 In this video I'm going to talk about reflecting points, so when we talk about reflecting points we're going to mirror them, one way to describe reflecting is to mirror, we're going to mirror them across the x-axis, we're also going to mirror them across the y-axis, I'll do that in the next slide. So now one thing about reflections, one thing about a mirror is that, one thing you've got to know is that if I'm reflecting across the x-axis right here is my x-axis, so this point is going to reflect across this x-axis. Now one thing that we know about this is that these points when they get reflected across the axis, the two resulting points, the original and the new one are going to be the same distance from your line of reflection. So what we're going to do is we're going to use that fact to help us to reflect this point across. So notice that the x-axis is right here which is two, it's two units away from this point, so the new point that I'm going to have is also going to be two units away from my mirror, away from my mirror is going to be two units away from my x-axis. So here's my new point that is negative three, negative two, so that right there is reflecting a point across the x-axis. It doesn't look too terribly difficult but one thing you've got to remember is that this distance here between the original point and the axis that we are reflecting across, this distance is two, this distance between the new point and the axis also has to be two. Got to remember that. So now let's look at some notation and then we're going to look at this in general. Give me a moment to move this around a little bit. We're going to look at this with the notation and then we're going to look at this in general. So if I want to reflect the points across the x-axis, so we start with our original point of negative three, two and now what happened to it? What did we do? What did we do to this point? We'll notice that the x-coordinate stays the same, so negative three, negative three, that's okay, but notice what happened to the y-coordinate, the two and then the negative two. Now if we had time for a couple more examples we might see this a little bit clearer but one way to get from two to negative two is that we are going to take two and we're going to multiply by a negative one. Okay, now writing it that way might be just a little bit confusing. Let me write it just a little bit differently. Let me write this just a little bit differently. Okay, because if you use multiplication we want to try to use it right. So negative one times two. There we go. How about that? Okay, so I took the y-coordinate and I multiplied by a negative one and what that's going to get me is the new point, my new point that I have down here which is negative three, negative two. Okay, so we can see with the notation how to get there. We can see with the notation how to get there. Okay, so in general, what does this mean? In general, so this means for just any point, for any point, if I want to reflect across the x-axis I'm going to take my x-y coordinates and what I'm going to do with it is I'm going to leave the x-coordinate alone. I'm not going to do anything to that, but I'm going to take the y-coordinate and I'm going to multiply it by a negative one. I'm going to multiply that y-coordinate by a negative one. Okay, so that's how I flip, that's in general how I flip points across the x-axis as I take the y-coordinate and I multiply it by a negative one. You could also think of it this way. You could also think of it as changing the sign, just going from a positive two to a negative two. That's another, that's a different way to look at it. Okay, so that is reflecting points across the x-axis. What about reflecting points across the y-axis? So mirroring across the y-axis, okay? Now, some of the rules are going to stay the same. So what's going to happen here is I have this point, which is one, two, three away from the y-axis. So when I take this point and reflect it, it's also going to be one, two, three points away from the y-axis. So this is where my new point's going to be. The coordinates of this new point are one, two, three, one, two. So three, two. So those are the coordinates of my new points. Again, the distance here is three and the distance there is three. The distance from my mirror, from my axis that I'm reflecting across is always going to be the same between the two points. Okay? So I reflect the point across the y-axis. What happened? So let's look at what happened. How did I get from this point, my original point, over to this one over here? So I took my point of negative three, two, and changed it. Now, notice here, again, looking at the numbers, the x and y coordinates, negative three to three, and then the y-coordinate, two to two. So it looks like the y-coordinate this time didn't change at all. It was the x-coordinate that changed a little bit. Now, very, very similar to last time on the last slide. It looks like it went from just a negative three to a positive three. So it looks like we just changed the sign. So it looks like we multiplied by a negative one, and we left the y-coordinate alone to get the new point, which is three, two. Positive three this time. Positive three this time. All right. Forgot to move that around. Give me a moment. Move that down there. There it's better. Okay, so looking at this, we have our original point three, two. We changed it by multiplying the x-coordinate by negative one, and we got the new coordinate of three, two. We can see that here in the picture over here. So in general, so for any sort of point, not just this point up here, for any point, it looks like we take our x, y-coordinate or x, y-coordinate, depending on how many you have, and we change them by multiplying the x-coordinate by a negative one. We multiply the x-coordinate by a negative one, and that gets us to reflect, to mirror across the y-axis. All right, that is reflecting points, both reflecting across the y-axis and across the x-axis.