 This video will talk about polynomial functions and the attributes of their graphs. So here we have a function, they want us to say to the degree, figure out what the N behavior would be, and tell them what the Y intercept is. So degree. Remember add those exponents and that will tell you the degree of this polynomial. So one plus three plus one is going to be five and then they want to know the N behavior. Well it's an odd degree and A is less than zero here because we've got this negative going on here. So that means it's going to start and then negative X's is going to start and the positive Y's are starting at positive infinity and it's going to go end being down at negative infinity. Remember these are the opposite. As X gets very small, Y gets very large. As X gets very large, Y gets very small. And then they want us to do the Y intercept. Well hopefully we've done enough of these that you remember that X equals zero. So we're really saying the opposite of zero plus one times zero minus two cubed times five times zero minus three. So we have a negative and then this is just going to be one. This is negative two cubed and this is really just negative three because five times zero is zero. So negative one times negative two cubed is going to be negative eight and then times negative three. Well negative times a negative here is going to give me positive eight times negative three which is going to be negative 24. Zero negative 24. Okay so now we have to we want to use some of this all together. Which graph matches the function? So X plus one quantity squared. Let's talk about the degree. The degree is going to be two plus one plus one to four. So that means that both ends have to go the same direction. It's a positive so this should be an up up. So A isn't going to work. B would. C doesn't because those are opposites and D would. So I only have these two that I have to worry about right now anyway. I've narrowed it down. Now what are my zeroes? My zeroes here are at negative one and it should be bouncing. And we have a negative three and that one should be going through. And then we have it at positive two and that one should be going through. T banks through and B means bounce. So at negative one it bounces right here and it doesn't bounce here. So I'm thinking it's B but let's make sure everything else works. X plus three goes through and that would be at negative three and it goes through. And at negative or positive two it goes through. So this is definitely going to be B. Let's look at this next one. The degree X I've got two plus one more is three plus two more is going to be five. And it's a A is greater than zero. So that means that we're going to have a down up situation. So down up is A and down up is C but it's not B and it's not D. So I've got these two to consider. Now we look at the zeroes. We have negative one and it's an even exponent so it's going to bounce. We've got negative three but it's an odd exponent so it goes through. And then we have two with an even exponent so it's going to bounce. Let's start at negative three. It should go through at negative three. It does for both of them. Negative one. This one looks like it went changed directions and this one looks like it bounced. So I'm thinking C. Keep going at two. It should also bounce and it doesn't bounce up here at A but it does at two down here. So we would say that this one is going to be C. Then it says assume that A is one so we don't have to worry about the A's. Write the function for the graph D in the same form, factored form. I've got that X plus three at negative three and it bounced so we'll say that's squared. And then we've got our X plus one to be that negative one and it goes through so we'll just leave it as one factor. And then we've got our X minus two at positive two and it also goes through so this would be a possible function. And I say possible because we're just thinking the minimum degree and those kinds of things. Now let's go the other way. Knowing what we have here let's sketch. So the degree here is three plus one so the degree is equal to four. A is greater than one so that means that we're going to have in behaviors that are up in both directions. The X intercepts are the zeros so let's figure those out. If I have three X minus four equals zero and X is going to be equal to four thirds. And if I have X plus one equals zero then X is going to be equal to negative one. That's my other zero or X intercept. And then the Y intercept I'll do it in a different color. But remember that's just going to be three times zero minus four times zero plus one quantity cubed. This is negative four times positive one cubed which is just one. So the Y intercept is going to be at zero negative four. And then we've done the zeros in our multiplicities. We can do that when we're graphing. So I'm going to graph. Here's my axes and I have X intercept at one and one third. So it goes somewhere around here. And I've got another X intercept at negative one. So it goes here. A Y intercept at negative four. So it goes right here. And I know I've got to start up and go through this negative one because this is an odd degree. And I have to go through my Y intercept and I also have to go through that four thirds. So start up and then go down to my graph and then it's going to go up and something like this. This is not a parabola. This is just a real rough sketch of what it looks like but this is not a parabola. Don't think that. Because of the way these were situated and then my other one was farther over it actually gave me a little bit different kind of graph.