 We can get even more information about the graph from the derivative as follows. First, there are two important cases. If f' of a is equal to zero, then the line tangent to the graph of y equals f of x and x equals a has slope zero. And so the tangent line is horizontal. And since the tangent line is a good approximation to the curve, then we know that the curve doesn't look like this or this, but may look something like this. Another important possibility for the derivative is that the derivative might not be defined. And so we might ask ourselves, self, what would make f' of a fail to exist? So remember that f' of a is the slope of the line tangent to the graph at x equals a. So one possibility is that f of a doesn't exist, so there's no point on the graph where x is equal to a. Or another possibility is that there is a point on the graph but the slope of the tangent line doesn't exist. And if the slope of the tangent line doesn't exist, that means the tangent line is vertical. Or there's one more possibility. Remember, the derivative is a limit and it's possible that the limit might not exist. And one way for this to happen is that maybe we have a point on the graph and maybe we have the slope of a secant line, but the limits disagree. If we approach the point from the left, we might get one slope. But if we approach the limit from the right, we might get a different slope. And so here's another possibility. So again, we can analyze our graph and produce a sketch of the graph of a function given some information about the function and its derivative. So again, it's useful to identify some points that the graph goes through and we'll set these points down. At x equals 0, the tangent line has a slope of 1, so we can sketch what that portion of the graph looks like. At x equals 4, the tangent line, wherever it is, has a slope of 0, so we can set down a placeholder point and run a line segment with slope 0 through it, knowing that we'll have to fix the location later. Likewise, at x equals 6, the tangent line, wherever it is, has slope negative 2, so we'll set down our placeholder point and run a line with slope negative 2 through it. And at x equals 8, the tangent line is either vertical or it doesn't exist. But since f of x is continuous, then the graph must have a point at x equals 8, so it will have a cusp at x equals 8. And so one possible graph of the curve might look like this. And we need to adjust the heights of our point so we get something that actually looks like a curve. And it's important to keep in mind that these three points are fixed because their locations are given by our function values. The other points can be higher or lower and we'll adjust them to form something that looks like a continuous curve. And so we might do something like this. Now all of the graphs we've produced have been essentially steak figures. And for those of limited artistic ability, such as myself, these steak figures are the epitome of artistic expression. And in fact, if we're drawing a curve, these steak figures are often enough to convey the essential features. And in general, mathematicians don't require curves to bend. And so a straight line is a curve. And the first derivative gives us a stick figure graph and for many situations and problems, that's enough. But what if we want a bendier curve, something that looks more like what people expect when we mention the word curve? In art, if you want to go from stick figures to the Mona Lisa, you need to learn about contrast and shading and tinting and so on. And in mathematics, we have to learn more about concavity. Concavity is a geometric concept. And there are several ways of introducing concavity, but because we're talking about tangent lines, let's talk about concavity in the context of these tangent lines. If I have a curve, that curve is concave up in some interval if the tangent line for any point in the interval is going to be below the curve. Alternatively, the curve is concave down in the interval if the tangent line for any point is above the curve. And finally, it's possible for the curve to change from one concavity to another if our curve has one concavity for some interval leading up to a point and a different concavity after that point, then the point itself is a point of inflection or an inflection point. One important note, how you speak influences how you think. The graph has concavity. The graph is either concave up or concave down. The function does not have concavity. You can't say that the function is concave up. That doesn't make sense. For example, let's take a look at the curve shown and identify where it's concave up, concave down, and any points of inflection. Since the definition of concavity relies on whether the tangent lines are above or below the curve, we should draw a few tangent lines. So we'll pick a few points on the curve, draw the tangent lines to those points, again not going too far to avoid cluttering up our graph, and then identify where those tangent lines are relative to the curve. So for these regions, the tangent lines are below the curve and the curve is concave up. In this region, the tangent line is above the curve and the curve is concave down. And since the concavity changes from up to down and from down to up, some place here in the middle we have a couple of inflection points.