 In this video, we're going to talk about rotations. So we've already talked about reflections, we've talked about translations, and this is the last one, which is a rotation. I think you're all familiar with rotating things, like a tire rotates, or if you think about like a merry-go-round or a tilt-a-whirl, those are all things that rotate. So a rotation, if you want to write this down in your notes, is a transformation that turns points around a fixed point and in a certain direction. And then the point that you are rotating around, that fixed point, is just called the center of rotation. So when we do rotations, we are going to do certain directions and specific angle measures. So you can see down here there are going to be a set of rotation rules. The three rotations that we're going to focus on are 90 degrees clockwise, which means 90 degrees in the direction that the clock turns, which is to the right, like this, 90 degrees counterclockwise, which is to the left, so this way, and a rotation of 180 degrees, which if you think about that, 180 degrees would be forming a straight line. So there are rules because it's really difficult on a graph to guess where 90 degrees is. We know what a 90 degree angle is, but when you do a rotation, it's easier to go by these rules than to have to estimate where you think 90 degrees is. So let's start with a rotation of 90 degrees clockwise. Your pre-image will be the point x, y, and once you do the 90 degree clockwise rotation, the new point, or the image, will be y negative x. A rotation of 90 degrees counterclockwise, if you start with your pre-image of x, y, the image will be the point negative y, x, sorry that looks like a point. And lastly, our rotation of 180 degrees, you're going to take your pre-image point x, y, and make both the x and y opposite, so negative x, negative y. Now in the next couple slides, you'll see examples of these rotation rules being used and how you can then, once you've done the rotation, you'll be able to actually see that 90 degree or 180 degree rotation. Okay, so for this, we're going to do a 90 degree clockwise rotation. So pause the video for a minute and put h, j, k, and l onto your graph, and I will do the same. Okay, so when we do our rotations, our fixed point is always going to be the origin. So what we want to do is we want to be able to take this four-sided figure and rotate it clockwise, so to the right, around the origin so that it comes out to a 90 degree rotation. So we're going to use the rule that we talked about on the previous page. And if you remember that is the pre-image becomes the point y, negative x. So if we look over here at our first point, 1, 0, what's going to happen for the image is that the y value of 0 is now going to be first and the x value of 1 is now going to be a negative 1. Go ahead and do that for j, k, and l, the images of j, k, and l. So for point j, the new image will be for 0. So the y value goes first and the x value goes second. Now at 0, you can't turn it to negative 0, so it's still just 0. For the image of k, it's going to be 5, negative 2. And for the image of l, it's going to be 1, negative 3. Now again, pause the video and I want you to graph each of these new image points and watch what happens. Okay, so we see what's happened is for example, point h, which is right here. If you go down to the origin, 0, 0, and then over to the new point, j prime, see how that forms a 90 degree angle? So using the rule, after you're done, you can in fact see that it is a 90 degree rotation. You could do the same thing with k if you think about going to the origin and then over to the image. You can see that if I draw that a little better, that that is in fact a 90 degree rotation. So using the rule helps you make sure that you've gotten the images in the exactly correct place. Okay, for this one, we're going to do our 90 degree counterclockwise rotation. So remember from the first page, the rule is that our pre-image, x, y, is going to become our image negative y, x. So if you haven't already, I want you to graph h, j, k, and l like I've already done over here. And so just take a minute and pause the video if you need to put your points on the graph. And then we're going to come over here and we're going to use the rule to change h into its image h prime using our rule. So the point 1, 0 is going to become the point 0, 1. So again, the y value is now going to be first and because we can't make 0 negative, we're just going to leave it 0, and then the x value is now second, so 0, 1. Our point j, 0, 4 will become negative 4, 0. So again, the y value we're putting first and we're putting that negative on it from the rule, and then the x value is second. The next point k becomes the point negative 5, 2, and the image for l would become negative 1, 3. And now I want you to take a minute and put these 4 points on the graph. Now you can see what's happened is this time we have rotated counterclockwise, so we've gone to the left. And if I want to check on my 90 degree rotation, let's just use j as an example. Here's the original point j and we went down to the origin because that's our center of rotation and look at what happens. If we go to the image, that is in fact a 90 degree angle. And because we've rotated around to the left, that is counterclockwise. Alright, the last example we're going to do is a rotation of 180 degrees. So if we remember the rule, in this situation when we do a 180 degree rotation, we don't have to specify counterclockwise or clockwise because 180 degrees, it doesn't matter which way you go, you're going to get to the same place. So our pre-image x, y is going to become our image of negative x, negative y. So let's take the 4 points, and again if you haven't graphed them yet, just pause for a minute and graph h, j, k, and l. And then we're going to come over here and we are going to use the rule to come up with the new coordinates. So our point of h, 1, 0 is going to become a new point of negative 1, 0. So in this rule, the x and the y stay in the same place, we're just making them opposite. So 1 becomes negative 1 and 0 stays 0 because there isn't a positive or negative 0. So for point j, 0 is going to stay 0 and the 4 becomes negative 4, k becomes the point negative 2, negative 5 for its image, and l has an image of negative 3, negative 1. So now take a minute and graph those 4 points. So what you can see has happened is we have rotated 180 degrees. So from j straight line to j prime. So we have rotated either this way 180 degrees or this way 180 degrees around the origin. Same thing with h, h to h prime forms a straight line, l to l prime, k to k prime. So whatever points you're looking at, all of those form 180 degrees or a straight line.