foreign [Music] today we're going to be talking about polynomial functions and rates of change now polynomial you know that poly means many right so we see that in Geometry with polygon and there's a lot of other times you see the word poly around it means many are much and if you have a non-constant polynomial which means it's it's more than just like y equals two that's constant right it's a function with many terms in the following form and this is in standard form and it's pretty General and it looks a little bit intimidating but it's not too bad essentially what's going on is we have a coefficient here in the front each coefficient it could be I don't say each coefficient is different they could be different most of the time they are and if you notice we have an X and there needs to be a whole number power on x in this situation we have x to the nth power and then look at the next term it's n minus 1 and then n minus two and all that means is when you have an exponent here when we put it in standard form they decrease so the highest exponent term exponent in is that a word but anyways the the term or the highest exponent goes first and then the second highest all the way down to the very last term which has x to the zero power but we don't need to write that out x to the zero power is just a constant number and that's a sub zero so let's give you an example right here so you can see what one looks like so here's a good example 7x to the fifth minus 3x squared plus 6X minus two and if you notice the exponents on each term decrease in order so we put the highest exponent term in front and then the constant if there is one or the lowest exponent term at the end of it we know that the leading term is this thing in the front which is the a sub n of x to the nth in this case it would be 7x to the fifth right I'm so this is called the leading term the polynomial has degree n well n is the exponent on that leading term so the exponent here is five so we could say from the example above the degree of this whole polynomial is five and then we would say that the leading coefficient the leading coefficient the coefficient is just a sub n it's just that number in front we don't include the X the coefficient is the number that's in front of the X so in this case the leading coefficient here in this polynomial just be seven so that is a quick review of our polynomial functions here leading coefficient we need to know degree remember the degree of the polynomials the degree of the highest term we always want to write that first and then in decreasing order with those exponents and that's called standard form but sometimes polynomial will not be written in standard form and you have to look at the variables with the largest exponent so one of the things teachers like to have their students do is rewrite the polynomial so I'm going to do that up here let's rewrite it and so when we do this we have to go through the polynomial looking at each term and you find the largest term that's got to go first it has the highest degree you have to be careful of that negative there it's a minus but it's really negative two right so that's going to come in front that's going to be our our leading term and our leading coefficient is going to be that negative 2 but then we go through each of these terms and we find the next term that has the highest degree on it the highest exponent so that would bring us over here to the 14 and that's a positive 14 so we would write plus 14x squared and you continue all the way down plus 6X plus 1. so as we said the the degree of the polynomials whatever the degree of the leading term is the highest degree of all the terms so in this case that would be six and the leading coefficient is that coefficient when the function's in standard form in this case it would be a negative two common student mistake they'll look back at the original function and say it's 14 but it's not it has to be in standard form so the leading coefficient is negative two so next we're going to move down to our local and relative extrema which are Maxima and Minima these are plural words extrema Maxima Minima if a polynomial switches from increasing to decreasing there will be a local or relative maximum that is the singular okay so maximum is when you have several maximums okay and then if it switches from decreasing to increasing there'll be a relative or local minimum what does this mean well all polynomial functions first off they are continuous so I'm going to draw a little function here that's a pretend polynomial function all right so let's suppose we have a function like this okay and polynomial functions are continuous which means that if you put your pencil on the paper you will not lift your pencil up and there's smooth edges there's nothing rough there's no points or anything like that now what this says here this definition is that if a polynomial switches from increasing which is this first part of the function to decreasing we call this a maximum okay a local or relative maximum means that it's a maximum the highest value in this area so in this local area this area right here this point right here we would call a maximum because it's the highest point in this area we also have one down here this is a local maximum if you look at it it's the highest point in the area you can think of it as being on top of a hill maybe or on top of a mountain wherever that is that is a maximum and this should make sense right because maximum means the greatest value it's at the top so in this area that is the greatest value and this area this is the greatest value and this same is true if it switches from decreasing to increasing then we say that is a minimum okay so a minimum would occur right here a local or relative minimum local and relative just means uh in this little area right here and you have to know both words because some teachers will use local and some will use relative so we'll use both as the course goes on we have another local minimum right here easy enough right so uh one question that comes up a lot if you have a function and its domain is restricted do you include an endpoint here that is part of the function is that a local minimum and based on our definition the answer is yes this is a local minimum because in the local area that is the smallest value in this area right here that's close to it now we also have a local minimum way down here do we have any maximums well yeah up here is a local maximum so this restricted function right here has two local minimums and it has one local maximum right here now we also have a different type of extrema global or absolute now I generally use absolute but sometimes you'll see the word Global and if you take all of the local Maxima the greatest valued maximum is called the global or absolute maximum and out of all the Minima the uh the smallest the least is called the global or absolute minimum now here we have a function this is not a polynomial function because you see it has like that little right there in it I don't know what that's called it okay but as we go through this function let's go through and find all the the absolute or the global Max and mins and then the relative or local maximums or minimums so the number three here says the absolute Min of this function is what okay so I'm going to look at it is there a value that is absolutely in the entire function the least value and right back here we just talked about that end point this value right here this value is negative two that is the absolute Min when x equals negative 6. okay so be very clear it's the Y value that is the Min all right and most frequently we tell where that happens when x equals negative six does this have an absolute maximum well some students would say right here but guess what that arrow means it goes forever so it goes up forever and how far does it go to Infinity so there's no value that it's the absolute Max so we're just going to cross that out and we will write none on that one right there now let's look at the relative mins well just because a point is an absolute Min doesn't mean that it's not also a relative Min because this is the smallest or the least value in this area so there is a relative Min at x equals negative six okay where else so I'm going to look right here this value in this area is the least value the smallest value so at x equals negative three what else we got right here x equals two are there any more I'm looking around I think that's it this is a local minimum local minimum local minimum I think we're good there relative Max okay a relative maximum remember it's like being on top of a hill so right here would be one at x equals negative four uh we climb the hill right here this is the top of a hill here yeah x equals one and then we would not include anything on the right because as we said earlier like there's no value here that's at the top it just goes forever so that basically defines our Global and absolute extrema from our local or a relative okay so now let's look at the characteristics of a polynomial function uh as long as we have a polynomial function that's not a constant a constant function BC a straight horizontal line right that's boring but as long as we have a non-constant polynomial we have this rule here if you have two zeros there must be at least one local extreme in between now let's find out why that is okay polynomial functions I told you this before they're either increasing or they're decreasing as long as they're not constant right so that means I'm going to make a little function right here we're going to come up it's increasing it goes through the point 1 0 we know that that is what we would call a zero or a root and then the only way that you can get to that three if you're increasing is if you start decreasing right and whatever else the polynomial function does but that increasing changing to decreasing that creates a maximum right here it's not the only possibility let me erase this we could also have a function that starts out decreasing right and we know let's suppose it's increasing here but then it's decreasing the only way to get back to that three is to increase again and that would create somewhere here there would be a minimum that's not also the only thing that could happen it is also possible that you have a polynomial function that really likes to dance and so it's increasing maybe decreases maybe increases in this case we would get two extrema right here a maximum and a minimum and so that gives us this rule here if you have two zeros there must be at least we're not sure but at least one local extreme in between those two zeros so here's an example using function notation if we have a polynomial function f and f of negative two is zero f of zero is five F of four zero and F of seven is negative one are there any guaranteed extrema and if so we're going to state where they occur so I would say what we got to find out where the zeros are the zeros business function are one and three where the zeros here the zero would be at x equals negative two that gives you a zero remember zero is where the Y value equals zero and then at x equals four that would be a zero and this other one F of zero that's thrown in that's put in there to throw you off a little bit that is a y-intercept but we could look at these two zeros we know between negative two and four there must be an extremum and we don't know what it is necessarily without some more information but we could write it in inequality notation or as an interval there must be at least one did I say at least one let me say at least one there we go that makes me feel a little more happy when I do it correctly at least one okay one more thing left here we're going to talk about even degrees uh mean that you have an absolute extrema somewhere if a polynomial has an even degree there is either an absolute Min or an absolute maximum even degree what does that mean I'm going to show you an example here what is an even degree let's start with our very basic one yeah y'all done this before how about Y equals X and it's got to be even so we'll do x squared y'all know what that means right that looks kind of like this okay one of the things we learned about our polynomial functions in Algebra 2 is that even degrees they have the same we call it end behavior like they either both go up or they're both going to go down the at each end of this function so if you have a positive leading coefficient like this is a positive one we didn't write it but a positive one right here that means you'll have an absolute minimum because on either end of it it's going to be increasing whereas if we have a negative leading coefficient remember what that does to this U and the quadratic flips it over so that means that we're going to have somewhere an absolute maximum let's pull that down over here okay so this only occurs with even degreed polynomial functions we know that somewhere even if it's even if the degree is crazy high and you have a lot of twists and you know Curves in it if it has a negative leading coefficient it might look something like this you know oh that is crazy ugly but somewhere along this polynomial function is an absolute maximum okay so we can look at these different functions and tell is there an absolute maximum or minimum for each function the first thing we need to determine is is it even degreed because if it was odd like X cubed you would have either end of it go to Infinity but we'll talk more about that later so here is there an absolute Max or Min we look at all of the different terms this is positive and it's even right so that'd be the same as x squared y equals x squared so 3x to the fourth yes that would have a minimum we would say okay and now we look at B we're going to find what do we got the degree here is 6 but it's negative right so that's more like this function here so we would say this function for Part B would have an absolute maximum remember because that negative is going to make it go down on both of the left and the right hand side but we'll talk about end behavior a little in 1.6 and part C what do you think about this one does it have an absolute minimum or an absolute maximum and the answer is no way man look at that we got an odd degree right there so I degree polynomial functions do not have absolute experiment but they might have local or relative they probably do right so just not absolute extrema okay one more thing to do and that is to talk about the point of inflection and really we kind of mentioned it before in the earlier lesson the point of inflection occurs at input values where the rate of change changes from increasing to decreasing so let's look at this function right here we see the slope is really high at this part right the slope oh my God it's going straight up so quickly and then the slope is decreasing if you notice how the slope is decreasing the slope is decreasing like a line right here would look like this oh Mr Kelly do your best and then a line if you move to to the right a little looks like this and then the line Looks like this if you notice the line is positive to zero to negative so this whole time the slope is decreasing until this point right here where the slope is here now it starts to tilt back the other way and go up right so we have a slope like this and then a slope of zero again and then the slope is positive so these areas where the slope is decreasing and then change to increasing that is called a point of inflection or an inflection point this part of the graph we would say is concave down whereas this part down here we'd say is concave up so this occurs when the graph is changing from concave up to concave down and some students have a hard time remembering which one is which so concave up is like a cup so this part right here is concave up and concave down is like a frown right so right here we got a frown it's looking pretty frowny right here is the front so it's concave down okay so those points where it changes from concave up to concave down that is called a point of inflection and I think that's all we have right here for you folks I wish you the best and hope you get the absolute maximum you can on your Mastery check this is Mr Kelly remember it's nice to be important but it's more important guys