 on polynomial functions in our Math 1050 College Algebra class. I'm Dennis Owelson and I teach mathematics here at Utah Valley State College. We're starting some new material today for the second exam and basically in this interval of several episodes we'll be talking about graphs of higher degree polynomial functions and also rational functions and a number of ideas that would be related to that. Today I'd like to begin with a number of objectives. First of all we want to look at the fundamental polynomial graph or graphs of fundamental polynomial functions. We'll also talk about the connection between zeros, roots and x-intercepts of a function. Then we're going to look at how we can approximate the extrema of polynomial functions, sometimes using a graphing calculator. So you'll want to use your graphing calculator today to follow along. And then we'll finish up by looking at a few applications of polynomial functions to real world problems. Now to begin with you remember in a previous episode we looked at about eight fundamental functions. There was the absolute value function, there was the squaring function, there was the greatest integer function. Well I want to add a few more functions to that list and let's begin with f of x equals x cubed. Now we've, oh let's see actually I'm going to begin with f of x equals x to the fourth. We've graphed f of x equals x cubed before. Now if I want to graph x to the fourth, what I would do would be to plot three target points and they're rather familiar target points that we've seen in other graphs. There's the target point at zero, zero. There's a target point at one, one and there's a target point at negative one, one. We've seen those same target points for f of x equals x squared. We saw those for the absolute value function. It's just a matter of knowing what curve to draw when you, when you graph these. Now the function f of x equals x to the fourth comes in steeper than the quadratic function. So it comes in, I'll draw it a little bit steeper than I would a parabola and then it flattens out a little faster along the, along the x axis. It doesn't actually touch the x axis until it gets to the origin but then it turns and it goes back up and it goes to the point negative one, one and it leaves going back up again. You notice this function is an even function. We talked about even function several episodes back. It's an even function because it has an even power there and its graph would be symmetric about the y axis. So if it goes up on the right, it also goes up on the left. If I were to compare it with the quadratic function, I'll write it just below here f of x equals x squared. There are the same target points but the difference is the quadratic function comes in not as steeply. It turns and it doesn't level off quite as fast so here the quadratic function would be above the fourth power function and then it turns and it goes back up and it also is an even function so it, it goes back out just as it did before. So these two graphs cross at only the three target points and no others. If I graph any other even power of x, I would get a similar graph. So suppose for example I wanted to graph the function f of x equals x to the tenth. Now this is another even function but it has the same target points at zero zero one one and at negative one one but but the difference is this one would appear to come in almost vertically. It isn't exactly vertical but it comes in very steeply. It flattens out very fast along here. It turns and it goes back up almost vertically again. There's a slight curve to that but it would be difficult for me to actually show that, that curve to you and this would be the graph of f. Just to show you how, how rapidly it goes up, suppose I come over here to two. If I found the function value at two, f of two would be two to the tenth power and two to the tenth power is over a thousand, it's a thousand and twenty-four. So at two I'd have to go up over a thousand to get onto the graph, that's how rapidly it's gone up. Same thing at negative two, f of negative two is a thousand and twenty-four also. So if I go to negative two I have to go up over a thousand to get on the graph. On the other hand, if I were to choose one-half right here, f at one-half that would be one-half to the tenth power which is one over two to the tenth and we said that two to the tenth was over a thousand so this is one over a thousand and twenty-four. So it's so close to the x-axis we really couldn't tell the difference between the x-axis and the point that's on the graph there. So it makes this very abrupt change from being almost horizontal to being almost vertical over there. Of course if I put in even larger, even powers I get even steeper graphs and even flatter graphs in the middle. Okay, now if I put an odd power on the polynomial, let's say we want to graph f of x equals x to the fifth. This behavior is very much like f of x equals x cube and f of x equals x. The target points would be, let's see if I substitute in a zero I get zero for the function value, if I substitute in one I get one and if I substitute in negative one I get negative one. We've seen these target points like we said on the cubic function and on the identity function f of x equals x. But the difference is because this power is bigger than the cubic function it comes in steeper than the cubic function and it flattens out a little faster than the cubic function would do and then it turns and it goes down. You notice that this is an odd function because of the odd power on the x and it has symmetry about the origin and what that meant was if you pick a point, for example this target point on the graph, if you go through the origin there is another point just like it on the same distance on the other side of the origin over here. So every point has a, shall we say, sort of a twin on the other side of the origin, same distance away. And if I were to increase this power from x to the fifth to x to the seventh, x to the ninth, x to the eleventh, I'd just get steeper graphs, flatter graphs, steeper on the way out. Now it would be difficult for us by hand to draw variations in that steepness so I'm not looking for you to be too precise on this but I want you to be aware of these relationships. Now on the odd function you notice that one end goes up, one end goes down. That's what we will refer to as the end behavior of the graph. On this end the graph is going up and on this end the graph is going down. The end behavior on an even powered function is that if it's only x to the fourth, x to the sixth, x to the eighth, the end behavior is that both sides go up. So if I refer to the end behavior I'm just saying what happens is you go further out on either end of the graph. Odd functions go up on one side, down on the other. Even functions exit the same way they came in. If they came in from up above they'll go back out up above. Okay, now what if I make a transformation of a fundamental polynomial function like this? Suppose we have f of x equals x minus two to the fourth power plus one. Well we've seen the same information for other fundamental functions. When I subtract two directly on the x that means move the graph two units to the right and when I add a one on the outside of the function that means move the graph straight up one unit. So when I graph it I'm going to shift my origin over two units and I'm going to go up one unit so my new origin is right here at the point two one. And my target points would be to go over one and up one and go to the left one and up one. And now I'll draw my basic fourth degree polynomial function so that it turns and it goes up rather steeply here and on the other side it turns goes up rather steeply rather steeply over here. This will eventually cross the y-axis but it'll cross the y-axis fairly high up. And over here this on the right hand side the graph goes up and it never comes back down again. So this is a transformation of my original function f of x equals x to the fourth. You know while we're at it let's just try calculating the y-intercept. To find the y-intercept who can remind me what do we do to find the y-intercept of a graph, Matt? Just set x equal to zero. Set x equal to zero so if I substitute in zero for x I'll get the y-intercept because that'll tell me where the will be an ordered pair where x is zero. Well if I substitute in a zero that's negative two to the fourth plus one and that says my answer is going to be positive because that's negative to the fourth power. Two to the fourth is sixteen plus one so I get seventeen. So that says I'd have to go up to seventeen on my graph before my curve would actually be crossing the y-axis. And just for the purpose of kind of reviewing this idea how would I find the x-intercepts? And you might say now wait a minute Dennis there aren't any x-intercepts look it doesn't cross the x-axis let's see if we can verify that algebraically. To find the x-intercepts what do we do Jeff? Set the equation equal to zero. Yeah to find the x-intercepts let me just write this down here. To find the x-intercepts I'll put the s in parentheses because sometimes there's only one. In this case it doesn't look like there really be any. What we do is we let y be zero and if I put a zero in for y or f of x then what we're doing is setting the equation or setting the expression equal to zero. And so this says that x minus two to the fourth power is negative one and here we have our contradiction. We have a real number raised to the fourth power it can't be negative one. This is impossible which means the original expression couldn't be zero which means there are no x-intercepts and that's exactly what our graph had indicated to us. There are no x-intercepts for this graph. So what I see visually in the graph I can support with algebraic computation and vice versa sometimes we get information algebraically and we can support it by drawing a graph. I'd like to take the graph of another fundamental function that's been transformed you might say. This time I'm going to call it little g of t equals two times let's make it a negative two times t plus one to the eleventh power minus three. Well if I were to multiply this out this would look like a very long polynomial that'd be an eleventh power, a tenth power, a ninth power all the way down to the constant term but it's easier to graph if I leave it in this form because you see this is a variation of, over here I'm sort of thinking of a variation, it's a variation of the function g of t equals t to the eleventh power. This is an odd function rather it's an odd power on t and that makes it an odd function so I know the target points and I know that it exits going up on the right and it exits going down on the left. So what changes will I make in my fundamental graph to graph this one? What changes will I want to make in this? You're going to go to the left one and down three. Okay go to the left one and go down three. What about the negative two? Well see now it's on the outside. It's a wider one. Matt what do you think? Well it's going to be, should it be upside down? Okay it's going to be inverted. And it will be stretched by a factor of two. Exactly it's going to be stretched too so I'll just make that longer to indicate not only are we going to flip it over we're going to stretch the graph okay so if you kind of get my little code there let's see if we can draw that graph now. So here are my coordinate axes. I'll have to call this the t-axis because I made this a variable t. And if I scale this off a little bit let's see we said we're going to go to the left one and down three okay that puts my new origin at the point negative one, negative three. I'll just indicate that over here negative one, negative three. And let's see now normally I would go over one and up one and I would go to the left one and down one but we have this inversion and stretch. So when I go over one I'm going to go down two and I get a target point right here. That's going to be at actually negative five. And when I go to the left one I'm going to go up two because I have flipped this graph over I've made a reflection out of it and now I think I'm ready to sketch the graph. Because I flipped it over this graph is going to be exiting going down and it's an 11th power so it's going to look almost flat and then it'll make a very abrupt turn and it'll go down very quickly. I'm sort of running out of room there for you to see that. On the other side it's not exactly horizontal but it's hard to tell the difference. It's just turning up and then all of a sudden it goes up very quickly and it crosses the x-axis just to the left of negative two. It goes up looking almost vertical. It isn't exactly vertical. Functions can't have vertical aspects, vertical portions in the graph. But this is a rough sketch of the function g. Now you see what's important here is we are now sketching some rather complicated looking functions. We're talking about an 11th degree polynomial, the 11th power of t plus one with a few other changes made to it. And we're coming up with a rough sketch of the graph by plotting only what three points? We plotted three points is all. And we don't have a totally accurate graph but we have a rough sketch. It's good enough for the applications that we're going to be looking at later in this course. Okay, now this time I'm going to take one more function and ask several questions about it. Suppose I want to graph, let's take f of x equals, let's see, I'm going to take x minus one to the fifth power plus plus three plus three. And this one is graphed in a manner very similar to the ones that we've just looked at. But I'm going to ask an additional question this time that I haven't asked about the others. This will be the x-axis because my independent variable is x. And this will be the y-axis. Now it looks like the changes we're going to make is we're going to move this graph to the right one and we're going to move it up three. So let's do that. I'm going to move it to the right one and I'm going to move it up three. So right one up three, that puts my new origin right here. And my target points, let's see, there's no stretch, there's no reflection. So I just go over one and up one. And let's see now, when I go to the left, should I go over and up one or should I go over and down one? Over and down. Over and down because this is an odd function. So I'm going to go over and down one right here. And so I come up with a graph that looks roughly like this. Now just by looking at that graph, what is its y-intercept? Two. Yeah, it looks like it's at y equals two. I'll make it an ordered pair and call it zero two. But this time I want to find the x-intercept. And so what I'll do is substitute in zero for f of x to find the x-intercept. I think maybe I can do that over here on this side to find the x-intercept. What I'll do is let y equals zero. And that says zero is equal to x minus one to the fifth power plus three. Now we can see in the graph that there is an x-intercept. And we can see that it's a negative number, but it's very close to zero. So that's what we expect to get here when we solve for x. So if I put negative three on the other side, if I subtract three from both sides, then we have x minus one to the fifth power is negative three. Now I'll need to take a fifth root. This looks kind of messy. I'm going to take a fifth root of negative three. You can take an odd root of a negative number and that will be x minus one. And there is a fifth power in here that I can factor out because this negative one times three, this negative one is the same thing as negative one to the fifth power. So its fifth root is a negative one, which I'll bring out. And then if I solve for x, I think I'll put the x on the left-hand side. If I solve for x, I need to subtract one from both sides. That's going to be minus the fifth root of three minus minus one. Minus the fifth root of three minus one. Oh, let's see, it looks like I copied something wrong there. That should be a minus. And so this should be a plus. Yeah, that was my copying error. Now, it turns out that the fifth root of three is slightly larger than one. So the negative fifth root of three is slightly smaller than negative one. And when you add one to it, you come up with a number that's just barely negative right there. You get this number right here. But it isn't a nice rational number. It's certainly not an integer, but it's an irrational number calculated to be this. OK, now the number that I found here, called the x-intercept, and this goes by several other names. An x-intercept is also called a zero or a root. And these words are used interchangeably, but normally in the context of other functions. So let's go to this first graphic that we have that we can call up here. And I put several pieces of information on this. So let's just look at the first statement. It says any function of the form, a constant a sub n times x to the n plus a constant a sub n minus one, x to the n minus one. And you notice the powers on x just keep going down until I get a sub one x plus a constant a sub zero. Any function that can be written in that form is called a polynomial function. For example, x cubed plus 2x squared minus 3x minus five. That would be called a polynomial function. Because I have basically a polynomial as the rule. And you notice that in that statement, I'm naming the function capital P of x, probably P for polynomial as opposed to f for function. So a lot of the problems in the book, they'll name the functions by P of x. Now in the second part, it says that if P at a number c is zero, then c is sometimes referred to as a root or a zero of the polynomial. A root or a zero. Because you see c makes the polynomial equal zero. So it's called a zero for the polynomial. Also, c is an x intercept of the graph of the polynomial function y equals P of x. And finally, x minus c is a factor of the polynomial P of x. Now, let me just show you an example of what that last part there refers to. Suppose I were to take the polynomial function P of x equals x plus 2 times x minus x minus 1. Now, if I multiply that out, this is a quadratic function. But actually, I want to graph it in this form and see what I can learn about this quadratic function. The graph is going to be a parabola. But there are two numbers that I could substitute in for x. There's a number I could plug in for x that would make this zero. And there's another number I could plug in for x that would make this zero. It looks like if I substitute in a negative 2, this factor is zero. And I get zero times negative 3 is zero. On the other hand, I could substitute in the number one. And I would get three times zero is zero. So I have two numbers that I could choose for x that would make this polynomial be zero, this polynomial function be zero. So I would say that x equals negative 2 and x equals 1 are zeros of the polynomial function or of the polynomial P of x. Because these are the numbers that will make the function value be zero. On the other hand, I would say that x equals negative 2 and x equals 1 are x intercepts of the graph of P of x. So now I know two x intercepts. On the other hand, x minus negative 2, that would be x plus 2. And x minus 1, those are factors of the polynomial P of x. Well, let's just take that last part. Certainly those are both factors because if you look at the way we wrote this in the beginning, x plus 2 and x minus 1, those are factors of the polynomial. On the other hand, both of these numbers are x intercepts. Because if I were to set this function equal to zero and solve for x, I would come up with x is negative 2 or x equals 1. And those would be the two x intercepts that I would get. And then finally, I'm using a word zeros. These numbers are both called zeros of the polynomial. And another word that's sometimes used here is they're called roots. Roots of the polynomial. Now, if I want to draw the graph of this function with this information, here's what I could do. I could just locate its x intercepts. And one of the x intercepts is at negative 2. And one of the x intercepts is at 1. There's 1, there's negative 2. And if I were to multiply my polynomial out now, I'm going to go back up here and multiply those two together. I get x squared plus 2x minus x of plus x minus 2. Here's the rule for the polynomial when it's written as a quadratic. And I recognize this as being a parabola, or the graph would be a parabola. So it's going to come down. It's going to go through this point. It's going to turn and it's going to go back up. The one thing I don't have is the vertex. So I know that the graph comes down like this, and the graph comes down like this. The one thing I'm missing is the vertex. But you know, we actually have a rule for finding the vertex. And if you remember that rule was x equals negative b over 2a. We had that just a few episodes back. And so in this case, x would be negative b, that's negative 1, over 2a, 2 times 1 is 2, so we get negative a half. And the y coordinate of the vertex would be p, evaluated at negative 1 half. Now let's see, that would be 1 fourth minus 1 half minus 2. 1 fourth minus 1 half minus 2. And I think if you total that up, you get negative 2 and 1 fourth. So if I go to negative 1 half, let's see, that was negative 1 right there. If I go to negative 1 half, and if I go down to negative 2 and 1 fourth, that would put me right about here. I would say that's the vertex. And so my graph turns and it goes up like this. And it turns and it goes up like this. I'm going to kind of even that out a little bit right there. This is the graph of p. Now the whole purpose of this example is to show you that I can begin to graph a polynomial function by locating its x-intercepts. And because it's a quadratic function, I can actually do more. I can find the vertex afterwards and I can locate the vertex and then I can complete my graph. Now when it comes to polynomials of higher degree than a quadratic, this is called a second-degree polynomial, but when I get a third or a fourth or a fifth-degree polynomial, I don't have formulas that will tell me where the vertex is. So what I'll do is only locate the intercepts and on that basis I'll sketch the graph. Let's take an example of that. Let's skip to the second graphic coming up here. Yeah, let's see. Now I have two examples here that I think we've already talked about these. Let me sketch these two examples before we go to the next one. The first graphic there says f of x equals x plus 2 cubed minus 3. And I want to graph that function. Let's see, and I would like to be able to draw on the screen. So there we go. So I want to draw this function directly below the statement of the function there. So this is the x-axis, this is the y-axis. And what are the changes I would want to make in that graph? To the left 2 and down 3. To the left 2, yeah, and got to go down 3. Okay, so to the left 2, down 3, I get my new origin right here. And this is a cubic function. So I know that I should go over and up one. I should go back and down one. And so my graph is going to look something like this. It's going to come down. It's going to level off there and it's going to go down. I don't know that I have the proper y-intercept there. I'm just kind of sketching, making a rough sketch of the graph. So this is the graph of that function, capital F. And when I go to the other graph, this has the same shifts, but it's a fourth-degree curve. I'll have to call this the t-axis. So I'm going to go to the left 2 and down 3. But because this is a fourth-degree function, I'm going to go over one and up one. And I'm going to go to the left one and up one. And now I'll draw a fourth-degree function, which should be slightly steeper than that one. So I'm going to have it coming in a little bit steeper. It turns a little bit flatter and its in behavior would have it going back up over on the other side. And that's the graph of that function. Okay, now let's go to the graphic after this one. Okay, in this next example, we have a function that is defined by the product of three linear factors, x plus 3, x minus 1, and x minus 2. So let's see how we'd go about graphing this. You know, if I were to multiply this out, this would be a cubic function, but it wouldn't be a simple power of a binomial, a cubic power of a binomial, but it's three different binomial factors. And so it would have a cubic, a quadratics, or a square term, a first-power term, and a constant term. And I wouldn't be able to graph it in that form, but I can graph it in this form. Here's how I'll do it. You see, there are three zeros for this function. Remember, those are the numbers that would make the function be zero. If I substitute in a negative 3, this factor's zero, and so the whole thing will be zero. So one of the zeros is at negative 3. What would be the zero for this factor? One. One, okay. And what would be the zero for the last factor? Two. That would be two. Okay. Now, remember, another name for a zero is a root. So these are also the roots, but also it tells me these are the x-intercepts. Well, now that's the word that I want to be able to draw this graph. We now have three x-intercepts at negative 3, at 1, and at 2. So with that information, I think we can sketch a graph that's relatively accurate, and we can certainly draw it very quickly. Here's how. I'm going to put my axes over here. And one, two, three, let's say that's four. And negative 1, negative 2, negative 3. There's negative 4. I'm not going to bother labeling the y-axis because I don't really know how high or low this function will go. I'm just going to get the general shape. And I'm thinking that I have an x-intercept at negative 3, and I have an x-intercept at 1, and an x-intercept at 2. And so looking back at the original polynomial function, if I were to multiply this out up here, I would be multiplying x times x times x. The very first term is going to be x cubed. Let me just write that out right here in that space. So my function, p of x, is going to begin with an x-cube, and then there will be some more terms after that. But the other terms are relatively insignificant for my purposes at the moment, but you see this is a positive x-cube. And if that first term has a positive coefficient, it means the graph on the right-hand side is going to go up. And so from this point, my graph is going to be going up like this. Or if I look at the reverse order, it's going to be coming down to this x-intercept. It's going to go through this point, through this intercept. It's going to turn, and it's going to come back to this x-intercept, and it's going to go up. But this time I'm going to draw it up quite a bit higher because we don't have to come back down to the x-axis until we get all the way over here. So I have room to go up before I have to come back down. It's like I have an appointment. I have to be here at negative 3. And then my graph keeps going down. So this is my very rough sketch of the polynomial function that we were given in this example. Now there's quite a bit to talk about here. Number one, when I draw my graph coming through this point, how far down should I go before I go back up? You see, other people might draw it much further down before they go back up. Well, for our purposes, it's relatively insignificant how far down you draw it. All I know is it goes down, and it goes back up, and this is the general shape. It may actually go much lower than what I've actually shown here, but without plotting more points, this is the best we can do, and we're trying to do this in a hurry. Now the reason I went up higher here than I went down here is because I had a span of only one interval, and here I have a span of four in that interval. So there's more time for it to go up, wander up before it comes back down. And it may actually be that this function should go up much higher than I've actually drawn it, but relatively speaking, I'm making that one go higher than I did this one to go lower over here. And then I come back, and I go back through this intercept on the way out. Okay, another question you might be asking is, Dennis, how did you know the function went up as opposed to going down just because that cubic term had a positive coefficient? Well, you see what happens if I go from, let's see, this is two right here. If I go over to three, four, five, six, et cetera, this term is going to dominate all the other terms that I see in this expansion. There will be a second-degree term, a first-degree term, and a constant term, but if the cubic term is positive, then when I start cubing numbers like three, four, five, and six, that's going to become extremely, shall we say, extremely positive, and it's going to cancel out any effects these other guys could have. Even if they're negative, this one's going to take over, it's the biggest power, and it wants to be positive, so I know that my graph eventually has to be positive up there. Same thing over on this side. I have the graph going down on the left. Now, why is that? Well, you see in general, when I go beyond negative three, this is negative three right here, if I go to negative three, negative four, negative five, what happens when you cube a negative number? It becomes very negative. It's going to become more and more negative, and it's going to dominate the square, the first power, the linear term. This one's going to take over, and this one says it wants to be negative, the other terms can't stop it, it's going to go negative, so it's going to go down over here. And this is the general characteristic of any third-degree polynomial function like this one. It goes up on one side, it goes down on the other. Now, if there had been a negative in front of that, I would have flipped this all over, and it would have been going down on the right, and it would have gone up on the left, and still, my in-behavior would be that they do opposites things on opposites sides because this is an odd power function. If the highest power had been a fourth power, they would either both go up or they'd both go down. That would be the in-behavior that I would see. You know, still another question that a person might ask is, Dennis, you said this function went up, but could it have done something like this? Could it have gone up and gone back down and then gone up like you said it was going to do? This would be impossible in this function because, look, it would create two more x-intercepts, and I don't have any more x-intercepts because we've identified all the roots. You remember back here we had negative 3, plus 1, and plus 2. Those are the only x-intercepts, so my graph can't turn and go back below the x-axis on the way up. And yet another point that a person might question about this is you might say, well, Dennis, could it turn up here and then go up? Is that possible? So that you don't have any x-intercepts, but maybe there's a little bit of an aberration in there. And the answer to this is yes, this is actually possible for some functions, but for the ones that we consider that are, shall we say, relatively simple, it's just three linear factors multiplied together, we generally don't get a lot of other fancy curves and reverses in our graphs. So without further explanation, I'll just say this is how the graph looks. Let's go to the next example, and I think we see further information, the next graphic has further information about polynomial functions. This refers to what's called the multiplicity of a root or a zero. Now if a factor appears only once, then we say that its multiplicity is one and the graph crosses the x-axis at that x-intercept. If a factor appears squared, or let's say to a fourth power, if the multiplicity is even, in this case I said if the multiplicity is two, what happens is the graph turns and it goes back from the x-axis. It doesn't actually cross the x-axis, but it turns sort of like a parabola and it goes back. I'll show you an example of this in a moment. And finally, if the multiplicity is three or any other odd number bigger than three, what happens is the graph levels off at the x-intercept and then it proceeds to cross the x-axis. Okay, let's do another example just like the one that I just did. I think I'll call this function q of x because we call the last one p of x. And this time I'm going to put a coefficient of two in front and then x minus three times x plus two times x minus one. Okay, several things I'd like to point out here. You notice each factor is given to the first power, x minus three. That's not square, that's just a first power. X plus two is to the first power and x minus one is to the first power. There's a word for this that's referred to the multiplicity of the factor. You remember there was a movie a few years ago called Multiplicity. Let's see, what was the name of the guy who sorted that to Keaton? Michael Keaton. Yeah, he was in a movie called Multiplicity. Well, this is called Multiplicity. And so I would say that the multiplicity is one because there's a first power there, first power there, first power there. I have only one of each one of those factors. Now you might say, well, what about the two in front? Well, this is a coefficient. That's not a multiplicity. That's going to cause a stretch. So it means that relatively speaking, my graph will be a little steeper than I would have drawn it otherwise. Okay, so the multiplicity of each factor is one in this example and that was the case in the last example as well. So what I'd like to figure out is the x-intercepts. Well, the x-intercepts are the same thing as the zeros are the roots. And I think I have three x-intercepts. Can anyone tell me what the three x-intercepts are? Three, negative two and one. Three, negative two and one, exactly. So I'm going to use those to draw the graph. So when I put the graph right here, I'll locate... Here's three. I'll put a dot on that one. And then here's negative two. I'll put a dot on that one. And then I'll put a dot right here. So I've got three, negative two and one. You notice I'm not going to bother marking off a scale on the y-axis because I don't really know how high this graph is going to go, how low it's going to go. I'm just trying to get a rough sketch. So I don't want to presume to know that I'm going to be able to tell you how high they'll be. Now if I were to multiply this out up here, I think I could just squeeze in the very first term. I would get x times x times x times two. That's going to be two x cubed. And then there will be some other terms after that. But what's important is that my lead term, my lead coefficient is a positive number, two x cubed. If it's positive, that tells me the graph goes up on the right. So I know that from this point my graph goes up on the right. So I'll just, that's probably a little bit too steep. I'll have it go like that. And it's going to come through this point. It's going to go down. I don't know how far. It's going to come back to negative one. And it's going to go through that point. It's going to go up. But you know it's going to go a little bit higher now than it did going down here. Can anyone explain why? Why that goes higher there than it goes down here? It has to do with the span. You notice there's only a two-unit span there. There's a three-unit span here. So I have more time to go away before I come back. So it wanders off a little bit further. But it does eventually come back here and the graph goes, should be kind of turning down right there. And this is a very crude sketch of Q. I don't know that this is actually how far down the graph would go. I don't know that's actually how high up the graph would go. It's just a rough sketch. What I do know is these three points are accurate. The end behavior here and here is accurate. And that's really all I'm looking for. Now if you stop and think about it, this is rather impressive, I think. Because we are now graphing functions that look rather complicated. This is a cubic function with a coefficient of two on it. And I've been able to make a rough sketch. Now you notice the coefficient of two I said was going to stretch it a bit. That's going to make it go up a little faster than I would have otherwise drawn it. That's not something you can do necessarily accurately portray in your graph, but just something that I can mention. So how would I have drawn it without the two there? I probably would have drawn just about the same shape, but technically it wouldn't have gone up quite as fast and it wouldn't have gone quite as low. It would have only gone down half as far as this one, because this has been stretched too. And here it would have gone up only half as much as I drew it because of the two. But those things are all relative to my graph. Okay, let's stop here. Let's go to the next graphic and look at multiplicity. Okay, now we have another example to sketch. This one looks even more complicated than the others, but it's actually no more difficult. This one says sketch the function u of x equals negative 2x squared plus, excuse me, negative 2x squared times x minus one times x plus two cubed. Now you notice that x is a factor, but it's squared. x minus one is a factor, but it's given to the first power, and x plus two is a factor, but it's given to the third power. Those powers are called the multiplicity because I actually have two factors of x. I have one factor of x minus one, and I have two, or three factors of x plus two. So I would say that the multiplicity of x equals zero is two. That's one of my x-intercepts. x-intercepts, zero. This one has multiplicity two. Another x-intercept is at one, and it has multiplicity one because it comes from a factor that's given only to the first power, and then the last x-intercept is negative two, and it has multiplicity three because I actually have three factors of x plus two. Now that has an effect on how I draw the graph. Let me just demonstrate this, and then I'll show you the rule on the next graphic. When I go to graph this function, I'm going to locate my three intercepts. There's the intercept at zero, there's the intercept at one, and there's the intercept at negative two. So I've highlighted those, and I'm expecting this function to come in from below rather than to come down from above, and the reason is if you look at the coefficient right here, the two is a stretch, and the negative means it's been flipped over. So it's going to be coming in from below. And by the way, I'm expecting to see it go out below as well because if you were to multiply this out, I think you would get negative two x to the sixth power. You would get an x to the sixth power, and then a fifth power, a fourth power, a third power on down. And let me just write that up here. I think that's going to be negative two, x to the sixth, and then some more. You see that even power tells me tells me that the end behavior is the same on both sides. So as this graph comes up, it approaches this intercept, and I'm going to have it level off and then turn and go back go back. Actually, I'm looking at the wrong, I was looking at the negative two. This is for one. Let me back up here. I'm going to have it pass right through the one. I'm going to have it pass right through the one, and then it turns and it comes back down. So as I approach the zero, you notice zero has multiplicity two. You remember up here it was an x squared. And therefore, right here, it's going to look sort of like a parabola. It's going to turn and it's going to go back up. It doesn't actually cross, it just touches the x-axis. And I'm going to have it go up even higher than the branch before. And that's because I have an interval of two, not an interval of one. So I can wander further away before I come back down. Now, as I approach the negative two, negative two comes from a factor with multiplicity three. So here, this is going to look sort of like a cubic function. It's going to turn, level off, turn, and go back down. So there are all these subtleties about what happens in the vicinity of an x-intercept. Right here it's basically, should we say, almost linear. It just passes right through. Then at zero it's almost parabola. It turns and goes back up. And then at negative two, it's almost cubic. It levels off, turns, and goes down. Now you notice both sides do go down like I expected. That's because this is a negative on an x-to-the-sixth power. And this is my rough sketch of the function, excuse me, that was called the function u. So this is a rough sketch of u. I've certainly sacrificed a lot of accuracy. I don't know the graph goes that high or that high. And that's all I'm looking for. It doesn't have to be a perfect likeness of it. Let's go to the next graphic about multiplicity of roots. This is a summary of what I've just mentioned in this last example. When a factor has multiplicity one, the graph merely crosses the x-axis in an almost linear fashion. When you have multiplicity two on an x-intercept, the graph turns back like a parabola. It's rounded and it turns back up as if it came in from below. And for multiplicity three, the graph levels off and then it crosses the axis like a cubic function. If we go back to the green screen here, I would like to graph the function that we were just looking at on the graphing calculator and verify that this is what the graph looks like. So let me just erase this and I'll write out the function once again that we were looking at. That was u of x equals squared times x minus one times x plus two cubed. Now, we just made a rough sketch. Let's see what the graphing calculator tells me it's going to look like. So if we could zoom in on the graphing calculator, I'll get it turned on here and I'm going to go to the button y equals so that I can enter this function and the function said negative two times x squared open parentheses x minus one close parentheses times x plus two but that factor was cubed so I have to enter a third power. This has come back down to the next line. So this is the function that I'm entering. Now to draw the graph, I have to pick a window. So I'll look to see what size window I want. For the minimum x, we know we wanted to go from negative two to plus one so let's say we're going to go from negative three to plus three. I think that should be wide enough. And for the scale I think one unit scaling should be all right. That means on the x-axis I'll see a little tick mark for every one unit. For the minimum y, well, I don't know how low it goes. Let's pick negative twenty. That may not be enough we'll see. And for the maximum y, let's put positive twenty. And for the y scale, let's go every five units we'll have a tick mark. Okay, so here's the graph. Oh, well now look what happened. There is something different about this that I hadn't predicted. You notice it does come up from below and it does go out going down. It does cross the x-axis at one, at zero, and at negative two. This function does pass right through one almost in a linear fashion. It goes up to a peak and right here it does turn sort of like a parabola. It turns and it goes back up. But look, it didn't go up as high there as it did over here. The reason is, I'm guessing it's because it had to come back and sort of level off over here at negative three. So it couldn't wander too far off because it had to come back down, level off. So it's sort of like a cubic function before it turns down. So we actually get a higher peak here than we had there. Now you may say, Dennis, that means your graph was wrong. So what are we supposed to do? Well, what I'm actually suggesting is if you draw the graph as I just drew it I would count that as being just fine because I think everything we drew made sense but it wasn't totally accurate. But remember we're not looking for total accuracy, we're just looking for speed. So generally we did have the right shape for this graph. Now let's go to an application to sort of finish this discussion and this will show you how people use graphs of polynomial functions to solve problems. So let's go to the applications graphic. Okay, in this problem it says an open box is formed from a 10 by 10 inch sheet of cardboard by cutting out squares from the corners and folding up the tabs. And then I ask several questions about this. Well, let's come back to the green screen and just illustrate what this problem is about and then we'll answer those questions. So imagine that I have a sheet of cardboard that is sort of looking at it at an angle. It's sort of laying down here. Let me move that a little bit closer to the center so you can see it better. So we have a sheet of cardboard and it's 10 by 10. 10 inches by 10 inches. So what I'm going to do is cut a square out of each corner and I'm going to fold these little tabs up. You see there's a little tab that sticks out there. There's a tab on the bottom. There's a little tab there and a tab over here. I'm going to fold those up and when I fold them up it's going to look sort of like this. On the next side this folds up. On the next side this one folds up and then on the back side it's going to fold up. What I get here is an open top box. And the question is what size tab should I cut out 1 inch, 2 inch, 3 inch, whatever so that this box has the maximum volume in it. So I'm looking for the maximum volume. Now you know if someone had asked me about this problem when I was a student in the early days when I was studying mathematics I would have thought that would be impossible. How can you know what is the maximum volume on each side of that box? But actually I think we can do this and we can do it rather quickly. Here's how. Imagine that the little square that I cut out is x on each side. x and x all the way around. Well then how much is left over for the length of that tab that I folded up right here? 10 minus 2x. There's 10 minus 2x and you know I think this is 4 and then how tall will that box be? x because if this was x wide then when I fold it up it will be x tall. So the volume of this box is going to be the length times the width times the height and that's x times 10 minus 2x squared. You see this is a polynomial function that represents the volume of that box. Now let's go back to the graphic first question that I asked in this application if we go back to that graphic said express its volume v as a function of x and that's what we've done here. Now in the second part in part b it says what is the domain of this function? So let's go back to the green screen what's the domain of this function? Now you see normally for a polynomial function you say that the domain is all real numbers but in this case we know that x can't be negative and also x can't be bigger than 5 because if x were bigger than 5 when I put an x on each n I would have more than 10 and that's the total length of the cardboard so it looks to me like the x's have to vary between 0 and 5. So this has a restricted domain not because the function inherently has that domain but because the application warrants it. Now when I go to draw this graph I would locate the x-intercepts and the x-intercepts appear to be at 0 and what's the x-intercept there? What makes that 0? 5 So I have 2 x-intercepts I have an x-intercept at 0 and I have an x-intercept at 5. Now when I multiply this out I'm going to get a cubic function in fact let me just erase this so we can see it and when I multiply it out I believe I think you're going to get a 4x cube so that tells me this graph goes up on the right because that's a positive and I know not only does it go up on the right it's going to turn like a parabola and go back up because 5 has multiplicity 2 so it's like a parabola there it's going to go up to a peak don't ask me right now where the peak is we're just drawing a rough sketch and it's going to come down and it's going to go through the origin because that had multiplicity 1 now we said the domain was restricted so I'm going to throw this portion of the graph away and I'm going to throw this portion of the graph away and what I have is my volume function which goes up to a peak, comes down but over on this side it levels off now you might guess that peak is exactly in the middle but it isn't. If I go to the graphing calculator I'll just graph this here and we'll see what we can learn about this function so I'm going to go back to y equals I'm going to enter x times 10 minus 2 x closed parenthesis squared there's my function I'm going to choose my window to be let's say we go from negative 1 to oops let's say excuse me 0 let's say we go from 0 to 5 and for the maximum minimum y, let's say we go to 0 and let's say we go up to about 250 and for my y scale let's say we're going to scale it every 50 units okay here's the graph this is like what I've drawn on the screen you notice it levels off over here at 5 and it comes in at 0 so where's that peak it looks like it's not in the middle it's closer to 0 so if I go to trace I'll just trace over and I'm going to calculate roughly where that peak is it's right about in here and it looks like I'm getting somewhere up around 74 maybe 75 74, 74, 73 I'm looking at this number so it looks like the maximum volume is up around 74 cubic units so I was able to get a rough sketch of the graph I went to a graphing calculator to find out where that peak was when you take calculus you'll find out more about how to determine those peaks thank you very much and I'll see you for the next episode