Hi. Welcome to module 5. We're going to talk today about how to use your derivative to graph a function. The first thing, I want to just look at this simple polynomial function, f of x equals x cubed minus 3x squared plus 2. We can't really easily factor this or find the roots of it. How could we go about graphing it? Well, we could take the derivative. We know how to do that. The derivative of this function is 3x squared minus 6x. Now, that I can factor. That's 3x, is in common, leaving an x there and a 2. I know the derivative is zero. The derivative is zero at x equals 0. That would make this part zero or x equals 2. Then let's just think about for a minute what it means for the derivative to be positive or negative. If the derivative is positive, what does that mean? Well, it means that my function is going uphill because that would be the slope of the tangent line is positive. If it's going downhill, the slope of the tangent line is negative and the derivative is negative, and where it's zero, somewhere it flattens out right there. It could do this and go on up, or it could turn around, but whatever case it is zero at that point. I know that my derivative is zero at x equals 0 and x equals 2. The other way I like to do this is to then divide up my domain, and I say, at zero, it's going to be zero and at two, it's going to be zero. What are my factors that go into my derivative? Well, my factors are 3x and x minus 2. This factor will be positive to the right of zero and negative to the left. You pick any number less than zero like negative 1, 3 times negative 1 is negative 3. You pick any number to the right, whether it's one or three or whatever, it's going to be a positive number. Then I do the same thing with this factor and this critical point. If I go to the right of two, like at three, 3 minus 2 is 1, and that's positive. To the left of two, like at one, it's going to be 1 minus 2 or negative 1, it'll be negative to the left. Now, a derivative is the product of these two things. It'll be the product of a negative with a negative which will be positive. Here, it'll be a product of a negative and a positive or negative, and here it'll be the product of two positives or positive. Now my function is increasing here, decreasing here, and increasing here. I will look in order to graph, based on that, I will then say, these are what I call my critical points so I want to look at what the value of my original function is at those points. I want to find what those two things are. F of 0 is going to be 0 minus 0 plus 2 or 2, and f of 2 is 2 cubed minus 3 times 2 squared plus 2, which is 8 minus 3 times 4 plus 2, so that's negative 2. I know I can sketch the graph of my function now because I know these are my critical points. At zero it's going to be a height of two and at two it's going to be at negative 2. I've got two points on my graph and I know that my function increases till it gets here, and then it decreases till it gets to this next point, and then it increases again. I have a basic sketch of my function f of x by looking at the derivative, seeing where that is zero, so finding out where it's positive and negative. Where it's positive, my function is increasing, where it's negative, my function is decreasing, and then it's increasing again because the derivative is positive again. That's the basics of how you use a derivative to sketch the graph of your function.