a

Introduction to Polynomial Functions

everybody today we are going to do a

brief introduction to polynomial

functions so the first thing we to talk

about is what standard form of

polynomial functions looks like so

essentially it's just a monomial or a

sum of monomials and it can be as long

or as short as you want it does have a

few requirements

first the exponents need to be whole

numbers which means they are positive

integers so there can't be any negative

exponents fraction or decimal exponents

the second requirement is that the

coefficients need to be real numbers

it's ok if they're positive it's ok if

they're negative it's okay for their

fractions decimals radicals they just

need to be real then there's a few

important parts to know about your

function the coefficient of the term

with your largest exponent is called the

lead coefficient the lead coefficient

will tell you a lot about the direction

of your graph depending on if it's

positive or negative

the next thing to know is that the

biggest exponent in your function is

called the degree and the degree is a

way to classify your function and it

tells you a lot about the shape of your

function and then another good term to

know is that if there is a number at the

end that does not have a variable it's

called the constant now in standard form

you order your terms from the largest

exponent to the smallest and then last

always comes the constant all right so

one thing you'll be asked to do today is

just to determine whether or not

something is a polynomial function to

write it in standard form and to state

its degree type meaning positive or

negative and the lead coefficient okay

let's look at number one so here I can

see that my exponents are out of order

so the first thing I'm going to do is

rewrite it so negative one point six x

squared

minus 5x plus 7 now since all of my

exponents are whole numbers and all of

my coefficients are real numbers I'd say

yes it is a polynomial function I can

see that the degree is 2 because that's

the largest exponent the I'll call it an

LC the lead coefficient is negative one

point six which means we have a negative

function and that is all ok let's look

at number two let's let's before we even

rewrite it let's notice something about

this function I notice that it has a

negative degree I'm sorry a negative

exponent so when you see that you

actually don't even need to rewrite it

in standard form because it is not a

polynomial function so we don't even

bother okay um let's look at the last

problem I notice that my exponents are

out of order so let's put 3x to the

fourth first then X cubed and then

negative 6x so this is a polynomial

function there's no negative exponents

and all my coefficients are whole

numbers I can see that the degree is 4

and I can see that the lead coefficient

is positive 3 so it is a positive

polynomial function okay so this is just

more of the technical part of learning

about polynomial functions more of like

how to write it and how to determine

different parts but now I want to talk

about some more interesting parts such

as what does the what do the graphs of

polynomial functions look like so here I

have 6 examples this is not what all of

the graphs what these degrees have this

is just a sample for each degree so one

thing I want you to notice is that that

the number of direction changes matches

the degree so if you have a degree of 1

your your graph only goes in one

direction but look with the degree of 2

it goes one two look with a degree of

three one two three there's three

directions if you have a degree of four

goes one two three four degree a five

one two three four five so on so forth

so that's a general way to get an idea

of what your graph could look like but

depending on the numbers the different

hills and valleys will look different

another interesting thing is to talk

about the end behavior of the polynomial

functions now as it might sound like end

behavior is what happens towards the

extreme ends of your graph meaning as X

gets really large as X approaches

positive infinity and as X approaches

negative infinity so there are a few

patterns that I want you to know I want

you to notice that when your degree is

odd your the end behavior graphs is

always in opposite directions okay but

when the degree is even the end behavior

of your graph is always in the same

direction okay so that's one

generalization that you can make about

all polynomial functions and this will

be true for all polynomial functions

another thing to notice is that when

your lead coefficient is positive your

graph ends up pointing up when it's

moving to the right or as X approaches

positive infinity

however when your lead coefficient is

negative your graph ends up pointing

down as you move to the right so that

lead coefficient kind of tells you how

your graph will act as X gets big or as

X approaches positive infinity now the

last thing I want you to notice is how

we write this the notation is going to

be a little bit new for you guys so

here's how you write or describe the end

behavior mathematically you'd say f of X

approaches positive and the

which is saying why is getting really

big as X is getting really big or as X

approaches positive infinity so and then

we'd say f of X approaches negative

infinity as X approaches negative

infinity that means that Y is getting

smaller when X is getting smaller okay

let's look at a different one this one

says f of X approaches negative infinity

as X approaches positive infinity that

is translated to Y is getting smaller

it's going down as X is getting bigger

on the other side you have f of X

approaches positive infinity as X

approaches negative infinity meaning

your Y value is getting bigger as your x

value is getting smaller so this is how

I would like you to describe the end

behavior so let's give it a try so if I

asked you to describe the end behavior

of the graphs one thing I like to do is

just sketch a little function nothing

won't even with the next y axis just so

I can see which way my graph is pointing

so I see that this graph has a negative

lead coefficient so I know it's going to

end up pointing down and I see that my

degree is 4 so I know it's going to look

it's going to make 4 direction changes

so 1 2 3 4 ok so how do we describe this

we would say f of X approaches negative

infinity as X approaches positive

infinity that's saying as X is getting

bigger your graph is going down and then

we would say f of X approaches negative

infinity as X approaches negative

infinity which means that as X is

getting smaller your graph is also

pointing down so remember for even

degrees you're at your end behavior is

always going to be the same now let's

try the second example so let's sketch a

graph it's going to be negative again

but my degree is 3 so go 1 2 3

so I'm opposite and behaviors so here

I'd write f of X approaches negative

infinity

as X approaches positive infinity which

means as X is getting bigger my graph is

pointing down or to the right my graph

is pointing down and I'd say f of X

approaches positive infinity as X

approaches negative infinity which means

as X is getting smaller or when I'm

moving to the left my graph is pointing

up alright last example here I have a

positive lead coefficient and my degree

is 5 so go 1 2 3 4 5 so once again it's

an odd function so my end behaviors are

going to be different this time f of X

approaches positive infinity as X

approaches positive infinity meaning

that going to the right my graph is

pointing up and this time f of X

approaches negative infinity as X

approaches negative infinity which means

to the left my graph is pointing down

alright at this time I'd ask you to

pause the video and give this problem a

try or these two now it's important that

you try these because writing in this

notation can be a little bit tricky so

give it a try

okay thank you for giving these a try so

you could see for each one I sketch a

picture of what my graph could look like

just to give me a better idea of which

way they're pointing so you can see

there for the first one my end behavior

is the same because it's even they're

both approaching positive infinity as X

gets increasingly large or small and for

a number two since it's an odd degree my

end behavior is opposite so going to the

right my graph approaches negative

infinity and going to the left my graph

approaches positive infinity okay the

last thing that you will be asked to do

tonight is to evaluate a polynomial

function now this is something that you

definitely know how to do I just want to

make sure you understand what they're

asking when they say to evaluate when X

is equal three they're just telling you

to substitute three plug in three so you

just go through and substitute a three

wherever you want Sonex and then just

simplify so f of 3 equals 162 minus 72

plus 15 minus seven so f of 3 equals 98

so that's all they're asking and this

can come in handy just so you can find

specific points of your function and

that is all for today