 The domain of a polynomial function is all real numbers, so we can put in larger and larger values of x. The end behavior is a description of what happens to the function values as x gets larger. It will be helpful to separate sine and magnitude. The sine of a number is either positive or negative. Meanwhile, the magnitude is either very large, far away from zero, or small, close to zero. And a useful idea to keep in mind is go big or go home. So, for example, a number like minus 30 trillion is negative and large. Meanwhile, a number like this is positive and small. And a number like 15, well, this isn't really worth talking about when we're dealing with end behavior. If we consider increasingly large but positive values of x, for example, 1,000, or 100,000, or 1, trivia check, quadrillion, we say that we are letting x go to infinity, and we write it as x arrow infinity. We can also consider increasingly large but negative values of x, like minus 1,000, or minus 100,000, or minus a whole lot, and we would write x goes to minus infinity. Now, to avoid getting bogged down in the details of the computations, we can focus on the sine and magnitude. And we might make a couple of observations here. First, we note that a big number raised to any power is a big number. A negative number raised to an odd power is negative, while if it's raised to an even power, it's going to be positive. And finally, a positive number raised to any power is positive. For example, let f of x equals negative 6x squared, and let's describe what happens to f of x as x goes to infinity and as x goes to minus infinity. Now, we'll do this two ways. First, we'll use some actual numbers. So as x goes to infinity, we consider larger and larger positive values of x. So for example, if x equals 1,000, we find that f of 1,000 is equal to... and if x equals 1 million, we find f of 1 million is equal to... and so it seems that as x goes to infinity, f of x becomes larger but negative. And so we say f of x goes to negative infinity. Next, let's consider what happens as x goes to negative infinity. So again, we consider larger and larger negative values of x, so we might take x equals negative 1,000, or x equals negative 1 million, and we find these values, and it seems that as x goes to negative infinity, f of x also goes to negative infinity. Now, while we could do these computations with actual numbers, a better way is to focus on the sine, positive or negative, and the magnitude, small or large. So as x goes to infinity, remember this means that x is becoming large and positive, then x squared also goes to infinity. And that's because the square of a large number is a large number and the square of a positive number is a positive number. Now, we actually want to know what happens to negative 6x squared and that's going to go to negative infinity because a negative number times a positive number is a negative number. And finally, equals means replaceable. And since f of x equals negative 6x squared, negative 6x squared is going to minus infinity, so f of x is going to minus infinity. And similarly, as x goes to negative infinity, x squared goes to infinity because again, while the square of a large number is a large number, the square of a negative number is a positive number. And again, negative 6x squared will go to negative infinity because that's a negative number times a positive number, giving us a negative number. And finally, equals means replaceable. And f of x, negative 6x squared, they're the same thing. And so f of x will also go to negative infinity. Now, if we're dealing with a more complicated polynomial, we're going to have to worry about the powers on x. And so this raises an important question as x goes to infinity, which is larger, something like 1 million x or x squared. Well, let's let x go to infinity and consider increasingly larger values of x. And so we'll check out some values. And we'll put in some values for x and then compare the values of 1 million x to x squared. So we might see what happens when x equals 10, except 10 is not a large value. So let's go larger, about 100. Well, 100 is not really a large value either. Let's go larger, about 1,000. And again, 1,000 is not really a large value. Let's go 1 million. Well, that's a somewhat large value. And so notice that while 1 million x starts out much larger than x squared, as x gets larger, x squared begins to catch up. And remember, x is going to infinity. So x just keeps getting larger. So if x is 1 trillion, then 10x will be and x squared will be. And we see that eventually x squared becomes larger than 1 million x. And since we're dealing with x going to infinity, eventually is all that really matters. This leads to the following idea. Given a polynomial, the dominant term is the term with the highest power. And the important idea here is that it is the only term that affects end behavior. We can ignore everything except for that dominant term. So for example, let's identify the dominant term in the polynomial and then describe what happens to f of x as x goes to positive infinity. So we look and we identify the highest degree term. So this is a fifth degree term. Here's a sixth. And here's a second degree term. And so the highest degree term is negative x to the sixth. So next, we see what happens as x goes to infinity. As x goes to infinity, x to the sixth goes to infinity. And negative x to the sixth goes to negative infinity. And so f of x goes to negative infinity.