 Another important skill that you want to have is the ability to interpret a graph correctly. And so the thing to remember is that any time we take a look at the graph of a function, it's a representation of some equivalences, some ideas and objects that are completely and totally interchangeable. So, first off, any graph is a collection of points, and at every point x, y on the graph of y equals f of x, well, because y is equivalent to is the same as f of x, then whatever the x value is substituted into the function should give you the y value. And what this means is every point on the graph, the y value, is the function value. And if you keep these two things in mind, graph interpretation is not too difficult. If you forget them, graph interpretation becomes a lot more complicated. So I have the graphs of two things. Here's y equals f of x in blue, and y equals g of x in orange. And the graph is shown, and I want to solve g of x equals zero, and I want to find where f of x and g of x are equal. So let's think about that. I want to solve g of x equal to zero, and I have the graph of y equals g of x. So that says that I want to find where y is equal to zero. So let's see, y is never equal, oh, that's right, this is the graph, f of x, raw graph. I want to find y equals g of x, so that's this red graph. So here, and here's right at this point, is where our y value is zero. And so that says that y is equal to zero, when x is equal to two, and there's our solution. How about where f of x and g of x are equal? Well, that means the y values are going to be equal, and so that means I want to look for the intersection point. And I follow the two graphs, my intersection points right around here, that's close to x equals three. Now, very important to notice here, I'm not looking for the y value of that intersection point. I'm not looking for anything about where y is, I want f of x and g of x, I don't want a y value, I want an x value. And so I see that the two graphs intersect around x equals three, and to indicate that it's an approximate value, I'll use the approximation symbol. Well, how about producing a graph from the description of something? So let's say we drive up to a friend's house, and we go through a couple of events. At first hour we drive to constant speed, and then we hit traffic, and we have to drive slowly for the next two hours. Traffic clears up, and we arrive at our friend's house one hour later. And let's see, I spent two hours visiting, and with no traffic the return trip takes one hour. So let's sketch a graph of y equals SOt, and we're going to make SOt the distance from home t hours after we begin driving. Now, based on the incidents described, you might draw the graph looking something like this. So we drive out, we do some stuff, we drive out, and then we return home. Well, it's not the right graph. Why isn't it? Well, consider what happens at any given time, say t equals two. So our t-axis tells us our time, our vertical axis, y is SOt, SOt is the distance. Our vertical values, our y values, correspond to our distance from home. So what happens at t equals two? Well, the thing to notice here is that at t equals two, I have not one, but two different y values. And what that says is that at two hours, after two hours, I am some distance from home, and at the same time I'm a different distance from home. That's not actually physically possible. So we know that whatever this graph represents, it is not the graph of this particular function. So let's begin with drawing our axes. So I have two axes to consider. I need a vertical axis which corresponds to the distance from home. I need a horizontal axis which corresponds to the time after I begin driving. So there's my horizontal, there's my vertical, and I might want to identify a couple of important points here. So what are those important points? Well, the first thing to notice is that something happens during the first hour. So I may want to mark that point at one hour. The next two hours, something happens, and let's see, I'm at one hour. So the next two hours is going to take me up to one plus two is three hours. One hour later, something else happens, so I'll go ahead and mark that four. I spent two hours visiting, so that's going to be another two hours past four. So that'll put me up at six hours. And then with no traffic, the return trip takes one hour. And that's going to take me one more hour, and that drops me up to seven. So here's a couple of important points at one, at three, at four, at six, at seven. Something happens that I want to take into account in our graph. And I'll fill in the intermediate points as well. And so we're graphing s of t, and s of t is the distance from home, then our vertical values, our y values, represent a distance. So our y values are going to change as a distance changes. So what's going on? Well, during the first hour, I drive at a constant speed, and so that means our distance is constantly and steadily increasing. So we'll draw a nice steady increase of distance of y, of our y values during that first hour. What happens next? Well, we get traffic for the next two hours, and we're going to drive more slowly. Now, we're still driving, so our distance will continue to increase, but not as quickly. And so that means our y values won't increase by that much, that quickly, but they'll increase by a lesser amount. So maybe I'll graph, we'll look something like that. And with no traffic, I'm going to continue to increase our distance. Traffic clears up, and one hour later we arrive, and our speed is going to go back up. I spent two hours visiting. Well, during those two hours, the distance shouldn't increase. We're actually visiting. We're not driving any place. So that means our distance, our distance, our s of t values, our y values aren't changing. Now, that is what's going on during this two-hour time period. So during a two-hour time period, our distance will not change, and that's going to be a flat horizontal section. And then finally, we drive home after one hour, and in order to drive home, our distance from home has to go from whatever it is back down to zero. So that's going to look like that. That's what we're going to do with this two-hour time period.