 Our first step in going back from the derivative to the graph of the function is to consider what happens to our function. So one possibility is our function might be increasing as x increases. What this means, this is usually shortened to if the function is increasing, but there's always this implied x is also increasing. If I substitute in a larger value of x, I should get a larger value of f of x. So the first idea to keep in mind is that this concept of function increasing is an algebraic concept. The larger the x value, the larger the function value. This translates into a geometric appearance as follows. If I consider the graph of y equals f of x, if x increases, we're moving left to right, if f of x increases, f of x is our y value, those y values are getting higher and higher. So the function increasing as x increases, algebraic concept, translates into geometric appearance, the graph rises as we go from left to right. Minor point of syntax, we don't and should never say that the graph increases. That gives us the totally wrong impression that maybe the graph is getting larger and it's going to go from something this size to something much larger. What we actually should say is the graph rises as we go from left to right, or again we can leave the direction of travel as understood. So we are always assuming we're talking about going from left to right on the graph or as x increases on the function. So we might just say the function is increasing, the graph is rising. So what does this look like? Well, this can look like this. So here as I go from left to right, the graph is rising, the y values are increasing, the function is increasing, but it's also possible that my graph might look something like this. And again, as I go from left to right, my y values are increasing, my graph is rising, my function values are increasing. What does this have to do with the derivative? Well, the derivative is the slope of the line tangent to the graph. So let's go ahead and draw those tangent lines at some point on the graph, here, for example. And if I look at those tangent lines, those tangent lines have a positive slope. And since the derivative is the slope of the line tangent to the graph, we can make the following conclusion. If the derivative is positive, if at some point the derivative is positive, then the function is increasing at that point. So again, function is increasing is an algebraic concept. The larger the value of x, the larger the value of the function. So whatever it is at x equals a, at x equals a plus a little bit, my function is going to be even larger. So there's my algebraic concept. The corresponding geometric concept is derivative is slope of the tangent line. So the graph is actually rising at x equals a. Now we usually say it's rising at x equals a a better way that you might look at it in terms of how we are moving left to right on our graph. You might think about it as the graph is rising through x equals a. So it's going through x equals a and continuing to get higher. I go through x equals a, I go through that point and the graph continues to rise. And the situation doesn't change too much if I look at decreasing functions. So again with the algebraic concept, the function is decreasing as x increases. Then the graph is going to fall as we go from left to right. And again it might look like this or it might look like that. But in both cases as I move from left to right, the graph is falling. As I move from left to right, the graph, the y values are decreasing. Sorry the graph is falling, the y values are decreasing and the function is decreasing. And my tangent lines again look like this and the slope is negative. So that gives us a corresponding result if the derivative at some point is negative. Then algebraically the function is decreasing at that point. And then geometrically the graph is falling at or again through x equals a. So as I pass through the point, the graph continues to fall. As I pass through the point, the graph continues to fall.