Algebra II unit 10 Lesson 1.5

Hello ladies and gentlemen, mister Wagner here to take you through unit ten lesson1.5, all right? No video from Kirk, so I'm going to take you through this one. I hope everyone was able to find it okay in the teacher tube account and let's get this thing going. So today we're going to talk about end behavior and polynomials. So let's take a look at the first thing we have here. So we all know polynomials are functions that have this form right here. Now this is a fancy way of writing it, right? X to the N plus B, X to the N minus one. So on and so forth. But all the powers are non-negative integers. So here's some examples down here. One, two, and three. Take a look at those. By now we should all know that those are powering the wheels. Now today, okay, we're going to be talking about their end behavior, okay? So far we've talked about their turning points and zeros a lot right where they're increasing decreasing so on and so forth. Today we're going to talk about N behavior. An N behavior is pretty much exactly what you think it was. So let's read the formal definition. The M behavior of a function is how it acts graphically as X gets very large, either positive or negative. Symbolically, now this is how we write it notation, it is how F of X, which is really our Y value, right? This is really why. Behaves as X goes towards infinity, or X goes to negative infinity. By this, we're saying our X values get larger. X values get larger. And this is our X values getting smaller, right? So going towards infinity, near X value is getting really, really large. Going towards negative infinity, meaning our X values get very, very small. Okay, so let's take a look at some examples. So exercise one, it says graph the function Y equals negative one half X squared plus three X plus four, okay? So take a second, look at that in your calculator. I don't have the cool calculator Kirk has, but I do have decimals, okay? Everyone's favorite graphing calculator. So I have it already typed in. I'm going to take a look at the graph. Let me zoom in here so I can see these numbers a little bit better. All right, so let's see. It looks like we have a turning point up here at. What's at three comma 8 and a half, okay? So let's graph that. So three comma 8.5, and let's say that's about here. Three comma E .5. Okay? Let's try to check out where the zeros are, and then we'll just kind of sketch it from there. So zeros, negative one and a little bit, and then at 7 in a little bit. So let's say about here and about there. It doesn't have to be perfect, just a sketch, right? And the function is a downward facing parabola. Which we already knew, right? Because there's this negative out in front. Okay. So now we're going to describe the end behavior of the function. All right, so this is what the function looks like or what happens to my Y values, as either my X values go towards infinity or negative infinity. Let's work with infinity first, meaning going to the right. So here are my X values. Let's see if we can make this look nice, right? Going towards infinity means going that way, right? That's my X value is getting larger towards infinity. Get rid of that. Okay? So as my X value gets larger, what happens to my Y values, guys? Well, they decrease right, see the shape of this parabola, my Y values are going down. Right? So we say, we can say a number of different things. Let's say as X gets very large, very large. Okay. The graph is pointing downwards towards negative infinity. The graph is pointing downwards. Towards. Negative infinity, right? Our values are getting smaller. So what we say notation wise, okay? We say F of X goes to negative infinity. FX is really my Y value. So my Y value is going towards negative infinity. Let's take a look at as X goes towards negative infinity. Now let's think about this. That's my X value is going to the left are getting smaller. So that's that direction, right? Okay, so what happens is my X values move to the left or get smaller. Same thing, right? It's a parabola so both sides have the same qualities. My Y value is decreasing again towards negative infinity, so I'm just going to skip all the words and I'm just going to write as X goes to negative infinity. F of X goes to negative infinity as well. Okay. Okay, so let's examine here. So we know that the N behavior of all quadratic functions, right? Quadratic meaning that there's an X squared term in there, is either to point upwards. So that would look like an upward facing parallel. That's terrible shape. And upward facing parabola. Where we're going towards infinity, or pointing downwards, AKA going towards negative infinity, right? Now, let's take a look at cubic functions. Okay, so here's two cubic polynomials, cubic meaning there's an X cubed. And the equation, right? Okay, here are two equations. F of X and G of X now, let's take a look at the end behavior for these cubics. Okay? Well, it looks like at least for F of X, one is going down and one end is going up. For G of X, same thing when it is going up when it is going down. All right, so how can we compare that to the N behavior of quadratics? Okay? Well, we just said that the end behavior of quadratics point in the same direction. Looks like for cubic state point in opposite directions. Okay, so we're going to say the ends point in opposite well, that's a terrible P opposite. Directions. Now it says why is the end behavior of F of X? The opposite of that of G of X, right? So we already said for F of X to the left, we're going downwards to the right we're going upwards. But now for G of X to the left we're going up to the right we're going down. All right, well let's study their equations, okay? We know what changes quadratic functions, right? We just said in the last part, they're either both going up or both going down. Oops. Right? They're either both pointing up, like we have here or down. Well, we know that depends on if it's positive or negative out in front. The same thing is going to be true of a cubic, right? Take a look at F of X, we have a positive three out in front as the leading coefficient for G of X we have a negative two as our leading coefficient. All right? So the difference at the reason that the N behavior is opposite is because the leading coefficients leading coefficients not to be able to fit that. Our opposite signs of one another. Our opposite signs of one another. So just how in a well, that's not how you spell another. Guys, tough day. Okay, one another. So with quadratics, right? If there's a positive in front, both are end behaviors are going towards positive infinity. There's a negative in front. They both go shorter negative infinity. Same idea kind of with cubics in the sense that if there's a positive coefficient, we have negative infinity to the left, positive infinity to the right. But if there's a negative out in front, we're going to have positive infinity to the left and negative infinity to the right. Remember, we're studying the Y values, not the X values. That's kind of an important note to make, so let's talk about that for just a second. When we talk about N behavior, we're only talking about the Y values, right? Because the X values are always going to go towards positive infinity to the right and negative infinity to the left. Agreed. But on my Y values could be different. My Y values go to our negative infinity to the left here in F of X, but in G of X they go towards positive infinity. So we're really only talking about Y values. Okay. I want to exercise four. So now we're going to talk about a quartic function, which is just a fourth degree polynomial. I have a four, as a coefficient out in front. Okay, so they are ready sketched this quartic function for us on the graph. Let's take a second and describe the end behavior. Well, as X goes to positive infinity, remember that means going to the right, what happens to my Y value? Well, hopefully we see that my Y value goes towards positive infinity as well. It's going up. So I'm going to say F of X goes to positive infinity. Now, as X goes to negative infinity, meaning moving to the left, what happens to my Y value, again it goes up, moves towards positive infinity as well. So B says does the M behavior of the chord function resemble that of the quadratic or the cubic function more? Okay, so take a second, think about that. Pause video, maybe if you want to think about it and go back. So the answer is quadratic, right? Because both of my N behaviors match each other. We saw in the cubic the N behaviors were opposite, and the quadratic they were the same. So that's what we're going to say here. We can write it in red. We're going to say that. Quadratic. Because. Both ends point in the same direction. In the same direction. Okay. Good. So now C says using the axes in the window indicated above, which is from negative ten to ten, and then my Y values are really big right, negative 1500 to 1000. Sketch this function, okay? Why do the ends of this quartic function point in the opposite direction? Okay? So I'm going to take a second and kind of sketch this out on desmos. I'm going to pause the video. But while you're doing that, think about why this function G of X, the endpoints with point in the opposite direction of Y up here. Okay? Thank you about what we just talked about. Okay, and we're back. That probably was half a second for you guys, but was a minute or two for me. So here's my function sketched, okay? So I'm going to go ahead and sketch that out on the graph that we have. Oh, I actually missed something. Now that I'm looking at it, good thing, look at that. Okay. So I got the minus one 54. Sorry. Still really not that much different. So it looks something like that. I think if we zoom in, get a little bit of a better look. But yeah, it basically looks like a downward facing parabola, right? So let's go ahead and sketch that. Here we go. So really without getting too crazy about it, it looks something like this, right? It's okay, we're not really interested in the intercepts right now, right? We're just intercept interested in the end behavior. So obviously the end behavior is directly opposite from the original quartic function, right? So hopefully you thought about why they would be opposite, okay? And the reason is that the leading coefficient is negative, okay? Same reason that we had with the quadratic function. So we can say that the ends point in opposite directions. Opposite directions. Now remember, I'm saying they point in opposite directions, not from. Itself, right? Both of the endpoints point down here. But from this original quartic function that we had. So they pointed opposite direction because. I'm going to say the leading. Coefficient. Leading meaning the first one, right? The one that goes with the X to the highest power. Which in this case is negative one, right? That is, let's say, has. A different. Sine than the leading coefficient in F of X, then the leading coefficient, sorry I have terrible handwriting. Terrible handwriting general, but even worse on here. Coefficient in F of X, okay? So again, our leading coefficient originally was two. Now it's negative one, okay? So my sine flip, meaning that both of my endpoints are going to flip. Okay, well, and behavior is not endpoints. Okay, let's take a look at the next one. Let's kind of try to fill some of these in. So in general, we're saying from the examples that we saw, the M behavior of a polynomial is determined by two things, right? The degree, which is the highest power, so we saw quadratics have a highest power of two. Cubics three and cortex four. And the leading coefficient, either the Glen coefficient being positive or negative. Now the number doesn't matter. It could be one X squared two X squared, 50 X squared. That doesn't control and behavior at all. Just the positive or negative. Controls that. Okay? So when the degree of the polynomial is even, okay? That's our X squared X to the fourth, even 6, 8. So on and so forth, you can put it in your calculator and look. All right, the ends of the polynomial, we said point in the same direction. Right? B, when the degree of the polynomials odd, okay? That's X cubed X to the 5th you can even look at. Even just think of F of X equals X, Y equals X, just looks like this. Okay? Well, what are the en behaviors, right? The ends of the polynomials point in the opposite direction. One goes up, one goes down. Even just X to the first is a good example of that. And then C so when the leading coefficient changes sign, meaning from positive to negative, negative to positive, right? The ends of the polynomial change direction. And that's true no matter if the degree is even or odd. If the sign changes, as we saw before, the ends, I guess, the end behavior of the polynomial is going to change direction. Okay? So exercise 6 I have these two graphs. So what I want you guys all to do is try exercise 6 a and B on your own. Okay? That'll be like your little part of the homework try these two out and then if you have questions about it tomorrow and click can certainly go over it with your teacher. All right, so I hope that was helpful. I hope I was close to as good as Kirk, even though my handwriting is terrible, but I hope you guys are enjoying your break and I'll see you guys when you get back.