Understanding Polynomial Functions and Their Characteristics

Welcome to Precalculus, video lecture number 14 on polynomial functions and models. So first definition of a polynomial. A polynomial function is one that has the following form. So you could write it as f of x equals a to the sub n times x raised to the n plus a sub n minus 1 times x to the n minus 1 plus so on and so forth up until a sub 1 times x plus a naught. where all of those a sub n, a sub n minus ones, all the way down to a sub zero or a naught are real numbers, and n is a non-negative integer. Okay, don't let this definition or the notation confuse you, okay? So let's look at just a couple examples before we dive into the actual material so you get used to looking at math definitions and not getting overwhelmed. So say I gave you the following polynomial. Say I wrote f of x equals 2x to the 5th minus x to the 4th plus 3x squared minus 7. Okay, so the notation here, a sub n, that represents the coefficient of x raised to the nth power. So if I asked you, what is a sub 5? That would be the coefficient of the term that has x raised to the 5th. power. So a sub 5 in this case would be 2. Similarly, a sub 4, that would represent the coefficient on x to the fourth, which we can see is negative 1. What about a sub 3? Do you see an x cubed in this polynomial? I don't. So a sub 3 would be 0. a sub 2, that's the coefficient on x squared. which is a 3. a sub 1, there's no x to the first, so that would just be 0. And then a 0 or a naught, your constant term is negative 7. And then notice it's just a way of referring to the coefficient. So the subscript here matches the exponent on all of the variables, okay? And the degree of the polynomial the highest degree of its terms. So in this case this polynomial would be degree 5 because that's the highest degree of the terms that are present. Okay also remember the domain of a polynomial function is all real numbers. You don't have any restrictions you can plug in any real number. Now when you're testing or checking if something is a polynomial or not the main characteristics that might be violated are the fact that the exponents here, n, has to be a non-negative integer. So what does that mean? None of the exponents can be fractions or negative numbers. No fractions or negative exponents. All right, so For the following exercises, we're going to determine whether the functions are polynomials or not. And then if they are, we're going to state the degree of the polynomial. And if not, we'll explain ourselves and say why not. So first one is f of x equals x times x plus 5. Now at first glance, this might look a little strange, but we can just distribute the x. And so now we have f of x equals x squared plus 5x. And the main thing I'm checking now are the exponents here on the variable. None of them are negative. They're both integers. So yes, this looks like a polynomial. Yes, it's a polynomial. Hooray! And then now we're going to state the degree. So remember the degree is the degree of the highest term or the power of x on the highest term. So the degree in this case is 2. Very good. Moving on. Next one we have is f of x equals the square root of x times the square root of x minus 2. Now, at first glance, notice the fact that square root of x is the same as x to the one-half power, right? And we know that's definitely not okay for a polynomial. This is not a polynomial because all of the exponents need to be integers. So this is not an integer exponent. Another issue is, remember we've talked about finding domains for RadX? and x has to be greater than or equal to zero. So there's restrictions on the domain anytime you have an even index for a radical function, and that's a problem for a polynomial because remember the domain of a polynomial is all real numbers. So again, you could tell, hey, something's up. This is not going to be a polynomial. All right, example c. f of x equals x squared minus 5x over x cubed. What do you think here? Well, two ways to go about it, but the answer is no. First of all, I notice we've got a restriction here, x cannot be zero, and that contradicts the condition for polynomials because the domain for polynomial is all real numbers. Now, why is that? Notice I can rewrite f of x as x squared over x cubed minus 5x over x cubed. And then if I simplify x squared over x cubed, that's 1 over x. That's the same as x to the negative first minus the second term is going to be 5x to the negative second. And we know that's a big no-no for a polynomial. No negative exponents are allowed. All right, last one. f of x equals negative 3x squared times x plus 5 cubed. All right, it looks good to me so far. I don't see any problems with the exponents. I don't see any fractions, no negative exponents. Now let's imagine if we were to multiply it out because we would have negative 3x squared and then x plus 5 cubed. You don't have to actually cube it, but you know that if you did, you would have x cubed plus blah, blah, blah, blah, blah, a bunch of other terms. Okay, so it looks like it's going to be a polynomial. What's the degree? What would be the exponent on x? What would be the highest degree term? If we were to multiply it out, you don't need to. So if I took negative 3x squared and multiplied by x cubed, I'd have negative 3x to the fifth plus a bunch of smaller degree terms. So the degree is five. So in this case, you could just add these exponents together, right? 3 plus 2, and there's the degree. All right, very good. Now we're going to move on and talk about power functions. So a power function of degree n is a monomial. So it only has one term, this power function. And it has the form f of x equals a times x to the n. So a can't be 0. It's a real number. n is a positive integer. Now if n is even, then the following are true about the power functions. So the graph is symmetric about the y-axis, the domain is all real numbers, the range is going to be 0 to infinity if a is positive, and negative infinity to 0 if a is negative, and the graph resembles the graph of y equals x squared, a parabola. So basically, if n is even, The power function either looks like a parabola opening upward or a parabola opening downward, depending on whether or not a is positive or a is negative. Now, if it's x to the fourth, that would increase more quickly. So what happens is it's just a little bit skinnier than the parabola x squared, but same idea. Okay, so they all have the same basic shape. x to the eighth, x to the 100th, so on and so forth. Obviously, it's going to have symmetry over the y-axis because if they resemble a parabola, which is an even function, they have symmetry over the y-axis as well. Now, if n is odd, then the following are true about power functions. So the graph is symmetric over the origin. Both the domain and range are all real numbers, and the graph resembles the graph of y equals x cubed. So hopefully you remember what x cubed looks like. This guy. So x to the fifth would also look the same. It would just be more skinny, because it's increasing and decreasing more rapidly. x to the 43rd, something like this. Okay, so they all have the same basic shape. That would be if a is positive. right and n is odd and then if a is negative and n is odd then It would look like the reflected version of x cubed. Something like that. Okay, I'll label up above two. This first one's if a is greater than zero, and this one's if a is less than zero. Very nice. So you can use the same key points that you use for x squared and x cubed when you're dealing with power functions, which you'll see in the following two examples. OK, so it asks us here, use transformations to graph each function. So we have f of x equals negative x to the fourth plus three. So what we're going to use for the parent function here is y equals x to the fourth. Alright, and I'm going to use the same key points for x to the fourth that I do for x squared because it's an even exponent. And those key points are negative 1, 1, 0, 0, and 1, 1. And remember, we got those back when we studied families of functions. If you need to review that video, I'll link it up here. Okay, so let's list out what the transformations are. I see two of them, do you? So first thing we need to do is reflect over the x-axis. And then the second thing we need to do is shift up three units. All right, so to reflect over the x-axis, I need to multiply y by negative 1. I'm not going to touch the x-coordinates. They stay the same. So they're going to stay negative 1, 0, 1. And then now my new y-coordinates are going to be negative 1, 0, and 1. And then I need to shift up 3 units. So I'm going to add 3 to y. So we have negative 1, 0, 1. And then my new y-coordinates are going to be 2, 3, 2. Okay, and that's it. Those are the final key points. So it will look just like a parabola if you're only using the three key points, x squared. You'll notice that it's different if you add more points. Okay, so if you were to add in maybe negative 2, negative 2 to the 4th is 16. So negative 16 plus 3 would be negative 13, all the way down here, where x squared would not decrease so quickly. So you don't have to include all these extra points, but just so you know, it's not identical to a parabola, it just resembles the shape, and we can use the same key points. This is extra bonus stuff. Okay, so let's list the domain and range. Why not? The domain is all real numbers from negative infinity to infinity. It is a polynomial after all. And then the range is gonna go from negative infinity, it goes down forever, up until 3. And I'm gonna include 3. That's the highest y-value that I see on the graph up here. All right, nice. Let's look at another example here. This time we have y equals or f of x equals one half times x minus one to the fifth minus two. So the parent function that I'm going to use is y equals x to the fifth. And since that's odd, I'm going to use the same key points as x cubed. So negative one, negative one, zero, zero, and one, one. These are the same key points as y equals x cubed. Okay, so what transformations do we see going on? I see one, two, three transformations. First one is we're going to compress by a half. So I'm going to multiply. y by a half. We use the word stretch if it's a number bigger than one being multiplied by the function because it's making it taller or skinnier and then we say compress if it's a number less than one because it shrinks it, squishes it down. Okay what else is going on? x minus one that means we're going to shift to the right one. And then the minus 2 means we're going to shift down 2. All right, let's get to it. So we're going to compress by a half, so let's multiply y. by a half. So x's are not going to change for now. I'll just leave them negative 1, 0, 1. And then my new y coordinates are going to be negative 1 half, 0, and positive 1 half. Now I can go ahead and do the other two transformations at the same time. So I'm going to add 1 to x to shift it right 1 and subtract 2 from y. And my final key points are going to be, let's see, I'll do the x's first. So I'm adding 1, 0, 1, 2. And then if I subtract 2 from y, negative 1 half minus 2, that's negative 5 halves. Then I have negative 2 and negative 3 halves. All right, very good. So let's plot these. It's going to have the same general shape as x cubed as well. So 0, negative 5 halves, 1, negative 2, and then 2, negative 3 halves. Wow, don't draw a straight line. Just remember what the shape is. so that you draw something that would resemble x to the fifth. All right, what about the domain and range? So the domain is going to go from negative infinity to infinity, and the range is the same. If the exponent is odd, then both the domain and range are going to be all real numbers. Okay, here we go. Now, we're going to look at more involved functions. If f is a function and r is a real number for which f of r equals 0, then r is called a real 0 of f. So if x minus r to the m is a factor of a polynomial f and x minus r to the m plus 1 is not a factor, then r is called a 0 of multiplicity m of f. So watch, let me give you a random function, hopefully this will help. So say f of x equals x minus 3 squared times x plus 1. Okay, so first things first, we say r is a zero, a real zero, if f of r equals zero. Now the way this is written, hopefully you can tell already what the real zeros would be. If I were to plug in... ... positive 3 into this function, I'd get 3 minus 3 squared times 3 plus 1, which is 0 squared times 4, which is 0. So 3 is a real 0. Can you identify another one? You don't even have to test it. If you can just look, that's awesome. Negative 1, right? The zeros of the factors are the zeros of the function. Now, notice the exponents are a little different. So we have x minus 3 squared. That means 3 is a 0 in this function twice, because x minus 3 squared really means you have x minus 3 times x minus 3. So we would say that the multiplicity of this 0 is 2, because it pops up there twice. the exponent on that factor. The multiplicity of negative 1 is just 1 because the exponent is a 1. Okay, so these are all basically equivalent. If r is a real 0 of a polynomial function, then r is an x-intercept, correct? You could write this as 3, 0 is an x-intercept, negative 1, 0 is an x-intercept. And also x minus r is a factor of f. Okay, they're all the same. If one's true, the others are true. So here's an example. Form a polynomial whose real zero and degree are given. So here they've given us three real zeros, negative two, two, and three, and the degree is three. So we have to come up with what the polynomial was. So let's think about this. If one of the zeros is negative two, then what had to have been a factor of the polynomial? It would have been x plus 2, right? And then if positive 2 is a 0, what else would have been a factor? It would be x minus 2, right? Just set each of those factors equal to 0, then you should get the 0 back. And then similarly, if 3 is a 0, then x minus 3 is a factor. And they want us to form a polynomial, so we could call this f of x, or let's call it p of x, to show it's a polynomial. Just put something. Don't leave nothing there. You got to put a f of x at least. Okay, would this be degree 3 if we multiplied it all out? It sure would, because I would have x cubed as the highest power of x in the end. Okay, now when we're multiplying it out, have... some sense about you, and notice that x plus 2 times x minus 2 is going to give us a lovely difference of squares. So let's go ahead and multiply those first. So you'll have x squared minus 4, and then now it's not so bad to multiply that by x minus 3. Okay, and we'll be done in the next step. So p of x is going to equal, here we have x cubed minus 3x squared minus 4x. plus 12. voila okay good let's try another so Here we have the zeros are negative 3, negative 1, 2, and 5, and it's degree 4. So our polynomial would have the following factors. So if negative 3 is a zero, just take the opposite. You're going to have x plus 3 as a factor. If negative 1 is a zero, again, take the opposite. x plus 1 is a factor. If 2 is a zero, x minus 2 is a factor. And if... 5 is a 0, x minus 5 is a factor. And yep, that would give me a degree 4 polynomial once we distribute everything out. Okay, so here we go. We get to have a foiling party. Let's just multiply these two and then these two. So we'll have x squared plus 4x plus 3 times x squared minus 7x plus 10. And then now let's distribute some more. Now watch how I do it because it keeps things organized. So this is going to be x to the fourth. This is minus 7x cubed plus 10x squared. And then now check this out. So when I distribute 4x times x squared, that's going to give me 4x cubed. So I'm going to list it underneath the negative 7x cubed. Then distributing here, that's going to be negative 28x. squared. And then here is plus 40x. And then last one is this 3. So 3 times x squared, you betcha I'm putting the 3x squared under the other x squared. Then I have negative 21x plus 30. And the cool thing about stacking it this way is now everything's lined up, all the like terms. So You can combine them so easily. Look at this. So we've got x to the fourth minus 3x cubed. 10 minus 28, that's negative 18 plus 3 is negative 15x squared. 40 minus 21, that's going to be positive 19x plus 30. Cool. So there's that one done. Good, good. Okay, enough with forming our own polynomials. Let's go back to talking about the multiplicity. Remember, that was the exponent on the factor for a zero of a polynomial function. And if the zero is of even multiplicity, even, then that means the graph of f touches the x-axis at that zero. So what does that mean exactly? What would that look like? So say here's the x-axis, and the zero is x minus r. to the n and this is even, then here's r, the graph touches it. So it touches and bounces back up. Now if the zero is of odd multiplicity, so this time you have x minus r to the n and this is an odd number, then the graph crosses the x-axis of that zero. So what would that look like? It crosses, it goes through to the other side. So here's r, it'll go through. Okay, now the points at which a graph changes direction are called turning points and in calculus these points are called local maximum or local minimum points. What does that look like? A turning point is where it switches from increasing to decreasing and we've talked about intervals of increase and decrease before but let me show you here. So say you have a graph Maybe it's decreasing, then it switches to increasing, then it switches to decreasing again. So this graph has two turning points, namely right here where it switches from decreasing to increasing, and then another one right here where it goes back to decreasing. And in calculus, you'll call this point down here a local min, and this one up here a local max. But for now, we're just going to call them turning points, where the graph turns around. changes its mind, switches direction. So we have some interesting facts about these turning points. Whatever degree the polynomial is, it has one less than that possible maximum turning points. What the heck does that mean? So if you have a degree three polynomial, at most it'll have two turning points because three minus one is two. If you have a degree eight polynomial, At most, it'll have seven turning points because 8 minus 1 is 7. It could have less than that, but that's the maximum possible, n minus 1. And then probably the most exciting thing is that for large values of x, either positive or negative, the graph of the polynomial f of x equals a sub n x to the n all the way down to the constant term, it just resembles the graph of the power function a sub n x to the n. So the behavior of the graph of a function for large values of x, either positive or negative, is referred to as its n behavior. This is big. So what does this mean? Well, power functions, remember, only have one term. So power functions have the form f of x equals a x to the n. So it might look like this. Let's write out a few. Maybe f of x equals 2x to the fifth, or f of x equals negative 3x to the eighth. And I know this would have the shape like x cubed because it's odd. This would have the shape like a negative parabola. And what this is telling me is if you have just a polynomial, not a power function, and it has a bunch of terms in it, so a sub n x to the n plus a sub n minus 1 x to the n minus 1, all the way down to the last terms, and say it has the form f of x equals 2x to the 5th. minus x to the fourth plus x cubed minus seven. If you zoom out far enough, it's going to look the same as 2x to the fifth because the end behavior is going to be the same. So you have to zoom out and then it's going to look the same. There may be some craziness or wiggles going on in the middle caused by these terms here, but the end behavior What it does in the long run, when you zoom out, is the same as whatever the highest term does. Similarly, if you had some other function, maybe f of x equals negative 3x to the 8th plus 4x to the 6th minus x to the 8th, whatever. All of these smaller terms don't matter as much. Whatever is the degree of the polynomial, the highest powered term, that's going to control the n behavior. So it's going to resemble the power function with that same value of a and the same exponent. So there may be some crazy wiggles, I don't know, I just made this up. There might be some wiggles, but it's going to look like that in the long run if you zoom out far enough. Okay, the reason why you'll learn later on in calculus when you study limits, but pretty exciting stuff. So here we're going to look at some examples now to wrap up everything and put it all together. And for each of the polynomial functions we're going to look at here, we're going to list the real zeros and their multiplicity. Then we're going to find the x and y-intercepts, determine whether the graph crosses or touches the x-axis at each x-intercept, then determine the maximum number of turning points on the graph, determine the end behavior of the polynomial, and then sketch a graph of the polynomial. All right, here we go. So first thing, we want to find the zeros. You know what I don't like? We should find the degree first. Let's do that. Okay, what's the degree of this polynomial? So don't actually multiply it out, but if you were to, what would be the highest power on x? You can just add those exponents. It's degree 5, right? So the zeros are x equals positive one-third and x equals one. You get that by just setting each of the factors equal to zero. You can do that mentally. We don't need to see it. And then let's figure out the multiplicity for each of these. So the multiplicity for x equals one-third is going to be two, because that's the exponent on that factor. And the multiplicity for x equals one is three, because that's the exponent on the factor. All right, that's done. Now let's find the intercepts. So x-intercepts are pretty easy because we have the zeros. We have one-third zero and one zero. What about the y-intercept? That's going to be a little bit more work. So this is the x-intercept. Y-intercept, I have to substitute in zero for x. So f of 0 is 0 minus 1 third squared times 0 minus 1 cubed. 0 minus 1 third squared, that's negative 1 third squared. So that's 1 ninth positive times negative 1. So negative 1 ninth. So the y-intercept is at 0 negative 1 ninth. Okay, now determine whether the graph crosses or touches the x-axis at each x-intercept. So remember, what you need to do is look back at the multiplicity. If the multiplicity is even, then that means it's going to touch the x-axis at the x-intercept. intercept. If the multiplicity is odd, then that means it's going to cross the x-axis at that intercept. So because it's odd, it crosses. And here, because this one's even, it touches. Okay, that was pretty painless. Determine the maximum number of turning points. So remember, the max number of turning points is equal to the degree, in this case, which is 5, minus 1. Always minus 1. So the max number of turning points is 4. All right. Now we need to determine the end behavior. So remember, the n behavior is going to match the n behavior of the power function of the highest degree term. So no, don't multiply this all out. Just imagine if you did multiply it all out. Remember, we said we would have... have x to the fifth as the highest degree term, and the coefficient's just a one, right? It's not like there's a two sitting out here or something. Okay, so x to the fifth, we know that's going to look like x cubed, and there's a couple different ways. to describe n behavior. One is using some limit notation, which you're going to see in calculus, so we'll introduce it now. So you could say as x approaches positive infinity, so as x gets really really big, meaning you go this way, what does f of x or y approach? Well, f of x is also approaching positive infinity, right? It's going up. And now as x approaches negative infinity, so x is going this way, right? So you're going down the graph here. f of x is also approaching negative infinity. Now that's a very precalc way of writing it out. In calculus, what you're going to write is the limit as x approaches infinity of f of x is equal to infinity. That's that first statement. And then you would say the limit as x approaches negative infinity of f of x is negative infinity. Okay, good. And then now let's sketch the graph. This is the most fun part. Use the end behavior to help you. And then we have to put all the intercepts on there. So what are our intercepts? We've got 1 3rd, 1, and negative 1 9th. Okay. So label everything. Here's the x-axis, here's the y-axis. From here, one of my intercepts is at 1, so this is 1 third. And then negative 1 ninth, you just go, hey, here's negative 1 ninth, it's my graph, I say so. Okay, y-intercepts here, x-intercept is here, x-intercept is here. Now look at the end behavior. So, as I approach positive 1, I know the graph is going to be doing something like this. And then go back to the multiplicity to help you. At 1, 0, it's going to cross. So, you better cross and go through to the other side of the x-axis. Now, we don't know where it's going to turn around. Somewhere it has to turn around because we have another intercept at 1, 3. Don't stress where this is. In calculus, you'll figure it out. So in Calc, you'll figure out where it turns, where that turning point is. For now, you can live a carefree life and just turn wherever your heart desires. My goodness. We're just throwing caution to the wind. Okay, so then once you get to one-third, you're going to check what was the multiplicity there. oh it's even that means it touches so don't go through to the other side you're just gonna come back around that's what it means to touch it bounces back off and then make sure you cross through that y-intercept and the graph goes down like that okay it shouldn't look so sharp hold on let me make it more curvy Oh, that's much better. I feel better about that. Okay. There we go. All right. And then just check, does the end behavior match? It sure does. That looks great. Okay. good let's look at another one this time we have f of X equals negative 2 times x squared plus 1 cubed whoo okay so what's the degree on this guy this is a little funky The degree is actually 6, because notice here we have x squared and then that's cubed. So don't multiply it out. Please don't multiply it out. It's x squared plus 1 times x squared plus 1 times x squared plus 1, right? With a negative 2 sitting in front. So you can see this would give you x to the 6th if you were to multiply it all out. Oh my goodness. Okay. What about the zeros? Well, I would set the factors equal to zero. All of them are x squared plus one, which would mean x squared has to equal negative one. Well, that's not possible. So there's no real zeros. What does that mean for us? That means there's no x-intercepts. It's not going to cross the x-axis. Oh, okay. Do we have a y-intercept at least? Yes, we do. Don't worry. So if I plug in 0, that's going to be negative 2 times 0 squared plus 1 cubed. That's all just 1. So this is negative 2. So the y-intercept is at 0, negative 2. Okay, what about the max number of turning points? Remember, it's always the degree minus 1. So in this case, the degree is 6. minus 1, that's 5. Okay, we're blazing through this one. Now we need the n behavior. So for n behavior, we want to identify the power function that this polynomial will represent. So we know it's degree 6, but in this case, we have a coefficient of negative 2 on there. So what does that mean for us? It's going to resemble y equals negative 2x to the sixth. What would that graph look like? Well, let's think. That power function has an even exponent, so it's going to look like a parabola. But since we have that negative 2 in the front, it's opening downward. We've got a sad little parabola. And then let's write the end behavior using limit notation. So the limit As x approaches positive infinity of f of x is going to be, notice in this case, as x gets really large, the function decreases towards negative infinity. So that limits negative infinity. And also, the limit as x approaches negative infinity of f of x is going to be negative infinity. Good. So we're going to try to graph it, but hold on a second. I only have one little point to plot. Did you notice that? That's not very much to go off of. Hopefully you noticed something. We can check for symmetry, and I'm going to replace negative x with x in this function. So f of negative x, or I'm going to replace x with negative x. Sorry, I said it backwards. That's going to be negative 2. times negative x squared plus 1 cubed. And that's going to give me negative 2 times x squared plus 1 cubed, which is f of x. So that means this whole function is even. So we have y-axis symmetry. Okay, so I know it's symmetric about the y-axis. So we'll just plot the y-intercept and kind of call it a day. Do you know what I mean? So here's the x-axis, here's the y-axis. The y-intercept was at 0, negative 2. And remember, it should look like a parabola that opens downward, and it's symmetric with respect to the y-axis. That's it! We didn't talk about touching or crossing the x-axis because there were no x-intercepts. Such a disappointment, huh? What can you do? Alright, is it clear though? See, there's no x-intercepts. It's opening downward, and since the y-intercept is at 0, negative 2, there's no way it's going to cross the x-axis there. Very good. Okay, got a couple more for you. So example C, f of x equals 5x times x plus 3 cubed. So what's the degree on this one? Yes, its degree is... Why don't you pause the video for a second, see if you can figure out the zeros, the multiplicity, the x and y intercepts, the n behavior, and then I'll go through the answer and we'll put the graph together. You don't have to worry about doing that on your own just yet. Okay, did you pause it? I hope so. Don't cheat yourself of this. So let's list the zeros. Okay, there's that lonesome little 5x, so that means x equals 0 is a 0. And then x plus 3 cubed means x equals negative 3 is a 0. What's the multiplicity of each of these guys? So this is 5x to the first, so that multiplicity is 1. And then the multiplicity on negative 3 is 3. Let's get the x-intercepts. So now we're going to list them as ordered pairs. So we have 0, 0. Negative 3, 0. Is it going to touch or cross the x-axis at each of these intercepts? We'll notice the multiplicity is odd for both, so it's going to cross for both. What about the y-intercept? Well, as soon as you notice 0, 0 is an x-intercept, it's also the y-intercept, so that's done. What about the max number of turning points? Ooh, so the degree is 4. So the max number of turning points is always the degree minus 1, which is 3. Very nice. Now let's take care of the end behavior. Okay, so what power function will this polynomial resemble? So the degree is 4, but the coefficient is going to be a 5. So it's going to resemble y equals 5x to the 4th. What does that look like? That looks like a parabola opening upwards. So now I can write out the end behavior. The limit as x approaches positive infinity of f of x would equal, so imagine you're walking on the graph forever in the positive x direction, you're going up forever in the y direction. And then the limit as x approaches negative infinity of f of x is also positive infinity. Okay, so here we go. Y-axis, X-axis. We've got intercepts at 0, 0 and negative 3, 0. And that's it. And it crosses the X-axis at both. And it looks like a parabola opening upwards. So it looks something like this. Remember, where this happens, don't stress. Just turn somewhere. It doesn't matter where because you're not in calculus. In calculus, you can be stressing about finding that minimum. So enjoy your freedom in these careless days. Okay, that's it. Looks good. How do you feel about it? Pretty cute little graph, huh? Okay. example D, this is the grand finale. I hope you love this one. So we have f of x equals negative x squared times x squared minus 1 times x plus 1 cubed. You guys watch out, this one is not ready to start working with. because it's not completely factored. x squared minus 1, I can keep factoring that. Oh my goodness, what are they doing to us? So this is negative x squared. x squared minus 1 is x plus 1 times x minus 1, and then I still have that x plus 1 cubed hanging out. And see, now I have an additional x plus 1. I want to group those together. So now f of x really completely factored is negative x squared times x minus 1 times x minus 1. times x plus one to the fourth. What if you didn't do that? Would all be okay? No, it would certainly not be okay because you might think, oh, the multiplicity is three on x plus one, but really it's four. See how dangerous it could get? Okay, so let's see, what's the degree on this guy? Imagine multiplying it all the way out. 1, 2, 3, plus 4, so 7. The degree is 7. Let's list the zeros. So if x squared is there, that means x equals 0 is a 0. x equals positive 1, and x equals negative 1. What is the multiplicity of each of these zeros? So on x equals 0, look at the exponent. The multiplicity is 2. On x equals 1, look at the exponent here. It's a 1. And on x equals negative 1, the multiplicity is going to be 4. Now let's determine if it's going to cross or touch the x-axis. So cross or touch. So if the multiplicity is even, it touches. If the multiplicity is odd, it crosses. So touch, cross, touch. What about the y-intercept? Let me list the x-intercepts too. So for the touch it's 0, 0. For the cross it's 1, 0. And for the touch it's negative 1, 0. Okay, and the y-intercept, well we have it already, right? Since 0, 0 was an x-intercept. Okay, max number of turning points, it's the degree minus 1, so 6. And then now let's figure out the end behavior. Okay, so you look at f of x and you say, if I were to multiply it out, what would be the highest degree term? It would be x to the seventh. But be careful, there's also a negative in the front. So the end behavior is going to resemble y equals negative x to the seventh. So it would look like negative x cubed, basically. So let's see, the limit as x approaches positive infinity of f of x. So as you're walking on the graph in the positive x direction, you're sliding downhill, right? Whee! So that limit is going to be negative infinity. And then if you went the other way, if you look at the limit, As x approaches negative infinity of f of x, now you're going up the graph. So the function, or where you're walking, is going towards positive infinity. Okay, beautiful. So let's put it all together. Oh, we have so many intercepts. This is going to be fun. Here's the y-axis. Here's the x-axis. I only need to put negative 1, 0, and 1 on there. Okay, so negative 1, 0, 0, 0, 1, 0. And remember, it's going to resemble negative x to the 7th. So I know as I approach... Negative 1, I'm doing it this way. And then you check the multiplicity right now. That's why we did it. So at negative 1, 0, it's going to touch. So you touch it and then turn back around. Go, oh, look at negative 1. It's looking so nice. Just kidding. Turn back around. And then somewhere you're going to have to switch. So it doesn't matter where, okay? If you want to be wild, why does it keep erasing? If you want to be all wild and go all the way up and then turn around. That's fine. Okay. It's up to you. So you're going to have to turn back around and head towards 0, 0 at some point. What's going on at 0, 0? Check the multiplicity. It's touching as well. Okay. So you go, oh, here's 0, 0. Just kidding. And then now somewhere in the middle, you're going to have to turn around. You go to 1, 0. And then finally, You're going to cross at 1, 0. Okay, you go through to the other side. Now, what if you messed up? What if instead of crossing, you thought it was a touch? This is wrong. Okay, this is the problem. Because... Look what's happening. This would resemble an even function, an even power function. It would have to be like x to the eighth, one, two, three, four, five, or x to the sixth or something. So you would know something went wrong because it's supposed to resemble x to the seventh or x cubed. So if you get any of those touches or crosses wrong, you should be able to catch it. Okay. Now things look beautiful. So look at our graph. How many turning points do we have? We have 1, 2, 3, 4. It said the max number we could have is 6. Is this a problem that we only have 4? No, it's within the max. Okay? So that's it. That concludes the lesson. I hope you enjoyed it. And we have more graphing fun coming your way. Next is rational function. So we're just taking these polynomials to the next level.

Welcome to Precalculus, video lecture number 14 on polynomial functions and models. So first definition of a polynomial. A polynomial function is one that has the following form.

So you could write it as f of x equals a to the sub n times x raised to the n plus a sub n minus 1 times x to the n minus 1 plus so on and so forth up until a sub 1 times x plus a naught. where all of those a sub n, a sub n minus ones, all the way down to a sub zero or a naught are real numbers, and n is a non-negative integer. Okay, don't let this definition or the notation confuse you, okay? So let's look at just a couple examples before we dive into the actual material so you get used to looking at math definitions and not getting overwhelmed.

So say I gave you the following polynomial. Say I wrote f of x equals 2x to the 5th minus x to the 4th plus 3x squared minus 7. Okay, so the notation here, a sub n, that represents the coefficient of x raised to the nth power. So if I asked you, what is a sub 5? That would be the coefficient of the term that has x raised to the 5th.

power. So a sub 5 in this case would be 2. Similarly, a sub 4, that would represent the coefficient on x to the fourth, which we can see is negative 1. What about a sub 3? Do you see an x cubed in this polynomial? I don't.

So a sub 3 would be 0. a sub 2, that's the coefficient on x squared. which is a 3. a sub 1, there's no x to the first, so that would just be 0. And then a 0 or a naught, your constant term is negative 7. And then notice it's just a way of referring to the coefficient. So the subscript here matches the exponent on all of the variables, okay? And the degree of the polynomial the highest degree of its terms. So in this case this polynomial would be degree 5 because that's the highest degree of the terms that are present.

Okay also remember the domain of a polynomial function is all real numbers. You don't have any restrictions you can plug in any real number. Now when you're testing or checking if something is a polynomial or not the main characteristics that might be violated are the fact that the exponents here, n, has to be a non-negative integer.

So what does that mean? None of the exponents can be fractions or negative numbers. No fractions or negative exponents.

All right, so For the following exercises, we're going to determine whether the functions are polynomials or not. And then if they are, we're going to state the degree of the polynomial. And if not, we'll explain ourselves and say why not. So first one is f of x equals x times x plus 5. Now at first glance, this might look a little strange, but we can just distribute the x. And so now we have f of x equals x squared plus 5x.

And the main thing I'm checking now are the exponents here on the variable. None of them are negative. They're both integers. So yes, this looks like a polynomial.

Yes, it's a polynomial. Hooray! And then now we're going to state the degree.

So remember the degree is the degree of the highest term or the power of x on the highest term. So the degree in this case is 2. Very good. Moving on. Next one we have is f of x equals the square root of x times the square root of x minus 2. Now, at first glance, notice the fact that square root of x is the same as x to the one-half power, right? And we know that's definitely not okay for a polynomial.

This is not a polynomial because all of the exponents need to be integers. So this is not an integer exponent. Another issue is, remember we've talked about finding domains for RadX?

and x has to be greater than or equal to zero. So there's restrictions on the domain anytime you have an even index for a radical function, and that's a problem for a polynomial because remember the domain of a polynomial is all real numbers. So again, you could tell, hey, something's up.

This is not going to be a polynomial. All right, example c. f of x equals x squared minus 5x over x cubed. What do you think here? Well, two ways to go about it, but the answer is no.

First of all, I notice we've got a restriction here, x cannot be zero, and that contradicts the condition for polynomials because the domain for polynomial is all real numbers. Now, why is that? Notice I can rewrite f of x as x squared over x cubed minus 5x over x cubed.

And then if I simplify x squared over x cubed, that's 1 over x. That's the same as x to the negative first minus the second term is going to be 5x to the negative second. And we know that's a big no-no for a polynomial. No negative exponents are allowed. All right, last one.

f of x equals negative 3x squared times x plus 5 cubed. All right, it looks good to me so far. I don't see any problems with the exponents. I don't see any fractions, no negative exponents. Now let's imagine if we were to multiply it out because we would have negative 3x squared and then x plus 5 cubed.

You don't have to actually cube it, but you know that if you did, you would have x cubed plus blah, blah, blah, blah, blah, a bunch of other terms. Okay, so it looks like it's going to be a polynomial. What's the degree? What would be the exponent on x?

What would be the highest degree term? If we were to multiply it out, you don't need to. So if I took negative 3x squared and multiplied by x cubed, I'd have negative 3x to the fifth plus a bunch of smaller degree terms.

So the degree is five. So in this case, you could just add these exponents together, right? 3 plus 2, and there's the degree.

All right, very good. Now we're going to move on and talk about power functions. So a power function of degree n is a monomial.

So it only has one term, this power function. And it has the form f of x equals a times x to the n. So a can't be 0. It's a real number. n is a positive integer. Now if n is even, then the following are true about the power functions.

So the graph is symmetric about the y-axis, the domain is all real numbers, the range is going to be 0 to infinity if a is positive, and negative infinity to 0 if a is negative, and the graph resembles the graph of y equals x squared, a parabola. So basically, if n is even, The power function either looks like a parabola opening upward or a parabola opening downward, depending on whether or not a is positive or a is negative. Now, if it's x to the fourth, that would increase more quickly. So what happens is it's just a little bit skinnier than the parabola x squared, but same idea.

Okay, so they all have the same basic shape. x to the eighth, x to the 100th, so on and so forth. Obviously, it's going to have symmetry over the y-axis because if they resemble a parabola, which is an even function, they have symmetry over the y-axis as well.

Now, if n is odd, then the following are true about power functions. So the graph is symmetric over the origin. Both the domain and range are all real numbers, and the graph resembles the graph of y equals x cubed.

So hopefully you remember what x cubed looks like. This guy. So x to the fifth would also look the same.

It would just be more skinny, because it's increasing and decreasing more rapidly. x to the 43rd, something like this. Okay, so they all have the same basic shape. That would be if a is positive.

right and n is odd and then if a is negative and n is odd then It would look like the reflected version of x cubed. Something like that. Okay, I'll label up above two. This first one's if a is greater than zero, and this one's if a is less than zero. Very nice.

So you can use the same key points that you use for x squared and x cubed when you're dealing with power functions, which you'll see in the following two examples. OK, so it asks us here, use transformations to graph each function. So we have f of x equals negative x to the fourth plus three.

So what we're going to use for the parent function here is y equals x to the fourth. Alright, and I'm going to use the same key points for x to the fourth that I do for x squared because it's an even exponent. And those key points are negative 1, 1, 0, 0, and 1, 1. And remember, we got those back when we studied families of functions. If you need to review that video, I'll link it up here. Okay, so let's list out what the transformations are.

I see two of them, do you? So first thing we need to do is reflect over the x-axis. And then the second thing we need to do is shift up three units. All right, so to reflect over the x-axis, I need to multiply y by negative 1. I'm not going to touch the x-coordinates.

They stay the same. So they're going to stay negative 1, 0, 1. And then now my new y-coordinates are going to be negative 1, 0, and 1. And then I need to shift up 3 units. So I'm going to add 3 to y. So we have negative 1, 0, 1. And then my new y-coordinates are going to be 2, 3, 2. Okay, and that's it. Those are the final key points.

So it will look just like a parabola if you're only using the three key points, x squared. You'll notice that it's different if you add more points. Okay, so if you were to add in maybe negative 2, negative 2 to the 4th is 16. So negative 16 plus 3 would be negative 13, all the way down here, where x squared would not decrease so quickly.

So you don't have to include all these extra points, but just so you know, it's not identical to a parabola, it just resembles the shape, and we can use the same key points. This is extra bonus stuff. Okay, so let's list the domain and range.

Why not? The domain is all real numbers from negative infinity to infinity. It is a polynomial after all. And then the range is gonna go from negative infinity, it goes down forever, up until 3. And I'm gonna include 3. That's the highest y-value that I see on the graph up here.

All right, nice. Let's look at another example here. This time we have y equals or f of x equals one half times x minus one to the fifth minus two. So the parent function that I'm going to use is y equals x to the fifth.

And since that's odd, I'm going to use the same key points as x cubed. So negative one, negative one, zero, zero, and one, one. These are the same key points as y equals x cubed. Okay, so what transformations do we see going on? I see one, two, three transformations.

First one is we're going to compress by a half. So I'm going to multiply. y by a half.

We use the word stretch if it's a number bigger than one being multiplied by the function because it's making it taller or skinnier and then we say compress if it's a number less than one because it shrinks it, squishes it down. Okay what else is going on? x minus one that means we're going to shift to the right one.

And then the minus 2 means we're going to shift down 2. All right, let's get to it. So we're going to compress by a half, so let's multiply y. by a half. So x's are not going to change for now. I'll just leave them negative 1, 0, 1. And then my new y coordinates are going to be negative 1 half, 0, and positive 1 half.

Now I can go ahead and do the other two transformations at the same time. So I'm going to add 1 to x to shift it right 1 and subtract 2 from y. And my final key points are going to be, let's see, I'll do the x's first.

So I'm adding 1, 0, 1, 2. And then if I subtract 2 from y, negative 1 half minus 2, that's negative 5 halves. Then I have negative 2 and negative 3 halves. All right, very good.

So let's plot these. It's going to have the same general shape as x cubed as well. So 0, negative 5 halves, 1, negative 2, and then 2, negative 3 halves. Wow, don't draw a straight line. Just remember what the shape is. so that you draw something that would resemble x to the fifth.

All right, what about the domain and range? So the domain is going to go from negative infinity to infinity, and the range is the same. If the exponent is odd, then both the domain and range are going to be all real numbers.

Okay, here we go. Now, we're going to look at more involved functions. If f is a function and r is a real number for which f of r equals 0, then r is called a real 0 of f. So if x minus r to the m is a factor of a polynomial f and x minus r to the m plus 1 is not a factor, then r is called a 0 of multiplicity m of f. So watch, let me give you a random function, hopefully this will help.

So say f of x equals x minus 3 squared times x plus 1. Okay, so first things first, we say r is a zero, a real zero, if f of r equals zero. Now the way this is written, hopefully you can tell already what the real zeros would be. If I were to plug in... ... positive 3 into this function, I'd get 3 minus 3 squared times 3 plus 1, which is 0 squared times 4, which is 0. So 3 is a real 0. Can you identify another one?

You don't even have to test it. If you can just look, that's awesome. Negative 1, right? The zeros of the factors are the zeros of the function.

Now, notice the exponents are a little different. So we have x minus 3 squared. That means 3 is a 0 in this function twice, because x minus 3 squared really means you have x minus 3 times x minus 3. So we would say that the multiplicity of this 0 is 2, because it pops up there twice.

the exponent on that factor. The multiplicity of negative 1 is just 1 because the exponent is a 1. Okay, so these are all basically equivalent. If r is a real 0 of a polynomial function, then r is an x-intercept, correct? You could write this as 3, 0 is an x-intercept, negative 1, 0 is an x-intercept.

And also x minus r is a factor of f. Okay, they're all the same. If one's true, the others are true.

So here's an example. Form a polynomial whose real zero and degree are given. So here they've given us three real zeros, negative two, two, and three, and the degree is three. So we have to come up with what the polynomial was.

So let's think about this. If one of the zeros is negative two, then what had to have been a factor of the polynomial? It would have been x plus 2, right? And then if positive 2 is a 0, what else would have been a factor?

It would be x minus 2, right? Just set each of those factors equal to 0, then you should get the 0 back. And then similarly, if 3 is a 0, then x minus 3 is a factor.

And they want us to form a polynomial, so we could call this f of x, or let's call it p of x, to show it's a polynomial. Just put something. Don't leave nothing there. You got to put a f of x at least. Okay, would this be degree 3 if we multiplied it all out?

It sure would, because I would have x cubed as the highest power of x in the end. Okay, now when we're multiplying it out, have... some sense about you, and notice that x plus 2 times x minus 2 is going to give us a lovely difference of squares.

So let's go ahead and multiply those first. So you'll have x squared minus 4, and then now it's not so bad to multiply that by x minus 3. Okay, and we'll be done in the next step. So p of x is going to equal, here we have x cubed minus 3x squared minus 4x. plus 12. voila okay good let's try another so Here we have the zeros are negative 3, negative 1, 2, and 5, and it's degree 4. So our polynomial would have the following factors. So if negative 3 is a zero, just take the opposite.

You're going to have x plus 3 as a factor. If negative 1 is a zero, again, take the opposite. x plus 1 is a factor. If 2 is a zero, x minus 2 is a factor. And if...

5 is a 0, x minus 5 is a factor. And yep, that would give me a degree 4 polynomial once we distribute everything out. Okay, so here we go. We get to have a foiling party. Let's just multiply these two and then these two.

So we'll have x squared plus 4x plus 3 times x squared minus 7x plus 10. And then now let's distribute some more. Now watch how I do it because it keeps things organized. So this is going to be x to the fourth.

This is minus 7x cubed plus 10x squared. And then now check this out. So when I distribute 4x times x squared, that's going to give me 4x cubed.

So I'm going to list it underneath the negative 7x cubed. Then distributing here, that's going to be negative 28x. squared.

And then here is plus 40x. And then last one is this 3. So 3 times x squared, you betcha I'm putting the 3x squared under the other x squared. Then I have negative 21x plus 30. And the cool thing about stacking it this way is now everything's lined up, all the like terms.

So You can combine them so easily. Look at this. So we've got x to the fourth minus 3x cubed. 10 minus 28, that's negative 18 plus 3 is negative 15x squared.

40 minus 21, that's going to be positive 19x plus 30. Cool. So there's that one done. Good, good.

Okay, enough with forming our own polynomials. Let's go back to talking about the multiplicity. Remember, that was the exponent on the factor for a zero of a polynomial function. And if the zero is of even multiplicity, even, then that means the graph of f touches the x-axis at that zero.

So what does that mean exactly? What would that look like? So say here's the x-axis, and the zero is x minus r.

to the n and this is even, then here's r, the graph touches it. So it touches and bounces back up. Now if the zero is of odd multiplicity, so this time you have x minus r to the n and this is an odd number, then the graph crosses the x-axis of that zero. So what would that look like?

It crosses, it goes through to the other side. So here's r, it'll go through. Okay, now the points at which a graph changes direction are called turning points and in calculus these points are called local maximum or local minimum points.

What does that look like? A turning point is where it switches from increasing to decreasing and we've talked about intervals of increase and decrease before but let me show you here. So say you have a graph Maybe it's decreasing, then it switches to increasing, then it switches to decreasing again.

So this graph has two turning points, namely right here where it switches from decreasing to increasing, and then another one right here where it goes back to decreasing. And in calculus, you'll call this point down here a local min, and this one up here a local max. But for now, we're just going to call them turning points, where the graph turns around.

changes its mind, switches direction. So we have some interesting facts about these turning points. Whatever degree the polynomial is, it has one less than that possible maximum turning points. What the heck does that mean?

So if you have a degree three polynomial, at most it'll have two turning points because three minus one is two. If you have a degree eight polynomial, At most, it'll have seven turning points because 8 minus 1 is 7. It could have less than that, but that's the maximum possible, n minus 1. And then probably the most exciting thing is that for large values of x, either positive or negative, the graph of the polynomial f of x equals a sub n x to the n all the way down to the constant term, it just resembles the graph of the power function a sub n x to the n. So the behavior of the graph of a function for large values of x, either positive or negative, is referred to as its n behavior. This is big.

So what does this mean? Well, power functions, remember, only have one term. So power functions have the form f of x equals a x to the n.

So it might look like this. Let's write out a few. Maybe f of x equals 2x to the fifth, or f of x equals negative 3x to the eighth. And I know this would have the shape like x cubed because it's odd. This would have the shape like a negative parabola.

And what this is telling me is if you have just a polynomial, not a power function, and it has a bunch of terms in it, so a sub n x to the n plus a sub n minus 1 x to the n minus 1, all the way down to the last terms, and say it has the form f of x equals 2x to the 5th. minus x to the fourth plus x cubed minus seven. If you zoom out far enough, it's going to look the same as 2x to the fifth because the end behavior is going to be the same.

So you have to zoom out and then it's going to look the same. There may be some craziness or wiggles going on in the middle caused by these terms here, but the end behavior What it does in the long run, when you zoom out, is the same as whatever the highest term does. Similarly, if you had some other function, maybe f of x equals negative 3x to the 8th plus 4x to the 6th minus x to the 8th, whatever. All of these smaller terms don't matter as much. Whatever is the degree of the polynomial, the highest powered term, that's going to control the n behavior.

So it's going to resemble the power function with that same value of a and the same exponent. So there may be some crazy wiggles, I don't know, I just made this up. There might be some wiggles, but it's going to look like that in the long run if you zoom out far enough. Okay, the reason why you'll learn later on in calculus when you study limits, but pretty exciting stuff.

So here we're going to look at some examples now to wrap up everything and put it all together. And for each of the polynomial functions we're going to look at here, we're going to list the real zeros and their multiplicity. Then we're going to find the x and y-intercepts, determine whether the graph crosses or touches the x-axis at each x-intercept, then determine the maximum number of turning points on the graph, determine the end behavior of the polynomial, and then sketch a graph of the polynomial.

All right, here we go. So first thing, we want to find the zeros. You know what I don't like?

We should find the degree first. Let's do that. Okay, what's the degree of this polynomial?

So don't actually multiply it out, but if you were to, what would be the highest power on x? You can just add those exponents. It's degree 5, right? So the zeros are x equals positive one-third and x equals one.

You get that by just setting each of the factors equal to zero. You can do that mentally. We don't need to see it.

And then let's figure out the multiplicity for each of these. So the multiplicity for x equals one-third is going to be two, because that's the exponent on that factor. And the multiplicity for x equals one is three, because that's the exponent on the factor. All right, that's done.

Now let's find the intercepts. So x-intercepts are pretty easy because we have the zeros. We have one-third zero and one zero. What about the y-intercept? That's going to be a little bit more work.

So this is the x-intercept. Y-intercept, I have to substitute in zero for x. So f of 0 is 0 minus 1 third squared times 0 minus 1 cubed.

0 minus 1 third squared, that's negative 1 third squared. So that's 1 ninth positive times negative 1. So negative 1 ninth. So the y-intercept is at 0 negative 1 ninth. Okay, now determine whether the graph crosses or touches the x-axis at each x-intercept. So remember, what you need to do is look back at the multiplicity.

If the multiplicity is even, then that means it's going to touch the x-axis at the x-intercept. intercept. If the multiplicity is odd, then that means it's going to cross the x-axis at that intercept. So because it's odd, it crosses. And here, because this one's even, it touches.

Okay, that was pretty painless. Determine the maximum number of turning points. So remember, the max number of turning points is equal to the degree, in this case, which is 5, minus 1. Always minus 1. So the max number of turning points is 4. All right. Now we need to determine the end behavior. So remember, the n behavior is going to match the n behavior of the power function of the highest degree term.

So no, don't multiply this all out. Just imagine if you did multiply it all out. Remember, we said we would have...

have x to the fifth as the highest degree term, and the coefficient's just a one, right? It's not like there's a two sitting out here or something. Okay, so x to the fifth, we know that's going to look like x cubed, and there's a couple different ways. to describe n behavior.

One is using some limit notation, which you're going to see in calculus, so we'll introduce it now. So you could say as x approaches positive infinity, so as x gets really really big, meaning you go this way, what does f of x or y approach? Well, f of x is also approaching positive infinity, right? It's going up. And now as x approaches negative infinity, so x is going this way, right?

So you're going down the graph here. f of x is also approaching negative infinity. Now that's a very precalc way of writing it out. In calculus, what you're going to write is the limit as x approaches infinity of f of x is equal to infinity.

That's that first statement. And then you would say the limit as x approaches negative infinity of f of x is negative infinity. Okay, good. And then now let's sketch the graph.

This is the most fun part. Use the end behavior to help you. And then we have to put all the intercepts on there. So what are our intercepts?

We've got 1 3rd, 1, and negative 1 9th. Okay. So label everything. Here's the x-axis, here's the y-axis. From here, one of my intercepts is at 1, so this is 1 third.

And then negative 1 ninth, you just go, hey, here's negative 1 ninth, it's my graph, I say so. Okay, y-intercepts here, x-intercept is here, x-intercept is here. Now look at the end behavior. So, as I approach positive 1, I know the graph is going to be doing something like this. And then go back to the multiplicity to help you.

At 1, 0, it's going to cross. So, you better cross and go through to the other side of the x-axis. Now, we don't know where it's going to turn around. Somewhere it has to turn around because we have another intercept at 1, 3. Don't stress where this is. In calculus, you'll figure it out.

So in Calc, you'll figure out where it turns, where that turning point is. For now, you can live a carefree life and just turn wherever your heart desires. My goodness. We're just throwing caution to the wind. Okay, so then once you get to one-third, you're going to check what was the multiplicity there.

oh it's even that means it touches so don't go through to the other side you're just gonna come back around that's what it means to touch it bounces back off and then make sure you cross through that y-intercept and the graph goes down like that okay it shouldn't look so sharp hold on let me make it more curvy Oh, that's much better. I feel better about that. Okay. There we go. All right.

And then just check, does the end behavior match? It sure does. That looks great. Okay. good let's look at another one this time we have f of X equals negative 2 times x squared plus 1 cubed whoo okay so what's the degree on this guy this is a little funky The degree is actually 6, because notice here we have x squared and then that's cubed.

So don't multiply it out. Please don't multiply it out. It's x squared plus 1 times x squared plus 1 times x squared plus 1, right? With a negative 2 sitting in front. So you can see this would give you x to the 6th if you were to multiply it all out.

Oh my goodness. Okay. What about the zeros?

Well, I would set the factors equal to zero. All of them are x squared plus one, which would mean x squared has to equal negative one. Well, that's not possible. So there's no real zeros. What does that mean for us?

That means there's no x-intercepts. It's not going to cross the x-axis. Oh, okay. Do we have a y-intercept at least? Yes, we do.

Don't worry. So if I plug in 0, that's going to be negative 2 times 0 squared plus 1 cubed. That's all just 1. So this is negative 2. So the y-intercept is at 0, negative 2. Okay, what about the max number of turning points?

Remember, it's always the degree minus 1. So in this case, the degree is 6. minus 1, that's 5. Okay, we're blazing through this one. Now we need the n behavior. So for n behavior, we want to identify the power function that this polynomial will represent.

So we know it's degree 6, but in this case, we have a coefficient of negative 2 on there. So what does that mean for us? It's going to resemble y equals negative 2x to the sixth. What would that graph look like?

Well, let's think. That power function has an even exponent, so it's going to look like a parabola. But since we have that negative 2 in the front, it's opening downward.

We've got a sad little parabola. And then let's write the end behavior using limit notation. So the limit As x approaches positive infinity of f of x is going to be, notice in this case, as x gets really large, the function decreases towards negative infinity.

So that limits negative infinity. And also, the limit as x approaches negative infinity of f of x is going to be negative infinity. Good.

So we're going to try to graph it, but hold on a second. I only have one little point to plot. Did you notice that?

That's not very much to go off of. Hopefully you noticed something. We can check for symmetry, and I'm going to replace negative x with x in this function. So f of negative x, or I'm going to replace x with negative x. Sorry, I said it backwards.

That's going to be negative 2. times negative x squared plus 1 cubed. And that's going to give me negative 2 times x squared plus 1 cubed, which is f of x. So that means this whole function is even. So we have y-axis symmetry.

Okay, so I know it's symmetric about the y-axis. So we'll just plot the y-intercept and kind of call it a day. Do you know what I mean? So here's the x-axis, here's the y-axis. The y-intercept was at 0, negative 2. And remember, it should look like a parabola that opens downward, and it's symmetric with respect to the y-axis.

That's it! We didn't talk about touching or crossing the x-axis because there were no x-intercepts. Such a disappointment, huh? What can you do?

Alright, is it clear though? See, there's no x-intercepts. It's opening downward, and since the y-intercept is at 0, negative 2, there's no way it's going to cross the x-axis there. Very good.

Okay, got a couple more for you. So example C, f of x equals 5x times x plus 3 cubed. So what's the degree on this one? Yes, its degree is... Why don't you pause the video for a second, see if you can figure out the zeros, the multiplicity, the x and y intercepts, the n behavior, and then I'll go through the answer and we'll put the graph together.

You don't have to worry about doing that on your own just yet. Okay, did you pause it? I hope so. Don't cheat yourself of this. So let's list the zeros.

Okay, there's that lonesome little 5x, so that means x equals 0 is a 0. And then x plus 3 cubed means x equals negative 3 is a 0. What's the multiplicity of each of these guys? So this is 5x to the first, so that multiplicity is 1. And then the multiplicity on negative 3 is 3. Let's get the x-intercepts. So now we're going to list them as ordered pairs.

So we have 0, 0. Negative 3, 0. Is it going to touch or cross the x-axis at each of these intercepts? We'll notice the multiplicity is odd for both, so it's going to cross for both. What about the y-intercept?

Well, as soon as you notice 0, 0 is an x-intercept, it's also the y-intercept, so that's done. What about the max number of turning points? Ooh, so the degree is 4. So the max number of turning points is always the degree minus 1, which is 3. Very nice.

Now let's take care of the end behavior. Okay, so what power function will this polynomial resemble? So the degree is 4, but the coefficient is going to be a 5. So it's going to resemble y equals 5x to the 4th.

What does that look like? That looks like a parabola opening upwards. So now I can write out the end behavior.

The limit as x approaches positive infinity of f of x would equal, so imagine you're walking on the graph forever in the positive x direction, you're going up forever in the y direction. And then the limit as x approaches negative infinity of f of x is also positive infinity. Okay, so here we go. Y-axis, X-axis.

We've got intercepts at 0, 0 and negative 3, 0. And that's it. And it crosses the X-axis at both. And it looks like a parabola opening upwards. So it looks something like this. Remember, where this happens, don't stress.

Just turn somewhere. It doesn't matter where because you're not in calculus. In calculus, you can be stressing about finding that minimum. So enjoy your freedom in these careless days. Okay, that's it.

Looks good. How do you feel about it? Pretty cute little graph, huh?

Okay. example D, this is the grand finale. I hope you love this one.

So we have f of x equals negative x squared times x squared minus 1 times x plus 1 cubed. You guys watch out, this one is not ready to start working with. because it's not completely factored.

x squared minus 1, I can keep factoring that. Oh my goodness, what are they doing to us? So this is negative x squared.

x squared minus 1 is x plus 1 times x minus 1, and then I still have that x plus 1 cubed hanging out. And see, now I have an additional x plus 1. I want to group those together. So now f of x really completely factored is negative x squared times x minus 1 times x minus 1. times x plus one to the fourth.

What if you didn't do that? Would all be okay? No, it would certainly not be okay because you might think, oh, the multiplicity is three on x plus one, but really it's four.

See how dangerous it could get? Okay, so let's see, what's the degree on this guy? Imagine multiplying it all the way out.

1, 2, 3, plus 4, so 7. The degree is 7. Let's list the zeros. So if x squared is there, that means x equals 0 is a 0. x equals positive 1, and x equals negative 1. What is the multiplicity of each of these zeros? So on x equals 0, look at the exponent. The multiplicity is 2. On x equals 1, look at the exponent here.

It's a 1. And on x equals negative 1, the multiplicity is going to be 4. Now let's determine if it's going to cross or touch the x-axis. So cross or touch. So if the multiplicity is even, it touches.

If the multiplicity is odd, it crosses. So touch, cross, touch. What about the y-intercept? Let me list the x-intercepts too. So for the touch it's 0, 0. For the cross it's 1, 0. And for the touch it's negative 1, 0. Okay, and the y-intercept, well we have it already, right?

Since 0, 0 was an x-intercept. Okay, max number of turning points, it's the degree minus 1, so 6. And then now let's figure out the end behavior. Okay, so you look at f of x and you say, if I were to multiply it out, what would be the highest degree term? It would be x to the seventh.

But be careful, there's also a negative in the front. So the end behavior is going to resemble y equals negative x to the seventh. So it would look like negative x cubed, basically.

So let's see, the limit as x approaches positive infinity of f of x. So as you're walking on the graph in the positive x direction, you're sliding downhill, right? Whee!

So that limit is going to be negative infinity. And then if you went the other way, if you look at the limit, As x approaches negative infinity of f of x, now you're going up the graph. So the function, or where you're walking, is going towards positive infinity. Okay, beautiful.

So let's put it all together. Oh, we have so many intercepts. This is going to be fun. Here's the y-axis. Here's the x-axis.

I only need to put negative 1, 0, and 1 on there. Okay, so negative 1, 0, 0, 0, 1, 0. And remember, it's going to resemble negative x to the 7th. So I know as I approach... Negative 1, I'm doing it this way.

And then you check the multiplicity right now. That's why we did it. So at negative 1, 0, it's going to touch. So you touch it and then turn back around. Go, oh, look at negative 1. It's looking so nice.

Just kidding. Turn back around. And then somewhere you're going to have to switch.

So it doesn't matter where, okay? If you want to be wild, why does it keep erasing? If you want to be all wild and go all the way up and then turn around. That's fine. Okay.

It's up to you. So you're going to have to turn back around and head towards 0, 0 at some point. What's going on at 0, 0?

Check the multiplicity. It's touching as well. Okay.

So you go, oh, here's 0, 0. Just kidding. And then now somewhere in the middle, you're going to have to turn around. You go to 1, 0. And then finally, You're going to cross at 1, 0. Okay, you go through to the other side.

Now, what if you messed up? What if instead of crossing, you thought it was a touch? This is wrong.

Okay, this is the problem. Because... Look what's happening.

This would resemble an even function, an even power function. It would have to be like x to the eighth, one, two, three, four, five, or x to the sixth or something. So you would know something went wrong because it's supposed to resemble x to the seventh or x cubed.

So if you get any of those touches or crosses wrong, you should be able to catch it. Okay. Now things look beautiful. So look at our graph.

How many turning points do we have? We have 1, 2, 3, 4. It said the max number we could have is 6. Is this a problem that we only have 4? No, it's within the max.

Okay? So that's it. That concludes the lesson.

I hope you enjoyed it. And we have more graphing fun coming your way. Next is rational function. So we're just taking these polynomials to the next level.

Transcript for:Understanding Polynomial Functions and Their Characteristics

Transcript for:
Understanding Polynomial Functions and Their Characteristics