 We often have information about the derivative, and we can try to sketch the graph of the function from the derivative. There's just one problem. Since the derivative of a constant is zero, the derivative omits information about any vertical shifts, so we can't place the x-axis. So, for example, let's say we have the graph of y equals f prime of x. Let's try and sketch the graph of y equals f of x up to a vertical translation, and then identify all local extreme values and inflection points, if any. So, there's a couple of important things to remember about the derivative. First, remember that if the derivative is positive, the function is increasing, and since we're talking about the graph, if the function is increasing, the graph is rising. Next, a useful idea in general, how you speak influences how you think. And so you want to be very, very, very careful about talking about the derivative and the graph, and make sure you identify which one you're talking about. And so we see that the graph is positive until x equals negative 2. Since this is a graph of the derivative, this means that the derivative is positive until x equals 2. So the graph of y equals f of x is rising until x equals negative 2. Now, remember we lose the information about the vertical translation, so we have no idea what y is, and it will be helpful if we draw a vertical line through x equals negative 2. And the idea is that wherever we start, we rise until we hit the line. So maybe our graph looks like this. Now, this dotted line just shows us where x equals negative 2, so we don't really need it anymore, and we'll get rid of it. We see that the graph is negative until x equals 4, and since this is a graph of the derivative, that means that the derivative is negative until x equals 4. So when we go to the graph of y equals f of x, derivative negative means that the graph is falling until x equals 4. And again, let's just draw a reference line through x equals 4, and we'll continue our graph dropping until we hit the line. And we no longer need the guideline. And in this last segment, the graph is positive, and again, since this is a graph of the derivative, the derivative is positive, and so the graph of y equals f of x will be rising after x equals 4. And again, think about this stick figure as a skeleton of the graph. It shows us a lot of information, and in fact, we can find extreme values. So we notice the graph is rising until x equals negative 2 and falling after. So there's a local maximum at x equals negative 2. And similarly, since the graph is falling until x equals 4 and rising after, there's a local minimum at x equals 4. So we want to talk about concavity. So remember if the derivative is increasing, the graph is concave up, and if the derivative is decreasing, the graph is concave down. So we see that the graph is falling until x equals 1. Since this is the graph of the derivative, that says that the derivative is decreasing until x equals 1. And so that means the graph is concave down until x equals 1. So let's put in a marker, and we want to draw a concave down graph that roughly follows the skeleton until we hit the dotted line. And so we might draw. Next, we see that the graph is rising after x equals 1. And again, since this is the graph of the derivative, the derivative is increasing after x equals 1. So the graph is concave up, and again we have to match the skeleton so our graph might look something like this. And since the graph is concave down until x equals 1 and concave up after x equals 1, then that tells us there's an inflection point at x equals 1.