 In this video, I'm going to talk about identifying transformations. This is mainly just a geometry video. The first thing that you should know about this video, you need to know what transformations are. The three transformations we can encounter are going to be translations, which is moving up, down, left, or right. Rotations, which are rotating clockwise, counterclockwise, either rotating either 90 degrees or 180 degrees is what we're going to work on, just the basics so far. And then also, last but not least, reflecting an image across an axis, either the x-axis or y-axis or some other axis. Okay, so if you don't know what those three are, you might have to read up on those three transformations before you started on this video. Okay, anyway, identifying transformations. I'm going to go over a couple of different problems, but this is the first one. We're going to identify the transformation. Then we're going to use arrow notation to describe the transformation. Now this arrow notation, not too very complicated. Don't worry about that. We'll do that at the very last. But again, we just want to identify the transformations. Okay, so I'm going to come over here to my image. This is the pre-image here. ABC is the pre-image. And then A prime, B prime, C prime is the image itself. So we have the pre-image here and then the image here. Okay, so how did this change? How do we go from ABC to A prime, B prime, C prime? Okay, how do we get from here to there? Okay, so now as I look at this, now I don't see a reflection of any sort. There's no, I draw a line in right here. There's no good line right there of reflecting back across C here and here. And then the B's over there. The lines are all over the place. Okay, so reflections are not going to work there. Okay, so reflection is out the window. Translations, that's another transformation that we have. Translations did is simply just move up, down, left and right. Now if you look at the orientation, I could move this down and to the left. Okay, so my pre-image is here. I moved down and to the left. I'm not sure if that quite works. Did A move down and to the left? No, A just moved straight down. C down to the left, yes. B down to the left. But they're all very different. One is down one and then left one, two, three. B was down one, two, three, four. One, two, three, four and then left two. I mean, the numbers are all over the place. Plus A just went straight down. If it was going to be a translation, all of the points would go the same left and the same right, the same up, the same down, whatever it was. Okay, so it's not a reflection. It's not a translation. The only thing we have left is a rotation. We can kind of see this here. So if we go from B to B, notice the rotation, notice the arc there. C to C, notice the arc there. A to A, notice the arc there. Those three, that can kind of give you a hint that this is in fact a rotation. Okay, so first step, identify the transformation. This is a rotation. Now we also have to identify which direction did it go. Now notice that the pre-image is here at the top and then the image is here on the left. So it went from, I guess we could look at B and then we could look at this B over here. We went this direction. Notice the arrows that I'm writing on here. Okay, now with those arrows, this tells me that I started with B and then went to B prime. I went to the left. So this would be a counter-clockwise, counter-clockwise type of rotation. But then I also need a degree measurement. Did this rotate 90 degrees? Did it rotate 180 degrees? There's a half circle, full circle, what was it? In this case, it didn't flip all the way around. Notice B is at the top. Now it's on the left. C is on the right. Now it's on the top. A is on the bottom. Now it's on the right. So this isn't a full flip. It was a full flip. Then A would go from bottom to top. B would go from top to bottom. And C would go, actually C would stay in the same spot. So this isn't 180 degree rotation. This is only a 90 degree rotation. So rotation counter-clockwise, 90 degrees. So that's the transformation. And then now I want to use arrow notation to describe the transformation. This isn't too overly complicated. Triangle ABC transformed. That's what the arrow means. Transformed into triangle A prime, B prime, C prime. And that's arrow notation. That's all there is to it. Okay, there's my first example to go over. The second example that I want to go over, if I can get to my menus here. The second example that I want to go over involves just a little bit of drawing. And there it is. Okay, so I have a figure that has vertices. Remember, vertices is corners. So I have a figure that has vertices of, here's A, here's B, here's C. Now, notice that there's one, two, three corners. So if I have three corners, it's probably going to be a triangle. But anyway, after the transformation, the image of the figure has the vertices A prime, B prime, and C prime. Notice I'm not reading the numbers quite yet. And what I want to do is I want to draw the preimage so I'm drawing both these triangles, as I'm assuming what they are. And then I'm going to identify the transformations. So a little bit of drawing and then identify what the transformation is. So let's start. Okay, here's A, which is one, negative one. Okay, so there's A, which is one, negative one. And I got B, which is two, three, one, two, one, two, three. There we go, there's B. And then C is four, negative two. One, two, three, four, negative one, negative two. Right there. All right, so there is C. Looks like my A is a little bit in the way. Let me reposition that just a little bit. Get out of there. All right. That's why we always draw with pencil, or in my case, a smart board. All right, so there's A, B, C. All right, so that's the preimage. Now I want to draw the image itself. Let's go back here. Here's A prime, B prime, C prime. Okay, so A is negative one, negative one. Negative one, negative one. There's A again. Got it right at the same spot I did here. B prime is negative two, three. So negative one, negative two, and then one, two, three. I'm starting to see what this might look like. See how the B's are across from each other. The A's are across from each other. I'm guessing that C might be down here. We'll see. All right. Negative four, negative two. One, two, three, four. Negative one, negative two. I even guessed right where it's going to be at. And there is the image. Okay, now remember, your preimage was your first one, and your image is the second one. One thing I need to put on here. B prime, C prime, and A prime. I'm going to put those dashes right there to make sure we know which one's the image and the preimage. All right, so there's the drawing, and then I need to identify the transformation. Okay, so as I look at this, well, it looks like I have a triangle on the left and a triangle on the right, and it looks like I have a mirror right here in the middle. Everything was reflected across this mirror right here. And those, the B's are the same distance from the mirror. The A's are the same distance from this mirror. And the C's right here are, again, the same distance from this mirror. If you see a mirror in a problem, this would be a reflection. Reflection. All right. Now, we could also define reflecting where. Where is the mirror? Okay, so right here is my y-axis. That's where the mirror is. So this is a reflection across the y-axis. Reflection across the y-axis. So not only do we have to define this reflection, we have to define where it's reflecting across. In this case, the y-axis. You could also think of the mirror. Notice the same distance, the A's from the mirror, the B's are the same distance from the mirror, and the C's are the same distance from that mirror. That's how you know where you're reflecting across. Okay, on to the next example for this video. All right, so in this example, let me scroll down a little bit, get rid of that. This example, find the coordinates of the, find the coordinates for the image of triangle ABC after translation of taking the x-y coordinates, transforming them by adding 2 to the x and subtracting 1 from the y, and then draw the image. This notation right here, it's nothing special. Just as I explained it, take the x-y coordinates, transform them, that's what the arrow means, transform them by adding 2 to the x's and subtracting 1 from the y's. Not too overly complicated. Okay, so this is going to be over here. This is going to be my pre-image. Now what I want to do is I actually want to write down all the coordinates for my pre-image. It makes it a little bit easier to work with. Now my A is, let's see, 1, 2, 3, 4, 1, 2, so that's negative 4, 2. And my B coordinate is negative 1, negative 2, negative 3, and then 1, 2, 3, 4, so that's negative 3, 4. And my C coordinate is going to be negative 1, 1, negative 1, 1. So what I want to do is I want to take all these points and I want to transform them. So what I'm going to do is I'm going to take my x, y coordinates, and I'm going to transform them by taking the x's and adding 2 to them and taking the y's and subtracting 1 from them. Okay, so that's what I'm going to do down here. So I'm going to transform these points. So I'm going to take the x coordinate, which in this case, my x coordinate is negative 4, and I'm going to add 2 to it. I'm also going to take the y coordinate, which is 2, and I'm going to subtract 1 from it. So I'm kind of doing this all at once. Notice, x coordinate, add 2 to it. y coordinate, subtract 1 from it, just like it says up here. All right, so what this does is this makes my a prime, this makes my new point, which is going to be 4 plus negative 2, or excuse me, 4 plus 2 is a negative 2, I'm getting ahead of myself, and 2 minus 1 is 1. So there's my new point that I have for a prime. I'll wait just a little bit to graph that. I want to do the rest of my math here. So bear with me. I'm going to do the rest of the math here. I'll take negative 3 plus 2, and take 4 minus 1. If I'm going too fast, you can always... Oh, I'm talking too much. I'm not doing the math correctly. If I'm going too fast, you can always rewind the video so that you can understand where all these numbers are coming from. Okay, so 3 minus 2 is negative 1, and 4 minus 1 is 3. So there we go. So here's my last transformation. Take negative 1 and add 2 to it, and take 1 and subtract 1 from it. That's an easy one. So my transformation for... I'll just do it again. C prime is going to be 1, 0. All right, so those are all my new points. Okay, so I'm going to take those new points. I'm going to plot them on my graph and see what it looks like. All right, so negative 2, 1. Negative 2, 1. Be careful here. Your image and your preimage might overlap just a little bit, which is okay. Which is okay. Not a big deal. All right. Let's see this one. It was A prime. All right, now I have negative 1, 3. 1, 2, 3. This is my B prime. I'll put my letter there for now. Again, I might have to erase it. We'll see. And then C is 1, 0. 1, 0, right there. There's my C prime. So boom, boom, boom. Okay. All right, so there's the preimage, which is in black, and then there's the image, which is in blue. So I got triangle A, B, C, and then triangle A prime, B prime, and C prime. So now I have to identify, I have to draw the image, and that's all I want to do. I just want to draw the image. One extra thing I'm going to do here, one thing that I really want to do is identify the type of transformation that we have here. So all right, so let's look at what all these points did. So I'm going to use different colors with these. So B went from here to here in that direction. Then A went from here to here in that direction. And then C went here to here in that direction. Almost looks like a three-dimensional shape if I look at it that way. So it looks like everything just slid kind of in the same direction. Now that word slide, if you have to slide something in this direction, that tells you right away that you have a translation. Translation. Okay, so this type of transformation changing the position of your shape is a translation. Translation I like to also commonly refer to as sliding the figure, moving it up, down, left, or right. And now you can see from this notation up here that you are in fact moving up, down, left, or right. We moved to the right two, and we moved down one. Right two, down one. You can see that from the image. Right two, and then down one. Right two, down one. Right two, down one. Oh, I messed up that one. Right two, down one. So there we go. All right, that is transformations. A couple of different examples of those. Hopefully this video was helpful. And yeah, I hope you enjoyed the video.