 Given an explicit function y equals f of x, we can sketch the graph over an interval. There's very little motivation to do so today with powerful graphing tools readily available, so it is far, far, far, far more important to be able to sketch the graph of y equals f of x from its derivative, from a graph of its derivative, or from other information. However, if you are given an explicit function, all of the same strategies apply. In particular, we care about where f prime of x is positive, negative, or zero, and similarly, where the second derivative is positive, negative, or zero. So if we have the formula for the function, we can still identify extreme values and inflection points. So we'll find the derivative, we'll find the critical values of the derivative, those are the things that make the derivative zero, and so we find x equals three, x equals negative two. So now we want to check the sign of the derivative over the real axis. So we'll throw down our number line and plot the critical values. So we see that at x equals negative two, and at x equals three, the derivative is zero. That's what the critical values are. And we want to find the sign of the derivative in each of the three intervals formed by the critical values. And so we find the sign of the derivative, positive, to negative, to positive. And so we see the graph of y equals f of x rises until x equals negative two, falls between x equals negative two and three, and then rises after x equals three. And this means it reaches a local maximum at x equals negative two and a local minimum at x equals three. But wait, there's more. We can find the second derivative, and we find the critical values will be x equals one-half. And again, we'll check the sign of the second derivative. At x equals one-half, we split the real line into two parts, and so before x equals one-half, our derivative will be negative, and after x equals one-half, the derivative will be positive. And so the graph of y equals f of x will be concave down until x equals one-half, and then be concave up after x equals one-half, and because it changes concavity, it will have an inslection point at x equals one-half. And this allows us to sketch the graph. So remember, the skeleton of the graph is based on the first derivative information. So we rise until x equals negative two, we fall until x equals three, and then we rise again. Then we can add some flesh to the graph, we're concave down until we hit x equals one-half, and then concave up afterwards.