 In the three videos that you're going to be watching tonight, we're going to be talking about reflections, translations, and rotations, and you're going to use these things to help you complete the flip book project, which we'll be talking about in class. And so we're going to start in this video with reflections, and hopefully we all understand that a reflection is usually what we see when we look in a mirror or sometimes a piece of glass or a window. Sometimes you can see a reflection in. What is a reflection? It is just a flip over something. So most of the time we're going to do reflections of objects over a line or sometimes reflected around a point. So you can see in this picture over here, we have figure A, B, D, E, and it is being reflected over this line. I think that's supposed to be an M. Can't tell. And what you see is that A sees itself on the other side of the line the same distance. So the distance from the line to point A is the same as the distance from the line to what we call A prime. Same thing for B and B prime, D and D prime. And then you'll notice that E is on the line. So therefore it doesn't see itself on any other side. It sees itself exactly at that same point because it's touching the line. So the other thing we have right down here is pre-image and image. The pre-image is the original. So for this would be A, B, D, E. And the image is the result of doing the reflection. And so we'll say the result of the reflection. And you'll notice that we label that with what we call prime. So it would be A prime, B prime, D prime, and E prime. This slide gives you a couple of examples of how to do a reflection. I mentioned that in the last slide that what you're going to do when you reflect something is generally you're reflected over a line. So if I look at this right here, here's my line that I'm reflecting over. And I'm reflecting this triangle H, J, K. So what you're going to do is you're going to say, okay, however far J is above the line, that's the same distance I want to go below the line and J will see itself on the other side of that line. Same thing for K. K, however far is below the line, that's how far above the line that K prime will be. Oops, I forgot the prime on J. And then for H, same thing, however far above the line that H is, you'll go the same distance below the line so H prime would be there. And so then when you draw your new figure, your image, it should be the exact same shape. It's just been flipped over the line. Now obviously that is a really rough sketch. We're going to be doing more exact examples in just a second. Over here we have another line L and we are reflecting a triangle this time from right to left. So we would say we'd have probably this point would see itself about there, this point be about there, and this point would be about there. So then when I draw that reflection, you have the triangle seeing itself on the right side of the line. This example is a reflection in the y-axis. You'll want to fill out the rule, which is when you reflect in the y-axis, your point xy becomes a new point negative xy, and you'll see why that happens in just a second. So let's look at point A. Point A is here and that happens to be the ordered pair 1, 2. Well if I reflect in the y-axis it means that I'm reflecting over, here's my y-axis, and this point A is one unit to the right. So the reflection will be one unit to the left. So I'm going to go ahead and put a prime right there and you will notice if you count that ordered pair it's negative 1, 2. And so it follows this rule that what happens is that the x becomes negative and the y remains the same. So this point would be negative 1, 2. Do the same thing for B. If we count 1, 2, 3, 4, 1, 2, 3, 4, the ordered pair for B is 4, 4. So what I'm going to do for the reflection is I'm going to count B is 1, 2, 3, 4 units to the left of the y-axis. So it's going to reflect 1, 2, 3, 4 units to the left. B is 4 units to the right so it's going to reflect 4 units to the left. And I kind of have my rule written on that. But this will be point B prime. Same thing for C. We'll count 1, 2, 3, 4, 5, 6 units to the right so therefore 1, 2, 3, 4, 5, 6 units to the left is going to be the reflection. So you'll notice when you reflect over the y-axis the points don't move at all up or down. They are just being reflected left or right. In this case all the points were reflected from the right side of the y-axis to the left side. In this example we are reflecting in the x-axis. So the x-axis is our horizontal axis, this one here, which means we're going to take these two points and they're going to reflect below. This point, since it's below the x-axis, is going to reflect above. So you're just going to use that idea of counting again. Point A is 1, 2 units above the x-axis. So when we reflect it it's going to be 1, 2 units below the x-axis. So I'll go ahead and put that point as A prime. Same thing with B. We have 1, 2, 3, 4 units above so I will go 1, 2, 3, 4 units below. And I will label that point B prime. And then C, like I said before, C is already below the x-axis so when it's one unit below the reflection is going to be one unit above the x-axis. And if I connect those points I have the exact same triangle. It's just been reflected over the x-axis. Now you'll notice up here is the rule that you take an ordered pair and it becomes x-y when you reflect in the x-axis. So just as an example again if we look at point A that's the ordered pair 1, 2 and if we look at A prime that's the ordered pair 1, negative 2. And so sure enough you see what happens is that this x is still the same. So x is 1, x is 1, but y has become negative. So here it was a positive 2 and now it's a negative 2. This example gives us an example of reflecting in the origin. We all know that the origin is 0, 0. Well when you do a reflection in the origin what happens is you take your ordered pair and the x and y all become opposite. So positive, positive would become negative, negative. So if we take point A for example which is the point 1, 2 it's going to become the point negative 1, negative 2. So here would be my point A prime. B is the ordered pair 4, 4 so reflecting in the origin it would become negative 4, negative 4. So here's B prime. And then C, 1, 2, 3, 4, 5, 6, negative 1 is the point for C. So what happens is the 6 will become negative, the negative 1 will become positive using this rule. So we will go 1, 2, 3, 4, 5, 6 to the left and up 1. So that's C prime. And when you connect those points what you have here is a reflection in the origin, 0, 0. The last example we have here is a reflection in the line y equals x. If we graph y equals mx plus B we can see that this has, this line has a slope of 1 and a y intercept of 0. So that's why you see it starting here at 0 and then it's up 1 over 1, up 1 over 1 to get the line y equals x. So this is our line y equals x. If we're going to do a reflection over this line what happens is, for example, this point B is going to see itself over here. This point C is going to see itself over here. D is already to the right of the line so it's going to see itself over on the other side. The rule that you use for a reflection in the line y equals x, you'll notice right here, is your ordered pair xy becomes the ordered pair yx. So let me show you how this works. Now A is going to stay on the line because remember a point that's on the line or if you think about a mirror, if you put your finger on a mirror the other side of your finger is in the same point. So A and A prime are both on the line. B, however, is the ordered pair 3, 1, 2, 3, 4, 5, 6, 3, 6. So what happens using this rule is the reflection is going to be the point 6, 3. So if I count over 6 and up 3, this is where the image of B is, B prime. If I get the ordered pair for C, that's the ordered pair negative 2, 1, 2, 3, 4, 5, 6 and so the reflection of C is going to be at 6, negative 2. So all you're doing here is you are taking this rule, you're taking the original ordered pair and just switching the x and the y. So negative 2, 6 becomes this point 6 negative 2. Lastly we will reflect D. So D is the point negative 1, negative 5 and so the reflection will be negative 5, negative 1. So that would be the reflection of D. And now if you connect A to B, B to C, C to D, whoops I missed and then back up to A, you have the exact same figure, it's just been reflected over this diagonal line. There will be a couple of examples on your practice where you reflect over other lines like y equals 2 or x equals negative 1. Those are no different than reflecting over the x and y axis. The only thing that changes is where your line of reflection is. So you will see examples of that in your practice.