Reflecting points across horizontal and vertical lines

We're asked to plot the image of point A under a reflection across the line L. So we have our line L here, and we want to plot the image of point A under reflection across line L. Well, one way to think about it is: point A is exactly one, two, three, four units to the right of L, and so its reflection is going to be four units to the left of L.

So if we go one, two, three, four, that would be the image of point A. We could maybe denote that as A prime. So if you're doing this on the Khan Academy exercise, you would actually just click on a point right over there, and it would show up, but this would be the reflection of point A across the line L.

Let's do another example. So here, we're asked to plot the image of point B under reflection across the x-axis. All right, so this is point B, and we're going to reflect it across the x-axis right over here. So to go from B to the x-axis, it's exactly five units below the x-axis: one, two, three, four, five.

So if we were to reflect across the x-axis, essentially create its mirror image, it's going to be five units above the x-axis: one, two, three, four, five. So that's where the image would be. Maybe we could denote that with B prime. We are reflecting across the x-axis.

Let's do another example. So here they tell us point C prime is the image of C, which is at the coordinates negative four, comma negative two, under a reflection across the y-axis. What are the coordinates of C prime? So pause this video and see if you can figure it out on your own.

There are several ways to approach it. It doesn't hurt to do a quick visual diagram. So that could be my x-axis, this would be my y-axis, and it's the point negative four, comma negative two. So that might look like this: negative four, negative two. So this is the point C right over here, and we want to reflect across the y-axis.

So we want to reflect across the y-axis, which I am coloring in in red right over here. So let's see: the point C is four to the left of the y-axis, so its reflection is going to be four to the right of the y-axis. So let me do it like this. Instead of being four to the left, we want to go four to the right, so plus four.

So where would that put our C prime? Our C prime would be right over there. And what would its coordinates be? Well, it would have the same y-coordinate, so C prime would have a y-coordinate of negative two. But what would its x-coordinate be? Well, instead of it being negative four, it gets flipped over the y-axis, so now it's going to have an x-coordinate of positive four.

So the coordinates here would be four, comma negative two. Four, comma negative two. You might have been able to do this in your head, although for me, even if I tried to do it in my head, I would still have this visualization going on in my house.

Okay, negative four, comma negative two: I'm sitting there in the third quadrant. If I'm flipping over the y-axis, my y-coordinate wouldn't change, but my x-coordinate would become the opposite, and I would end up in the fourth quadrant. And that's exactly what happened: my y-coordinate did not change, but then my x-coordinate, since I'm flipping over the y-axis, it became the negative of this—the opposite of negative four, which is positive four.