About this video
The video reflects quadrilateral JKLM over the line y = -x. Testing the point (1, 2), which lands on (-2, -1), helps recall the rule (x, y) -> (-y, -x): swap the coordinates and make both negative. The rule is then applied to each vertex.
You'll be able to
- describe a reflection over y = -x in coordinate notation as (x, y) -> (-y, -x).
- use a test point to recall or check a transformation rule.
- graph the image of a polygon after reflecting over y = -x.
Key words
- reflection
- A flip of a figure over a line so each point lands the same distance on the other side.
- line y = -x
- The diagonal line through the origin going down from left to right, where y is the opposite of x.
- coordinate notation
- A rule like (x, y) -> (-y, -x) that tells you where every point goes.
Transcript
- [Instructor] We're told to graph the image of quadrilateral JKLM, right over here, JKLM after a reflection over the line y = -x. And then we wanna describe the reflection in coordinate notation. So pause this video, have a go at it, then we'll do this together.
All right, so first let's visualize the line that we're reflecting over. The line. y = -x looks like this. y = -x looks like this. Some of you might already know the transformation, but I am going to try to remember what it is and not just tell you what the transformation is.
So if we're reflecting over this line y = -x, let's imagine this point over here, the point zero, comma... Actually, let me do a point that actually has a non-zero coordinate here. So let's say this point right over here, which is the point 1, 2. Let's just think about where it would end up.
So if we were reflect over this line, you'd go straight to that line and you'd go the same distance across this line. So you're going to go right over to this point right over here. And what is this point? This is -2, -1. And this helped me remember. I didn't prove this, I didn't prove what the transformation is, but it helped me remember what it is 'cause what just happened?
The new x-coordinate is the negative of the old y-coordinate. And the new y-coordinate is the negative of the old x-coordinate. So if I were to write this in coordinate notation, my new x is the negative of the old y, and my new y is the negative of the old x.
And you could validate that with other points if you like. Some of you might have already known this as the transformation when you reflect over the line y = -x, but I always like to try it out with a point before I go all in. So now let's just do that transformation for each of these points.
So let's start with this point right over here, which is at the point 5, -3. Well, if we take the negative of the y, that goes to positive three, positive three, and then the negative of the x is negative five. So this will map to this point over here. So I'll write that -3, or sorry, positive three.
+3, -5. All right, three more to go. Let's do this point. This is the point two, comma, and this is getting a little bit messy, but I'll try to keep track of things. This is the point 2, -6. So I'm going to map that. I'm gonna swap these and make them negative.
So it's gonna be positive six and negative two. So it's going to go over here. All right, two more to go. This point over here is -1, -6. Well, I swap them and make them the opposite. So then I go to 6, 1. 6, 1 is going to be right over there. And then last but not least, let me do this in this...
Well, lemme do this, and finding the colors is the hardest part, right over here. This point is at -1, -3. So if you swap them, it's gonna go to positive and make them negative. It's gonna go positive three and then positive one. So it's gonna go to this point right over there.
And then we just connect the lines. So we get that, we get that, we get that, and then we get this. And there you have it. We've graphed the quadrilateral JKLM after reflection over the line y = -x. And you can visually see that, that it is the reflection over that line.
And so this transformation served us well.
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