 Another important feature of geometric objects is known as symmetry. We say a geometric object has symmetry if a transformation of part of the object can be used to produce all of the object. The type of symmetry is determined by the type of transformation. For example, let's consider these symmetries of a line. So if I have a line, I can take part of it, and I can translate it to produce the rest of it. Since we can take part of a line and translate it to form all of the line, the line has translational symmetry. How about the circle? One of the things we can do is we can rotate part of the circle to produce the whole circle, and so the circle has rotational symmetry. Now, most discussion of symmetry centers around the idea of a reflection symmetry, so let's consider the following. Suppose a graph is symmetric about the y-axis. So let's consider a point on the graph with horizontal coordinate x and vertical coordinate y. When x, y is a point on the graph, then, because the graph is symmetric about the y-axis, we can reflect across the y-axis to get another point on the graph where our horizontal coordinate is negative x and our vertical coordinate is still y. So the coordinates are negative x, y. Now, since x, y is on the graph, it satisfies the equation. And since minus x, y is also on the graph, it also satisfies the equation. Conversely, if both x, y and minus x, y satisfy the equation, the graph is symmetric about the y-axis. So let's determine whether this graph is symmetric about the y-axis. So suppose x, y is a point on the graph, then we know that x and y satisfy the equation. What we'd like to know is if negative x, y also satisfies the equation. Well, let's check it out, making our replacement, and we'll do a little bit of algebra. Now, we know this statement is true, and this is the same thing, so we know that this statement is also true, which means that negative x, y also satisfies the equation. So negative x, y is also on the graph. And we can summarize our results. For any point on the graph, we know that the equation x to the fourth minus x squared equals y cubed plus y is true. Since minus x, y also satisfies this equation, minus x, y is also a point on the graph, and the graph is symmetric about the y-axis. What if our graph is symmetric about the x-axis? When x, y is a point on the graph, so is x negative y. And again, since x, y is on the graph, it satisfies the equation of the graph. Since x and negative y is also on the graph, it also satisfies the equation. And so conversely, if both x, y and x negative y satisfy the equation, the graph is symmetric about the x-axis. So let's see if our graph is symmetric about the x-axis. Again, suppose x, y is a point on the graph, then we know that the x and y coordinates have to satisfy the original equation. We want to know if x negative y also satisfies the equation. So let's substitute and check. And while we know that this equation is true, we don't know whether this equation is true. We have no guarantee that this equation is true. And I don't know about you, but without that guarantee, I'm not willing to commit. And so we might summarize our result. For any point x, y on the graph, we have x and y satisfying the equation of the graph, but since x and negative y does not in general satisfy this equation, x negative y is not in general a point on the graph, and the graph is not symmetric about the x-axis. And let's do some final reflections. Suppose the graph is symmetric about the origin. When x, y is a point on the graph, so is negative x and negative y. Since x, y is on the graph, it satisfies the equation. Since negative x, negative y is also on the graph, it also satisfies the equation. And so if both x, y and negative x, negative y satisfy the equation, the graph is symmetric about the origin. So let's put those all together to determine the symmetries of the graph, y squared equals x cubed plus x minus 8. So again, let's suppose x, y is a point on the graph. Remember, for any point x, y on the graph, y squared equals x cubed plus x plus 8, guaranteed true. We'll check to see if negative x, y is on the graph. So we'll substitute and simplify a little bit. We have this guaranteed true statement, but it doesn't look like this, so we have no guarantee that this is a true statement, so the graph is not symmetric about the y-axis. What about symmetry across the x-axis? We'll check to see if x and negative y is on the graph, so we'll substitute and simplify. And since we know that this statement is true, this, which is the same statement, is guaranteed true, so the graph is symmetric about the x-axis. And finally, we'll check for symmetry about the origin. Suppose x, y is a point on the graph. We'll check to see if minus x minus y is on the graph. So we'll substitute and simplify. Again, we're guaranteed that this equation is true, but this equation is nothing like that, so we don't have a guarantee of this, so the graph is not symmetric about the origin.