 So a useful thing to learn about is graph symmetry. Well, that's not really true. Graph symmetry isn't all that useful. It's only useful if you want to do things the easy way. If you want to do things the hardest way possible, you can ignore graph symmetry. But maybe we want to do things in an easier way, so let's take a look at what this is all about. So when I graph, so here's the thing to remember. Anytime we take a look at the graph of any sort of function, we're looking at all points x, y that satisfy the equation of the graph. In some cases, it is possible that if we know one point is on the graph and what that means is that point, the x and y values, satisfy the equation of the graph, then I can immediately determine whether another point, x prime, y prime, is also on the graph. And this is the basis of symmetry. So there's three important cases of symmetry. First, suppose that for every point that's on a graph, so I have some point x, y that's on my graph of y equals half of x. Suppose that I also know that the point negative x, y is on the graph. So if I'm at the point x, y, this is on the graph. If I also know this other point negative x, y is also on the graph. So that's the same y coordinate, same height, but my x coordinate is the negative of what it was. It's flipped across the y-axis. Then if this is true for every point, wherever I am, if I have a corresponding point on the other side of the y-axis, then what I have is all the points on my graph on this side are reflected and show up as points on the graph on the other side as well, and so our graph is symmetric about the y-axis. Likewise, suppose I have, suppose I know that for every point on my graph, x, y, wherever it is, I have another point also on the graph x negative y. Same x coordinate, but my y coordinate is flipped. And so what I've done is I've gone from a point above the axis and I've flipped down to a point below the axis. So suppose this is also true for every point on the graph. So again, I have some point and I have a corresponding point on the other side of the axis. I have a point and there's another point on the other side of the axis. And again, what we see is that every portion of the graph that's up here has a portion of the graph that's down there. And so what's above the axis looks exactly like what's above the axis and so my graph has been reflected across the horizontal axis, the y-axis. And I say that the x-axis and I say that the graph is symmetric about the x-axis. And then finally suppose that for every point x, y on the graph. So here I am, I have a point x, y on the graph and let's say I know that there's another point minus x minus y also on the graph. That's the negative of the x value, the negative of the y value. So if I originally went out and up, I'm going to go back and down the same distances. So if I have a point over here, I have another point over there. And again, if this is true for every point on the graph, then wherever I am, there's another point. This is out x of y. There's another point back x down y right there. And again, for any other point, xy over x of y, back x down y and there's going to be another point there. And here what I have is I have a set of points over here and I know there's going to be a reflection of those points on the other side. And we say that this is symmetric about the origin. Well, how do you find symmetry? Well, to determine whether a graph of an equation is symmetric about the y-axis, x-axis, or the origin. Well, paper is cheap, so let's write down the equation of the graph. And the important thing is that for any point that's actually on the graph, they have to satisfy the equation of the graph. So whatever we write down, we should regard this as a true and unequivocably true statement about a relationship between x and y. So the next thing I'm going to do is I'm going to take my equation for the graph and I'm going to replace x with negative x. And if I end up with something that is still true, then I know that negative xy, my x-coordinate, negative x, my y-coordinate is the same, I know that this point is also on the graph of the equation. And the relationship between these two points, xy on the graph, negative xy also on the graph, tells me that the equation is going to be symmetric about the y-axis. Now I'll restore the original equation, I'll go back to the original equation and this time I'm going to replace y with negative y. And again, if the statement is still true, if I still get a true statement, then the point x negative y is also on the graph. So again, xy is definitely a point on the graph. If x negative y also satisfies the original equation, then this point is also on the graph. And the relationship between these two points, x-coordinates the same, the y-coordinates are opposites. And so that tells me that the graph is going to be symmetric about the horizontal, the x-axis. And then finally, I'll restore the original equation again. And this time I'll replace x with negative x and at the same time y with negative y. And once again, if the statement is still true, then the thing I know about the point negative x negative y is that it's a point on the graph. And that tells me the equation is symmetric about the origin. So for example, let's say I have the equation xy equals 1. Let's see what sort of symmetries we have. So I'll set down the equation xy equals 1. And again, I want to regard this as a true statement that for any point on the graph, I know that the product x times y, the x-coordinate multiplied by the y-coordinate, is going to be equal to 1. Now I do want to replace x with negative x in the equation. And I want to determine whether this is a true statement. And at this point, I have to do some fantastically difficult algebra. And I have to determine whether this statement is true. And, well, here, I know this statement is true. xy is equal to 1. I know that's true. And that says that negative xy equal to 1, that can't be true. If xy is equal to 1, negative xy can't be 1. So I know the statement is false. And that tells me that the graph is not symmetric about the y-axis. Well, I'll do the same thing. I'll replace y with negative y in the original equation. So here's my original equation, x times y equals 1. I'll replace it with negative y. And again, the question is, is this going to be a true statement? So after some fantastically difficult algebra, I then determined that, well, again, if this is false, I might as well take advantage of the work that I've already done. If this is false, this obviously the same statement also has to be false. And so, again, I know that this point x negative y can't be on the graph. And so once again, I know the graph is not symmetric about the x-axis. Finally, I'm going to replace x with negative x and y with negative y in my original equation and see if I get a true statement. So here, x I've replaced with negative x, y I've replaced with negative y. And I have the fantastically difficult algebra to evaluate. And well, I know this statement is true. So the question is, what do I know about this statement? And, well, it's the same statement. So it's a true statement. And that tells me that the point negative x negative y is also on the graph. And that tells me my graph is symmetric about the origin. I'll have another graph, y squared equals x cubed minus 4x plus 7. So again, I'll set down the equation. And I'll note that this is in fact a true statement. And I'll replace x with minus x and see if this is also a true statement. So let's take a look at that. Well, this is x becomes minus x, x becomes minus x. Everything else stays the same. After some fantastically difficult algebra, I get this statement. And, well, I know y squared is x cubed minus 4x plus 7. So that means y squared can't be this other expression. So this is a false statement. And the graph is not symmetric about the y-axis. This time I'll replace y with negative y. So again, negative y goes there. Everything else stays the same. After some fantastically difficult algebra, I get this statement here. And I know that this statement is a true statement. So I know that the statement is true, which means that the graph is going to be symmetric about the x-axis. And finally, I'll replace x with negative x and y with negative y. And I'll do the algebra. And I get this statement. And I already know this statement is false. So this statement is also false. It's the same statement. So the graph is not symmetric about the origin. And I might just summarize my results. The graph is symmetric about the x-axis, but not about the y-axis or the origin.