 So let's talk a little bit more about graphing from the derivative. So again, one possibility here, I may have some graph that I'm interested in, some function y equals f of x, and somehow I've obtained the derivative. I've either found it by using the derivative rules, or again, in many, many, many, many, many situations, what we can acquire information about usually often corresponds to information about the derivative, and again, what we are interested in is information about the function. So maybe I have the derivative, and I want to get some idea of what the function looks like. So a good starting point, again, is any place where the derivative is positive, negative, zero, or undefined, and again, for positive, negative, finding where it's zero or undefined is important, because the only time we can ever change from positive to negative is by either passing through a point where we have derivative zero, or where we have derivative undefined. So we don't yet need to differentiate, because what we're actually given is the derivative. So the question is, where is the derivative zero, where is this equal to zero? So it's never undefined, because it's a polynomial, and the solutions to derivative equal to zero, it's product equal to zero, and so I can solve product equal to zero by individual factors equal to zero. So that occurs at x equals four and x equals negative eight, and I'll go ahead and plot those on a number line. And so here I have my numbers, negative eight and four. My notes, derivative is zero at these two places. Now these two points divide the number line into one, two, three intervals, and so I do want to find what the sign of the derivative is in the remaining intervals. So I'll figure that out. Test point works very well here, out here, greater than four, x equals, I don't know, 400 billion. This is a positive number. In the middle here, x equals zero. If x equals zero, then this is going to be a positive number. And over here, if x equals, oh, I don't know, negative 10 trillion, then derivative is going to be negative. So I have the signs of the derivatives, where it's positive, negative, zero, nowhere undefined so I don't have to worry about that. Now we'll draw the stick figure. So here derivative negative to positive to positive again. My stick figure sketch, derivative negative says my graph is falling, derivative positive said the graph is rising, derivative positive again says the graph continues to rise. So my stick figure sketch is going to look something like that. And if at this point we call time, that's all you have the time to do, that's a good first draft. So again, this is your stick figure, this is the rough, rough, rough sketch of the graph. What we can do is we can flesh out some of the details, but this is a very good starting point. Now where do those details come in? Well, the details come in using the second derivative, which means I have to find the derivative of the derivative. So I have the derivative, there's the important thing to remember, and I want to differentiate it to get the second derivative. And worth reiterating, the second derivative really is the second derivative of the original function. So I've differentiated it once and got this. If I differentiate it a second time, I just have to differentiate this once. A little analysis goes a long way. If I know the value of x, find x plus 8, hold it, find x minus 4, square it, then multiply the two terms together. This thing that I'm about to differentiate is a product, which means we have to apply the product rule. And without going into any of the details, my derivative looks like this. Again, everywhere defined. So the only question is, where is this thing equal to zero? And if you want to make your life easier, one of the things you might notice here is that both of the terms here have factor x minus 4. So I'll factor that out. And I'll do a little bit more algebra. And I now have product equal to zero. And I can solve this. The solutions here are x equals positive 4 and x equals negative 4. So I'll plot these. I've already plotted x equals 4, but negative 4 is a new place. And again, these are places where the second derivative is equal to zero. So I'll take my notes on that. And again, I've now divided the number line into 1, 2, 3 intervals. So I can find these signs in each of those three intervals. And again, if I want to use a test point out here, test point x equals 500 billion. Useful to have the factored form here. If x is 500 billion, I get positive times positive. So out here, second derivative will be positive. In here, useful test point, that's between minus 4 and 4. So useful test point x equals zero is right in the middle. And that'll give me second derivative negative. And here, x equals, I don't know, negative 10 trillion. Again, substituted into here. And we find the sign of the second derivative in those three intervals. It's going to be positive at the start, negative in the middle, positive at the end. Now it's helpful if we actually have the signs of both derivatives at the same point. So someplace over in this interval, first derivative is negative. And in this interval, remember, you don't change the sign until you hit either a zero or an undefined value. So here, the second derivative is going to be positive. Here, the second derivative is also going to be positive. Likewise, here I know the second derivative, and the question is what's the first derivative? Well, again, we don't change the sign unless we hit a zero. So everywhere in here, first derivative is positive. So I know that here, the first derivative is positive. Here the first derivative is positive. And then we can fill in our extra spaces there. And that changed to a negative 3. That should actually be a negative 4. That gets fixed in the next picture, but so let's see. So what happens? Well, the graph starts with second derivative positive, first derivative negative. So remember, second derivative tells us that the curve is concave up. So in that first interval, our graph looks like part of this. Now which part does it look like? There's three distinct parts. There's this part where we're falling. There's this part where we're making a turn. And then there's this part where we're rising again. And what's going to decide us on which of the three parts we look at is the first derivative. So second derivative positive, part of this. First derivative negative tells us we're looking at the falling part of the graph. So we're going to look at just that first portion. So we're going to look at that first part of the curve, that section there. And all I'm going to do is I'm going to clip this and move it so it maps onto this falling portion of the graph. So there we go. All right. Well, let's take a look at the next part of the graph. So let's take a look at the next part of the graph. So the graph continues. Still the second derivative is positive. So we still look like part of a concave up curve. But now let's take a look at that first derivative. In this section here, the first derivative becomes positive. So I want the portion of this curve that has positive derivative, and that's going to be the right-hand side. And again, I'm going to clip this and move it onto the curve. And that actually stretches out until first derivative positive, second derivative positive. This actually stretches out until here. So I'm going to stretch that out a little bit so that it looks a little bit better. Again, the rough line, the straight line sketch is sort of our quick stick figure version of the graph. We're rounding it out, giving it a little bit more curvature, making it look more like a graph that we're familiar with. What about the next part of the graph? So here, our second derivative ends up being negative. And so we know that in that next part of the graph, we look like a concave down curve. Again, three parts to this. We have the rising part, we have the flat part, and then we have the falling part. And what distinguishes those three parts is the first derivative. So in the next portion, second derivative negative, first derivative positive. Second derivative tells us we're looking at part of this. First derivative positive tells us we're looking at this part, the rising part. So I'll pick that out. I'll clip and move it, and the graph looks something like that. And then the last part of the curve, concave up, positive derivative. So graph is concave up, positive derivative, that section. And I'm going to clip and move this section from here over to there. And there's our refined sketch of our graph. It looks something like this. Falling for a while, rising after that, and it kind of does a couple little bubbles along the way.