 In this video, I want to do some more examples of graphing polynomials without the use of a graphing calculator whatsoever, just using the polynomial and its factorization. Now the first thing I like to think of is I actually want to think of what's the leading term, what's the constant term, and so we can multiply these out in our head. If we want the leading term, we're going to take all of the biggest powers of x's and put them together. So we're going to get a 2x squared times an x times an x. That's going to be a 2x to the fourth. The middle terms, I mean they don't affect the graph as much, it doesn't make any bearing on the y-intercept or the in-behavior. If I look at the constant term here, your constant term, you're going to get a 0 because I mean after all, if you expand upon this, this really looks like 2 times x plus 0 squared. You're going to get a 0 times a negative 4 times a 1. The constant term in this situation is going to be 0. So what this tells us is that our function, its y-intercept is actually going to be an x-intercept. So we're going to go through the origin on this graph. Because the leading term is 2x to the fourth, our function is going to have the basic shape of a quartic polynomial. It kind of looks like a parabola, although it might look a little bit flatter, but who cares about the flatness because of the x-intercepts, that part will look very different. We only care about the in-behavior. So we see that on the right-hand side, it's going to go up. That is, as x approaches infinity, y will approach infinity. And on the left-hand side, it will also point up because it's an even degree monomial. As x approaches negative infinity, y will approach infinity still. So we have the in-behavior. Let's investigate the x-intercepts, the roots of the polynomial. We already mentioned that x equals 0 was a root that came from the 2x squared. If you look at x minus 4, that gives us a root of positive 4, you switch the signs. And if you look at x plus 1, that gives us a root of negative 1. Now we want to focus on the multiplicities here. So we have a 2, we have a 1, we have a 1. So this tells us we have a multiplicity of 2 at 0, 1 at 4, and 1 at negative 1. So 4 and negative 1, these are odd multiplicities, which tells us that the function will cross the x-axis at 4 and negative 1. And then at 0, it'll just touch the x-axis because we have an even multiplicity. Now when graphing these things, I often use the y-intercept as sort of like a starting point to go from here. Since we don't have the, we don't have a y-intercept, there's two options. You could just pick a different point other than the, other than an x-intercept or in this case a y-intercept. So like you just pick x equals 1 and see what happens there. Or this is also a time to see what happens when you get close to the origin. So like, like we can, we can investigate what happens as x gets close to 0. When x is close to 0, f of x will look like the function where you plug a 0 in for all of the x's except in the x-squared. You're gonna get a 2x-squared times 0 minus 4 times 0 plus 1. I mentioned in the previous video that I don't, I don't usually use this technique very often. But when the function passes through the origin, it's something I do a lot because we don't have a y-intercept to use because it coincides with an x-intercept. And plugging an x equals 0 is a very painless process. So we end up with negative 8x squared. So our function is gonna look like a downward pointing parabola when we are close to the origin, right? When we're at 0, 0, it's gonna look like a downward-facing parabola. And so that information we're now, with that information we're ready to graph our function. We're gonna go through the origin at x equals 0, we have an x-intercept at 4. And we also have an x-intercept at negative 1. So we might get something like this. Notice our in behavior goes upward. So since we have a downward-facing parabola, right? We have something like the following. Let's do the left-hand side first. The parabola points down at the origin. But we have this x-intercept that we have to hit at negative 1. So somewhere between 0 and negative 1, it's gonna turn, come back up to x equals negative 1. At negative 1, it's gonna cross and go to the other side. And then we're gonna go up, up, up, up, up, up, up to match with the in behavior we expect on the left-hand side. On the right-hand side, what's happening? Well, we have this downward-facing parabola. I'm gonna try to clean this thing up a little bit. That looks a little bit better. On the right-hand side, we have this downward-facing parabola. But it's gonna have to bend back to go towards the x-intercept at 4. So we would anticipate our function to look something like the following. It's gonna have to be a turning point when you go from 0 to 4. At 4, we cross the x-axis to go up. And then we have to go up and up and up to match with our in behavior. So this is what our function's gonna look like. We see there's two turning points somewhere, but there's one turning point between negative 1 and 0. There's another turning point between 0 and 4. These are gonna be local minima. They're exact locations. We don't know where they are. Because 0 has an even multiplicity, we do know there's a local maximum at x equals 0. In terms of inflection, there's gonna have to be some inflection points right here and right here, somewhere between the local minima and 0. Where they are, I don't know. We don't really care about that right now. We just want the basic picture of this graph. Now, if we were to take a look at a computer-generated image here, you would get something like the following. Which you're like, well, that goes really, really deep there. Let's zoom out a little bit. You can see that with this one, the local minimum is actually really, really far below the function. This one, again, is drawn to scale in this situation. When we were graphing this, we didn't have enough information. Let's start plugging in lots of points into the machine to turn them out. It might not be obvious where this local minimum is. And we wouldn't be so obvious that it's so far below. We're not looking for that at this moment. We want to figure out how does the function turn and bend. We don't actually care about the exact depths of these local extrema right now. Because like I mentioned in the previous video, at this moment, we don't know how to find the extrema. We know roughly where they'll exist, but we don't know how tall or deep they're going to be. We need calculus to help us out with that. And so for that reason, we're going to be ignoring that. When I drew the picture here, I drew this one kind of matched up. But I drew this one much more shallow and far less steep than it was right here. That's okay. We just want the intuition of what's going on with this picture. We don't need a picture perfect yet. We just want the intuition. Let's do one more example here. Let's take the polynomial p of x to equal negative x cubed times x minus 1 squared. If you've been following this series, I'd actually encourage you to pause the video right now and try this example on your own to see what's going on here. Because I think you have the skills to do it. But we'll come do it in just a second. So let's graph this one, p of x equals negative x cubed times x minus 1 squared. The first thing I like to do is think of the in behavior. If we multiply this thing out, we're going to take a negative x cubed times an x squared. That'll give us a negative x to the fifth. That's our leading term. And our constant term, well, again, you'll see that this one actually is x minus 0 cubed right here. So you're going to take 0 times 1. This thing doesn't have a, well, it's 1 except it's 0. The constant term here is going to be 0. And so this does inform our picture. We're going to go through the origin. Our in behavior, because it's an odd monomial, but it's negative, this actually tells us that as x approaches infinity, normally the graph points upward, but because of the negative sign right here, we've actually reflected it downward. So as x approaches infinity, y is going to approach negative infinity. So we expect that the bottom, we'll be pointing the bottom right. On the other hand, as x approaches negative infinity, odd functions always do the opposite. So if one side is pointing down, the other side needs to be pointing up. So as x approaches negative infinity, y will approach infinity, which means that our in behavior is going to be pointing up on the left-hand side. Now let's investigate the x-intercepts. Looking at the factors, we have an x-intercept at 0 and an x-intercept at 1. So we're going to get that x equals 0 and 1 as our x-intercepts. What about their multiplicities? Well, x equals 0 came from x cubed, which shows up three times. Odd multiplicity means that our function is going to cross the x-axis at the origin. And when you have x minus 1 here, that shows up twice. So the multiplicity of 1 is going to be 2. And so because of that, our function is going to be touching the x-axis, because we have an even multiplicity. So let's mark up our x-intercepts. We're going to have 0, we need a 1. So I'm just going to put some space right here and get a 1. Think about our multiplicities here. It's nice to have a test starting point. I like to use the y-intercept. You can just pick any point if you wanted to to go from that. You can try like x equals negative 1. You could plug it into the function to see what happens. Personally, I'd like to do as little arithmetic as possible. So what I'm going to do to help the next thing to get started here is I'm going to actually figure out what happens when x goes to 0. So if there is a y-intercept other than 0, I'm going to use that y-intercept to get started connecting the dots there. If the y-intercept is 0, I'm going to figure out what happens as x approaches 0. So I'm going to take p of x here, and it's going to be approximately the same thing as negative x cubed. But for every factor other than x, we're going to plug in 0. So we get 0 minus 1 squared. This would simplify to be negative. You're going to get 0 minus 1, which is negative 1, squared is a positive 1. So you're going to get negative x cubed. That's what it's going to look like near the origin. So as we're near the origin, you're going to get something like this, negative x cubed. Well, on the left-hand side, the in behavior tells me I need to go upward, which is great. So you're going to go off towards infinity on the left. And then on the right, notice that this is crossing the x-axis at the origin. On the right, we have to connect and touch the point, x equals 1, that's an x-intercept. So at some point, we're going to have to bend upwards and touch the x-axis, but not cross it, because it's an even multiplicity. So we're going to touch the x-axis and then come back down, pointing to the bottom right, which would then match up with the in behavior we expect over here. So you see something like the following graph. It was up on the right-hand side, came down, bends back upward, and it comes back down like this. This is the basic shape of our function. If we remove the x and y-axis there. Let's take a look at the computer generated image. We see something like this. Well, unlike the last example, we had a really deep y-intercept, our local minimum. Here, the local minimum is really shallow. It barely, barely dips down. I mean, it does, we can zoom in, of course. It barely dips below at all. And so again, our point is not to determine where the, how deep or tall the local maxima or minima are going to be. Our point is not to determine where the points of inflection are. Just using behavior near the intercepts and the in behavior to give us this picture of the polynomial. And it does, we're doing really good. We have enough of information about the graphic here from this behavior right here. And so this is how we're gonna be graphing polynomials in this unit. Like I said, is it perfect? No, but without the final piece called calculus, we actually can't improve this without just calculating 10,000 points. And that's not what we wanna do. We don't want to graph a function just by connecting a million points together. That's computationally very inefficient. What we're able to do is with very little effort, we can get a fairly accurate picture of these polynomials. And that's the type of polynomial graph we wanna see in this chapter.