our three x-intercepts
are (0,0), (65.5,0), and (85,0).

That indicates that at these three points,
the y-value is equal to 0.

And so those are the values of x
when the y is 0.

Now, there's one value that's
already on the screen

that also serves as another value.

And that is our (0,0).

It doesn't always go through the origin.

So your x-intercept will not always
be your y-intercept like it is here.

But if it does go through the origin,
then the x-intercept will be the same

as the y-intercept.

And of course the y-intercept
is when the x-value is 0.

So these are some of the
key features.

You may recognize some from
different functions in the past.

There may have been some that
were relatively new to you this time,

but we're going to be using these
as we move throughout this module

and into future ones.

So get ready and look for that vocabulary.