 This video is going to talk about characteristics of functions and graphs, and the first thing we want to talk about is the symmetry of a graph. Not all graphs are symmetrical, but if they do, they're either going to be symmetrical about the y-axis or the origin, those are the ones we're going to look at. So this tells us f of negative x is equal to f of x. Remember function notation really means this is my input and my output is the f of x. So this is really saying I put a negative x in and I should get the same answer as if I had put a positive x in. What does that really mean? It means if I had x, y, then if I put a negative x in for the input, I should get that same output, that same y value. And if you look at the graph, that's what happens here. We have negative 2, 1, and we have positive 2, 1. So this x has changed, but the y stayed the same. The equal distance away from the y-axis, each point, symmetrical point. So that makes this symmetry with respect to the y-axis. And if you think about it, if you put a mirror here on the y-axis, then you would see the other half of the graph in the mirror. It's a mirror image across that y-axis. Well, symmetry with respect to the origin says f of negative x is equal to the opposite of f of x. That means if I put in a negative x, then I should get out the opposite of what I got when I had a positive x put in. What does that mean? If I have x, y, then if I put a negative x in, I'm going to get the opposite of my y. It was y with a positive x, so now it's going to be a negative y with a negative x. So this means that it changes both signs. And if you take your finger and put it on the origin here and spin your paper 180 degrees, you're going to end up seeing the exact same graph. That's how you would think about looking to see if something was really symmetrical with respect to the origin. All right, so let's try and see if we can actually graph with respect to the y-axis. So that means, remember with the y-axis, that means it only changed the x's. So we have negative 8 will become positive 8, but I still have a negative 5. Positive 8 would be right here, down to negative 5. Negative 6, 0 would be positive 6. I'm going to change the x, and 0 doesn't change signed anyway. So I'm going to have a positive 6, 0. And then we have negative 5, which becomes positive 5, but I still have a 3. So 5, 3, okay, the negative 3, 3 would become positive 3, 3. So we would have this one here. And then 0, negative 3 stays 0, negative 3, because 0 doesn't change signed. So that's still this point, and if I connect all my dots sort of, then I can see that it is pretty close to being symmetrical. And it should actually be a bit of a straight edge. Let's try it with the origin. This means that you change both signs. Negative 8 is going to become positive 8, and negative 5 is going to become positive 5. So 8 and positive 5, and we have 6 and 0, changing the signs on both here. We get positive 5 and negative 3. So positive 5 and negative 3, and we have changing both signs. We have positive 3, negative 3. And then we have this 0, which doesn't change signed, but the negative 3 becomes a positive 3, and we have this point here. So if I connect all my points, this one's a little bit harder to see that it is symmetrical about the origin, because it's not even connected anywhere. But if you spun your paper, you would see exactly the same graph if you do it 180 degrees. All right, so while we're looking at graphs, we want to think about inequalities and how those work with our graph. We're looking at something like this, f of x is greater than equal 0, less than 0, greater than 0, something like that. So on our graph, anything above the x-axis here, the y-values are greater than 0 because they're positive. And anything that is below the x-axis is going to be y is less than 0 because that's our negative values down here for a given x. And then when we have on the x-axis, then we have y is equal to 0. Our inequality over here is asking us to find greater than or equal to 0. So that means that we want to be on, because that would be equal to, and above the x-axis. So what does that show us? Well, on the x-axis, I have negative 2, and I also have positive 2. But above the x-axis, I have this part of my graph. I want to know what x's satisfy this. So the x is an element of, and this is just one point. So we say negative 2, union, and then we start at 2, but anything larger would also satisfy our inequality. So bracket 2 to infinity. All right. So when we talk about increasing and decreasing, we're really talking about, as I go from left to right, does my graph go, do my y's keep getting bigger and bigger, or do my y's keep getting smaller and smaller? So here they want us to look at these first two points. And if I look at those two points, I want to compare their y's, because that'll tell me whether I've increased as my x's got larger or decreased. So when I look at y2 compared to y1, which is really what this is saying, the output of x2 compared to the output of x1, this one, x2, the output looks like it's 1, and the output of x1 looks like it's negative 3. Then we would say that f of x2 is greater than f of x1. Now if we look at a graph, and it says we want to look at these next two points, so x2, y2, x3, y3, if I compare y3, it looks like it's at 1, and y2, we already said it was at 1. So then we would say they're equal. This is what we would call constant. It's not increasing, it's not decreasing, it's constant. And this here was increasing, and if we do x4 to x3, we can see the x3 we know to be 1, and x4 looks to be negative 1, so it's less than or it's decreasing. Write the interval of increasing constant and or decreasing of the graph. And if we look at a graph from left to right, we start here and we go up. These y's are getting bigger and bigger, so we're increasing. And we look here, and we also see that we're increasing at the very end of the graph. So we can say that x is an element from negative infinity up to, and we'll call that negative 2. And then it starts back up, and we'll call that maybe 1 half, and that will go on to infinity, a constant. There is no place where this graph is the same for several x values, so this one's not applicable. And then we look at where it's decreasing, obviously that's what's in between my two increases. It starts at this point here, and it ends at this point here. So we're going to say that x is an element between negative 2 and 1 half. Now notice that I've always used parentheses, because it's increasing at that end point and decreasing, now at the same time, so we just make it a parentheses. And then we also want to look at the end behaviors, and this is just the way that we'll talk about this more later. But right now we're just going to look at it, notice that on the left hand side of this graph, if I look at it, we could say that it is going down on the left, and we could say that it's going up on the right. So we could call that an up kind of end behavior. If it had been the opposite, it would be an up-down, and it could also be up-up, it could be down-down. So just describe what you see on either end. Okay, so the last characteristic we're going to talk about is the maximums and minimum values. The y-coordinates, that's important to know, these are going to be y-coordinates, not x-coordinates. So everything else we talked about, we've been saying x is an element of. But in the y-coordinates here, we would say that the peaks would be the maximum value. So here, we would say that the max would be y equal, and it looks like it would be 1. And if we did the valleys, we're called the minimums. So these points down here, okay? And min, and y is going to be equal to, and it looks like it's at negative 3. Then it tells us all peaks and values can be evaluated as local. All of them are going to be local or relative maximums or minimums. It depends on how you see them written, and the first time you learn them probably. I learned them as local. So we'll say max and l min for local and max and min. And then there's this global. Global or absolute is the largest or smallest y value over the entire domain. We're not going to include infinities, but we look at this graph and we have lowest point on this graph. This function does not go any lower than negative 3. So we would say that we have, and I learned it as absolute, an absolute min. Because that's the lowest point at y equal negative 3. Okay, if I had this graph right here, but then it's got a max and it's got a min. But it does not have an absolute max and min because it goes to infinity on both ends. It's down, up. And if you have a down, up situation, you're never going to have an absolute max or min.