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