Reflections: graph to algebraic rule | Transformational geometry | Grade 8 (TX) | Khan Academy

We're told that quadrilateral A'B'C'D' is the image of quadrilateral ABCD after reflection. So we can see ABCD here and A'B'C'D' right over here. What we want to do is figure out a rule for this transformation. So pause this video and have a go at that by yourself before we do this together.

Just as a reminder, a rule for a transformation will look something like this: it's saying for every (x, y) in the pre-image, for example ABCD, what does it get mapped to in the image? And so it's going to tell us, well, how are these new coordinates based on x and y?

There are a couple of ways we could do that. We could just think about each of these points; for example, point A, and then what happens when it goes to A', and see if we can come up with a rule that works for all of them.

For example, point A is at the point (5, 6). Let's see the image when it goes to A'. It looks like it's at (-5, -6). So the x-coordinate stayed the same if I just look at this point, but the y-coordinate became the negative of it. That makes sense because when we do this reflection across the x-axis, it makes sense that our x-coordinate stays the same but that the y-coordinate, since it gets flipped down, becomes the negative; it becomes the opposite of what it was before.

So my candidate for this transformation for the rule here is that x stays the same and that y becomes the opposite. But we could do that with a few more points just to make sure that that holds up.

For example, we could look at point B in the pre-image, which is at (-6, 5). If this rule holds up when we do this reflection, B' should be at -6, making the y the opposite of this, so it should be at (-6, -5). If we go to (-6, -5), that is indeed where B' is.

You can validate the other points if you like, but this should just make intuitive sense: the x-coordinate stays the same, but the y-coordinate becomes the opposite.