Intro to end behavior of polynomials

What I want to do in this video is talk a little bit about polynomial end behavior. And this is really just talking about what happens to a polynomial if as x becomes really large or really, really, really negative. For example, we're familiar with quadratic polynomials where y is equal to ax squared plus bx plus c. We know that if a is greater than 0, this is going to be an upward opening parabola of some kind. So it's going to look something like that, the graph of this equation, or of this function, you could say. And if a is less than 0, it's going to be a downward opening parabola. We've spent less time with third degree polynomials, but we've also seen those a little bit. So for example, if you have the third degree polynomial, y is equal to ax to the third plus bx squared plus cx plus d, if a is greater than 0-- I don't want to use that brown color. If a is greater than 0, when x is really, really, really negative, this whole thing is going to be really, really, really negative. And then it's going to increase as x becomes less negative. It's going do something-- it might do a little bit of funky stuff in between. But then as x becomes more and more and more positive, it will become more and more and more positive as well. So it might look something like this when a is greater than 0. But what about when a is less than 0? Well then, just like here, we would flip it. We would flip it so that if a is less than 0, when x is really negative, you're going to multiply that times a negative a and you're going to get a positive value. So it's going to look something like this. And then it's going to go like this. It might do a little bit of this type of business in between. But then its end behavior, it starts decreasing again. It starts decreasing. So when we talk about end behavior, we're talking about the idea of what is this function? What does this polynomial do as x becomes really, really, really, really positive and as x becomes really, really, really, really negative? And kind of fully recognizing that some weird things might be happening in the middle. But we just want to think about what happens at extreme values of x. Now obviously for the second degree polynomial nothing really weird happens in the middle. But for a third degree polynomial, we sort of see that some interesting things can start happening in the middle. But the end behavior for third degree polynomial is that if a is greater than 0-- we're starting really small, really low values-- and as a becomes positive, we get to really high values. If a is less than 0 we have the opposite. And these are kind of the two prototypes for polynomials. Because from there we can start thinking about any degree polynomial. So let's just think about the situation of a fourth degree polynomial. So let's say y is equal to ax to the fourth power plus bx to the third plus cx squared plus dx plus-- I don't want to write e because e has other meanings in mathematics. I'll say plus-- I'm really running out of letters here. I'll just use f, although this isn't the function f. This is just a constant f right over here. So let's just think about what this might look like. Let's think about its end behavior, and we could think about it relative to a second degree polynomial. So its end behavior, if x is really, really, really, really negative, x to the fourth is still going to be positive. And if a is greater than 0 when x is really, really, really negative, we're going to have really, really positive values, just like a second degree. And when x is really positive, same thing. x to the fourth is going to be positive, times a is still going to be positive. So its end behavior is going to look very similar to a second degree polynomial. Now, it might do-- in fact it probably will do some funky stuff in between. It might do something that looks kind of like that in between. But we care about the end behavior. I guess you could call the stuff that I've dotted lined in the middle, this is called the non-end behavior, the middle behavior. This will obviously be different than a second degree polynomial. But what happens at the ends will be the same. And the reason why, when you square something, or you raise something to the fourth power, you raise anything to any even power for a very large-- as long as a is greater than 0, for very large positive values, you're going to get positive values. And for very large negative values, you're going to get very large positive values. You take a negative number, raise it to the fourth power, or the second power, you're going to get a positive value. Likewise, if a is less than 0, you're going to have very similar end behavior to this case. For a polynomial where the highest degree term is even-- so this is a is less than 0-- your end behavior when a is really, really, really, really negative, this thing is going to be really, really, really positive. We're going to be multiplying it times a negative, so it's going to be really, really, really, really negative. So it'll look like this. And likewise, when x is really, really, really positive, you get the same thing. Because you're going to be multiplying a positive times a, which is negative, and in between it might be doing something like that. But its end behavior, you see, is very similar to a second degree polynomial. So if you ignore this, its end behavior is very similar. Now the same is true for a fifth degree if you were to compare it to a third degree. And the overall idea here is what happens to this value when we get really large x's or really small x's? Are we taking it to an even power? In which case for either really negative values or really positive values we're going to get positive values. And then it depends what our coefficient a is. Or are we taking it to an odd power? So the general idea-- and actually let me just do a fifth degree, just to make the clear. So if I had something of the form y is equal to ax to the fifth plus bx to the fourth plus-- and it just went all the way-- I won't even have to write it. This thing, if a is greater than 0, would look something like this. Its end behavior is very similar to a third degree polynomial where a is greater than 0. At the end it would do this. Now, it might do kind of some craziness like this. I have to get this right. So 1, 2, 3-- it might do some craziness like this in between. But then for really large x's it will look the same as ax to the third when a is greater than 0. So once again, very, very similar end behavior when a is greater than 0, and very similar end behavior when a is less than 0. It would look like this. At the ends at a negative value it will be positive because this part is going to be really negative. But then it's going to be multiplied by a negative to get a positive. And for really positive values of x, it will be negative. Because once again, this a term is going to be negative. And then what it does in between-- at least for the sake of this video-- we're not really thinking about. So the big takeaway here-- and this is kind of a little bit of a drum roll here, we're talking about end behavior-- if you're looking at an even degree polynomial it's going to have end behavior like a second degree polynomial. If you ignore what happens in the middle, what happens at really negative values of x and really positive values of x is going to be very similar to a second degree polynomial. And if your degree is odd, you're going to have very similar end behavior to a third degree polynomial. You might do all sorts of craziness in the middle, but given for a given a, whether it's greater than 0 or less than 0, you will have end behavior like this, or end behavior like that.