 In this video, we'll talk about graphing polynomial functions and talk about end behaviors. End behaviors are what happens to the graph at very small and very large values of x. So as x goes to negative infinity, they're getting very, very small. And if we look at this graph, as it goes this way, the y values are going to positive infinity. And as x is getting very, very large, we see that if we were to hit our graph, y would also be going to positive infinity. We would say this is what we call an up, up. I mean, we've talked about that before, but now we know that they're both headed toward positive infinity. So we want to describe the end behaviors here. And we have, as x gets very small over here, we can see that it is getting, the y values are getting very large. So y is going to go toward positive infinity. And as we go toward a large x's, we have to go down to hit our graph. So y would be heading toward negative infinity. And we would say that that would be an up, down, end behavior. In this case, as x gets very small, we'd have to go down to our graph. y would be going to negative infinity. And again, this is when x is going to negative infinity. And when we get very large x's, x is going toward positive infinity. Then we would have to go up on this side of our graph. So we go to positive infinity. And this is what we would call a down, up graph. So let's talk about these end behaviors. An even degree polynomial. And don't let this get this mixed up with even function, symmetry. This just means that we have a 2, a 4, a 6, an 8, a 10 as our leading term or degree. It's not the symmetry. So if a is greater than 0, as x goes to negative infinity, y goes to positive infinity. And as x goes to positive infinity, then y goes to, that's an even degree, so that should be a positive infinity. In other words, my ends go the same direction. If a is less than 0, so we go to negative infinity, well then it's going to start on the left hand side. It's going to be down at negative infinity. And when we get to big, very big x's, our y is still going to be the same going to negative infinity. So we would say the ends are the same. That's for even degree. So what about odd degree? Again, that's our leading term degree. So if a is less than 0, then we have x is going to negative infinity, so is y. And as x gets very large, then y is also getting very large. So these are the same. The direction x goes, y goes. And if a is less than 0, so you've got a negative number in front of your coefficient, leading coefficient, then you have, as it gets very small, y will be very large. As x gets very large, y will get very small. So these are opposite. So let's identify what we have here in this negative 0.25x to the fifth plus 4x to the cube minus 8. This is what we care about. We have the degree says that it's 5. A is less than 0 because it's a negative 0.25. And this means this is odd. So we're going to come up here to the odd ones. A is less than 0, then that means that I'm going to have, from the left, it's going to start up, where it goes to positive infinity, and it's going to end in negative infinity or down. Now let's turn and see how turning points affect the polynomial graphs. And that's just where the graph changes direction. So if p of x is a polynomial of degree n, then the graph will have at most, there should be a t there, at most n minus 1 turning points. It's also true that if the graph of the function p has n minus 1 turning points, then the degree of that function must be at least n. So let's determine the degree of this polynomial. So let's talk about the turning points now. Here's one turn, and here's a second turn, and here's a third turn. So there are three turns when we have three terms. And that is saying that n minus 1 is equal to 3, and minus 1 is the at most number of turning points we could have. So n must be a degree 4. We know degree 4 makes sense because it's an even, and both ends are going up. If it went one more turn, then the end would be going down, and we would no longer have an even function. So if we have a graph, and we talk about its multiplicities, we talked about that in the previous section. I want to draw a graph here before we do this exact problem. But if I have a degree 3, and remember we talked about that it had to have, if it's a degree 3, it would have to have exactly three zeros. Well, here's one for sure, but there's only one right here. But remember that multiplicities add to the degree. So we could say that then if this one has only one zero, then this one here could be two. So when it says the zero of even multiplicities bounce off, it means this one's below the x-axis. It hits the x-axis and comes right back down below the x-axis. And odd multiplicities go through. So we start over here to the left, and we're below the x-axis. We hit our zero, but we continue on above the x-axis. So that would be an odd multiplicity. So let's look here at our graph. We have a zero, and it looks like it's going to be at negative two. And if you look at it, it looks like it goes right through. We're up here, and then we're down here. So it's going to be a multiplicity of one, or we at least know it's odd. And then we have another zero at zero, and that one started below and then it ended below. So that was going to be something like two, or at least it's even. And then finally, we have one at positive two, and at positive two, it goes right through. So we would say that would be one or odd. Now if you read on your paper, it talks about these even and odd multiplicities. But it says in either case, the graph will be more compressed near the zero of larger degrees. So you see this one was kind of small here, but this one actually looks a little elongated, like it got squashed a little bit. This probably is not one. It's probably more like three or five. I'd say three maybe. And that's because it's just going to get close to zero a lot faster because we have a higher degree on it. Alright, going to the next one then. Let's look at these. We have, where are our zeros? They are at negative two, and they are at, it looks like three, and at five. And let's talk about even or odd. At negative two, it's going to be even. At three, it goes through the x-axis, so that would be odd. And then at five, again, it starts above and then stays above, so it's going to be even. And let's give those some values. At negative two, you can see that this is definitely compressed. So I wouldn't say that's a good degree two. I'd say that's at least a degree four. And then when we do three, it definitely goes through and it looks straight. It might be a little compressed there, but not like the one before. So we'll say one. And then at five, this one looks a lot less compressed than the one before, so we'll call that two. So let's see if we can do, then, talk about attributes of these functions. And we have one on your paper that was not here, and that is the end behavior. So the end behavior for this one, I'm going to call EB, the end behavior here is down, up. So that means that the degree is going to be odd. And it also tells us about A. If it starts down in the negative x direction, then that means A is going to be greater than zero. Remember, with A is greater than zero. When x is negative, so is y. When x is positive, so is y. It goes in their multiplicity here. So the minimum degree, well, let's count. We have four, five, six, seven. So the minimum degree would be seven because of the multiplicities, multiplicities added. And the domain and range for this one, since it starts in the negative direction and ends in the positive direction, we would say that both are all reals. If we go back and look at the one before, the end behavior, EB, would be up, up. I'm just going to use here. So up, up. That means that the degree is going to be an even degree. And we've already listed the zeros and their multiplicities. And then we want to talk about the minimum degree. Well, if we looked at the turns, we have, so there's one, two, and three. So the minimum degree would be four. And that would have been if we had our one, two, three, four. It's probably more than four, but it's the minimum degree. And then the domain in this case is going to be all real. So I'm just going to use a double backer. And the range only goes down this far. It goes up forever. So it's going to go to positive infinity, but it's going to start at, and I'll call that maybe negative 4.75. We're just approximating it to positive infinity.