Hey, everyone. Whenever we worked with inequalities and we graphed them, we saw that our graph could continue on to infinity, whether it be positive infinity or negative infinity. Now the graphs of polynomial functions are going to do the same thing, continue on to infinity, whether it be positive or negative. But we're no longer working with simple inequalities, so how will we determine what the graph of our polynomial is doing, going to a positive or negative infinity? Well, it might sound like it's going to be complicated, but I'm going to show you how it's actually just going to be 4 possible things that could be happening on the ends and we only need to look at one single term in our polynomial function to determine this. So let's go ahead and jump in.
Now, this behavior on either end of our graph is very creatively named the end behavior, and it's simply referring to what f of x, our graph, is doing far to the left side. So, looking at our graph as it goes far to the left, that's as x approaches negative infinity in that arrow notation, or as f of x goes far to the right side, so going all the way to positive infinity in that arrow notation as x approaches positive infinity. Let's go ahead and look at the 4 possible things that could be happening.
Now, looking at each of these, the behavior in the middle of the graph is going to be different based on the function. So here we're really only concerned with what's happening on the ends and something different could be happening in the middle, again, depending on your function, waving up and down or doing a number of things. But let's just focus on our end behavior. Now, like I said, there's only one term in our polynomial function that we need to consider, and that is going to be the first term when our function is written in standard form. So that's our leading term.
And we're going to consider 2 things; the first of which is going to be the leading coefficient of our polynomial function. Now the leading coefficient we're going to look at the sign, and if the sign is positive, then the right side of our graph is going to rise. So f of x will approach positive infinity and if it is negative, the right side is instead going to fall, so f of x will approach negative infinity. So, let's look at our graphs up here and determine whether we're dealing with a positive leading coefficient or a negative leading coefficient. So on my first one, looking at that right side, my right side is rising. That tells me that my leading coefficient is positive. Now, over here, I have my right side falling down. So that tells me that my leading coefficient is instead negative.
Now, on my third example over here, it's going up. So it's rising. That means that my leading coefficient is positive. And then lastly, I am falling over here. That tells me that my leading coefficient is negative. Now a way that I like to think about this is, if my leading coefficient is positive, that's good, it's going to rise up. And if my leading coefficient is negative, that's bad, it's going to fall back down.
Now that we've looked at that right side, what's happening on the left side? So the other thing that we're going to look at in that first term is going to be the degree of our polynomial and whether it is even or odd. So if that degree is even, then the ends are going to be the same. So that means that if one side is rising, the other one has to be rising as well. And if they are odd, the ends are instead going to do the opposite thing, have the opposite behavior.
So in my polynomial right here, if my degree is 5, that's an odd number. That tells me that my ends are going to do the opposite thing. Now something that I like to use to remember this is "odd opposite," kind of sounds a little bit similar. Odd, op if they're odd, they're going to be the opposite. So let's go ahead and go back to our examples here. So looking at that left side, here both sides are rising. They have the same behavior. That tells me that n, my degree, is even. Now, here again, they're both falling, so they are again the same, so n is again even.
Now, moving to my third possibility, they are opposite. One side is rising while the other is falling. So odd, opposite, n is odd. And then lastly, my right side over here is falling. My left side is rising. Again, opposite odd opposite. So my degree is odd. So those are my four possibilities.
Let's go ahead and look at some polynomials and try to sketch their end behavior. So looking at this first example here, f of x is equal to −4x6+x3+2. So looking at this, I only want to consider the first term. So the first thing I'm going to look at is my leading coefficient, which here is negative 4. Now that leading coefficient is negative. That's bad. It's going to fall down on that right side. And then the other thing I want to look at here is the degree. So my degree here is 6. That is an even number, so it is not odd opposite; it is even and the same, so my ends are going to have the same behavior. Now to sketch this, I'm going to consider that my right side is falling. Go ahead and indicate that. And then if my ends are the same, my left side has to be falling as well.
Now remember, we're not focused on what's happening in the middle, so you could really connect them in any way. But here I'm just going to connect them with a parabola because we're not concerned with that yet. So let's look at one final example here. Here we have f of x is equal to 2x3+x. Now again, we only want to consider our leading term, our first term in standard form, which is already in standard form. So looking at that first term, my leading coefficient is a positive 2. So since it is positive, that's good, my right side is going to rise up. And then looking at my degree, my degree is 3, which is an odd number, odd, opposite. So my ends are going to have opposite behavior. So let's go ahead and sketch this. So, again, my right side is going to rise. Let's go ahead and draw that. It's rising up. And then, my ends are opposite, so I know that the left side has to do the opposite and fall back down. And I'm just going to connect this in the middle. Remember, we're not yet concerned with that. So that's all you need to know about the end behavior of a polynomial; let's go ahead and get some practice.