 Recall the notion of end behavior. So the end behavior of a function is the behavior that the function exhibits as it goes to the end of its domain. Now for your standard function, if your domain is gonna be all real numbers, negative infinity to infinity, end behavior means we're asking the question, well, what happens as x goes towards infinity and what happens as x approaches negative infinity? So we go to the far right of the graph and the far left of the graph. What is the y coordinate doing as these extremes happen? Now, for a polynomial function, it turns out that for large values of x, and I'm here talking about the absolute value of x. So whether we go to the far right or the far left, direction doesn't matter. A polynomial graph will resemble its leading term, which notice the leading term, of course, is just a monomial. So if you have a polynomial f of x, as the absolute value of x approaches infinity, it gets large, whether that's to positive infinity or negative infinity, I don't care. As the absolute value of x approaches infinity, our polynomial will be approximately the same thing as its leading term, a times x to the end, which this, of course, is a monomial. So the end behavior of a polynomial will just resemble the end behavior of a monomial. And we've seen some possibilities as we studied monomials in the previous lecture. So for example, if we have an even monomial, the basic shape is gonna look something like this or something like this. And this will be dependent upon, you know, which direction is our polynomial concaving. And so if you have an even, you're gonna either have that as x approaches plus or minus infinity, you're gonna have that y approaches infinity. So you can have it at the function points up on both sides. The other possibility is x approaches plus or minus infinity, y is gonna approach negative infinity. So those are the possibilities when it comes to an even monomial, either both sides point up or both sides point down. And this will be dependent upon the leading term. I should say the leading coefficient. When the leading coefficient is positive, that means there's no reflection that happened to the monomial. And so the standard monomial will concave up. On the other hand, if your leading coefficient is negative, that means there's been a reflection across the x-axis. And therefore your monomial, your leading term will be pointing down. And then the general behavior of your polynomial will also be pointing down. That's if you have an even degree polynomial, like two, four, six. On the other hand, if your monomial has odd degree, then it has one of two pictures. You either get a picture that looks something like this or you're gonna get a picture that looks something like this. These are your two possibilities. And now in the first possibility, as x goes to infinity, then y will go to infinity as well. And as x goes to negative infinity, then y will go to negative infinity as well. This occurs when you have an odd monomial and your leading coefficient is positive. Because this shape right here is the standard behavior of an odd monomial. Nothing changed, so it's gonna be pointing up on the right and pointing down the left. Now, if your leading coefficient were negative, that means there would be some type of reflection across the x-axis that happened, which is gonna change the end behavior. Now, as x approaches infinity, y is gonna approach negative infinity. So it points down on the right-hand side. And as x approaches negative infinity, y will approach positive infinity. So as x points to the left, y is gonna point up. And these are the only four possibilities you get when it comes to the end behavior of a polynomial. It'll depend upon the end behavior, the end behavior that is dependent solely upon the leading term. So once we identify the leading term, we just wanna figure out what's the degree even or odd, and then look at the leading coefficient as a positive or negative. Those two bits of information is all that we need, is the leading coefficient positive or negative, and is the degree positive or is it even or odd? That's what we need to figure out. So let's look at some examples. Let's determine the end behavior of the function f of x equals negative two x to the fourth plus five x cube plus four x minus seven. Now, like we said on the previous slide, the end behavior is determined entirely by the leading term. The leading term is the power of x that is largest. And so that's gonna be a negative two x to the fourth. Make sure you grab the sign. So as the absolute value of x approaches infinity, we see that f of x will be approximately the same thing as negative two x to the fourth. So we see that the degree of this polynomial is even. So this tells us that we're either gonna point up on the left and right hand side, or we're gonna point down on the left and right hand side. It's either both up or both down. Now, but also we see that the leading coefficient here is negative. And so that tells us we're gonna point down in both situations. So the end behavior is gonna look something like the following, or as you see graphed over here, it'll point down on the right hand side. So as x approaches infinity, y will approach negative infinity. And as x approaches negative infinity, y will approach negative infinity as well. So the graph will point down on both sides. And this is typical for even monomials either point both up or point both down. Even monomials will always point in the same direction. It's the odd ones that'll flip flop. Let's look at another example. The polynomial g of x here is given as 2x squared minus 5x cubed plus 3x to the fifth. We need to identify the leading term, which in this case is gonna be 3x to the fifth. The leading term does not have to be the first term listed in a polynomial, it's gonna be the biggest term listed. And so as x, the absolute value of x approaches infinity, g of x will be approximately the same thing as 3x to the fifth. And the reason why we're saying that g of x is approximately the same thing as 3x to the fifth is that as x gets really big in absolute value, these other terms are gonna become peanuts in comparison to 3x to the fifth. It's kind of like saying, I'm gonna go take a bucket of water, maybe have like one of those five gallon buckets you get at the hardware store, and you fill it with water then you throw it in the ocean. Did you stop anything right there? Or the other hand, we're gonna take buckets of water out of the ocean to stop the concern of coastal rising, right? Well, if I go take my five gallon bucket and scoop out the ocean, you're not gonna notice any difference in the ocean. Even if I go take a dump truck and you dump like a truck of water into the ocean, or you take it out, you're not gonna see any difference. The ocean is just so much bigger compared to those other measurements. It's gonna be peanuts in comparison. And that's the idea right here that as x gets bigger and bigger and bigger and bigger, sure, x squared might be fast growing, x cubed is even faster growing, but comparing x squared to x to the fifth is like comparing the size of an elephant with the size of the moon. The order of magnitude is so much different that the other terms are insignificant in comparison to that leading term. So as x gets big, the function will look like three x to the fifth, for which we can notice that the leading coefficient here is positive and we can notice that the degree right here is odd. So since the degree is odd, it's either gonna point up on the right and down on the left, or it's gonna point up on the left and down at the right. The standard picture is pointing up on the right. Therefore, since we're positive, no reflection happens, that's the picture we would expect. And that's what we see over here. The graph will point up on the right and down on the left. So we see that as x approaches infinity, y will approach infinity. And as x approaches negative infinity, y will approach negative infinity as well. And this is the type of thing we're trying to describe when we talk about in behavior. Now in behaviors only describing what happens on the end of the graph, what happens on the far right and what happens on the far left. But what happens in the middle? Well, it turns out that the moment we don't know, it could do a lot of different things. It could do something funky like this, right? Maybe that's a possibility. We're gonna fill in the gaps at a later moment. Be aware that for the moment, we only care about what's happening on the ends, the far right and the far left. Let's take a look at one more example. This time let h of x equal three x squared times x minus one times x plus four cubed. Now in this situation, the leading term is not exactly apparent because the polynomial is factored. I want to point out to you that the factored form is actually a gift because in order to determine some of the behavior in the middle of the graph, we're gonna need to know the factored form. We wanna know things like x intercepts and such. So the factorization is gonna be a benefit to us. Finding the leading term is not gonna be such a big deal. Basically, we're gonna take the biggest powers that we can see and multiply them together. So the leading term of h, this is gonna be three x squared. The first factor gives us that. The second factor gives us an x. Make sure you grab the coefficient here. The final factor, it's an x plus four cubed. So if you were to multiply that out, it's gonna give us an x cubed. So if you were to multiply this thing all the way out, you're gonna get this three x squared times x times x cubed. Putting that together, we get three x to the sixth because we can add together all the powers. Two plus one plus three is the sixth power. And then the coefficient is three, in particular it's a positive three. So we see that the leading coefficient is gonna be positive. It's important that we grab the leading coefficient and then the exponent is even. So that tells us that because it's even, it's either gonna point both up or it's gonna point both down. And because it's a positive leading coefficient, no reflection happened. And so by default, the right-hand side should be pointing up. So we see the picture we have here on the right. As x approaches infinity, y will approach infinity. And as x approaches negative infinity, the left-hand side, y will also approach infinity. And so we now see that in behavior, you might have to multiply out the polynomial. You just look for the biggest terms in each of the factors. Remember the exponents as well and the coefficients.