Reflections: graph to algebraic rule
About this video
Quadrilateral ABCD is reflected to A'B'C'D'. Comparing A(-5, 6) with A'(-5, -6) shows the reflection is across the x-axis, so x stays the same and y becomes its opposite: (x, y) -> (x, -y). Point B confirms the rule.
You'll be able to
- read coordinates of a point and its image from a graph.
- write the rule for a reflection across the x-axis.
- use a rule to predict an image point and check it on the graph.
Key words
- reflection
- A transformation that flips a shape over a line, like a mirror image.
- x-axis
- The horizontal number line on a coordinate grid.
- opposite
- The number with the same size but the other sign, like 6 and -6.
- image
- The shape after a transformation, labelled with primes.
Transcript
- [Instructor] We're told that quadrilateral A prime, B prime, C prime, D prime, is the image of quadrilateral, ABCD after a reflection. So we can see A, B, C, D here and A prime, B prime, C prime, D prime, right over here. And 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. So 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, A, B, C, D, what does it get mapped to in the image?
And so it's gonna tell us, well, how are these new coordinates based on x and y? And there's 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 prime, and see if we can come up with a rule that works for all of them.
So for example, point A is at the point -5, 6; -5, 6. And let's see, the image when it goes to A prime is at... It looks like it's at -5, -6;
- 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. And 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, it 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, is at -6, 5; -6, 5.
And if this rule holds up, when we do this reflection, B prime should be at, we should hold the x the same, -6. And we should make the y the opposite of this. So it should be at -6, -5. And if we go to -6, -5 that is indeed where B prime is. And 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.
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