 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 satirro, the derivative omits any information about any vertical shifts. So we can't place the x-axis. So suppose we have the graph of y equals f' of x. So there's three ideas we want to keep in mind. First, if the derivative is positive, the function is increasing. If the function is increasing, then the graph is rising. And most importantly, how you speak influences how you think. And in particular, you want to be very, very careful about saying it is rising, falling, increasing, or decreasing. And in fact, as a general rule, lose the word it from your vocabulary. For example, we could say that it is positive until x equals negative 2, but we won't, because what we really mean is that the y values are positive until x equals negative 2. And since y equals f' of x, then the derivative is positive until x equals negative 2. And so the graph of y equals f of x is rising until x equals negative 2. So let's start somewhere and show a rising graph until we hit x equals negative 2. Then it, I mean the y values, are negative until x equals 4. Again since y equals f' of x, then the derivative is negative until x equals 4. And so the graph of y equals f of x is falling until x equals 4. So we can continue our graph by falling until we reach x equals 4. The y values are positive after x equals 4. Since y equals f' of x, then the derivative is positive after x equals 4. And since the derivative is positive, the function is increasing, and so the graph is rising after x equals 4. This gives us what we can think about as the stick figure version of the graph, and you can think about this as the skeleton of the graph. And since the graph is rising until x equals negative 2, and falling after, there is a local maximum at x equals negative 2. And since the graph continues to fall until x equals 4, and rises after, there is a local minimum at x equals 4. How about concavity and inflection points? 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 might observe that the graph is falling until x equals 1. And since this is the graph of the derivative, this means the derivative is decreasing until x equals 1. And since the derivative is decreasing, the graph of y equals f of x is concave down until x equals 1. Now remember the stick figure is the skeleton of the graph, so we'll draw a concave down graph that follows the skeleton. Similarly, the graph is rising after x equals 1. This is the graph of the derivative, so the derivative is increasing after x equals 1. And since the derivative is increasing, the graph is concave up after x equals 1. So let's draw a concave up graph that follows the skeleton. And since the graph is concave down until x equals 1, and concave up after x equals 1, there's an inflection point at x equals 1.