 This lesson is on attributes of graphs. The first thing we're going to learn about is notation, because when we do increasing and decreasing, we're going to use a different type of notation than you might normally have used before. We are also going to talk about maxima and minima and concavity. So let's look at this notation. I believe you already know about inequality notation. You would say x is less than 2, which means all real numbers less than 2, or x is greater than negative 1, which means all real numbers greater than negative 1. Or we could have this inequality, x is between negative 4 and 0. We could also do these with equal twos. We could say x is less than or equal to 2, or x is greater than or equal to negative 1, or x is between and equal to both negative 4 and 0. Now let's change that to something we more commonly use in this class. The x is less than 2 is written in parentheses negative infinity, comma 2. And then we close it with the parentheses. Again, this is read x is less than 2. The next one with the parentheses negative 1, comma infinity is read x is greater than negative 1, just like the one with the inequality notation. And the last one that we have here is x is between negative 4 and 0, and is read the interval between negative 4 and 0. And that's negative 4, comma 0. Now with the equal 2, anytime we use infinity, we never use the bracket. So we always use a parentheses. So it's parentheses negative infinity, comma 2 with a bracket. The bracket makes it equal to 2. So this is read, just like the one with the inequality notation, for all numbers less than or equal to 2. The next one has a bracket at negative 1, which means it's going to be equal to negative 1. So this one reads all numbers greater than or equal to negative 1. And for this last one, we have the brackets on both sides. This interval notation reads all numbers between negative 4 and 0 inclusive. Let's go on to other ideas, relative maxima and minima. The relative maxima or minima are a high point or a low point over an interval on a graph. The relative maxima is read as f of a, so some y value is greater than or equal to f of x, some numbers of y over an interval. And of course this is called local or relative maximum. And if we drew a graph of something like this, we could see that the relative maximum here is that, or the relative maximum on this one is here. We could also have a relative max that has some sort of a point, and that would give us a relative max as long as it's the highest point over an interval. The relative minimum reads some value for y that is less than or equal to other values of y over an interval. Again, this is a local or relative minimum. And in the same vein, we just need it to be the lowest point over an interval. So this is the lowest point over that interval. This is the lowest point over that interval. And of course we could have any other type of graph and still have it the lowest point over an interval. Now this next idea, absolute maxima and minima is similar to the relative, but what we are looking for is the highest point or the lowest point over the entire curve. So if we are looking for the absolute maximum, we want our y value, our f of a, to be greater than or equal to all y values over the entire curve. So what would be an absolute maximum? Well an absolute maximum on this curve would be this point right in here. These are both relative maximum, but this one is the absolute. And of course we could do something else and do another graph that looks like this and comes down like that, and of course that's the relative maximum. Or we could even do a curve that increases to some point over an interval and this here would be the absolute maximum. What about absolute minimum? Well this is when the y value is less than or equal to any y value over the entire curve. And the same idea is relative except it is the smallest value. So if we have a curve that looks like this, this would be your absolute minimum. If we have a curve that looks like this, this has an interval from a to b here, then this would be your absolute minimum. Let's go on to concavity. What is concavity? A curve is concave up if it looks like a bowl holding water. So it looks something like this. It could look like that, it could be completely concave up or slightly concave up. But as long as it's holding water in some way, we think of it as concave up. A curve is concave down if it looks like a bowl spilling water. So it looks like this. And you can see water would spill out of a bowl if it were tipped like that, or if it were tipped like this. As we get on with derivatives, we will have a more precise definition of concavity that has to do with derivatives. But right now we are just looking for curves that look concave up or look concave down. And our last idea here is point of inflection. A point of inflection is the point at which a curve changes concavity. That means the curve changes from concave up to concave down. And it would be a graph that goes from concave up to concave down. Holding water, spilling water, point of inflection. Or from concave down to concave up. Spilling water, holding water, point of inflection. Concave down, concave up. This ends our lesson on attributes of graphs.