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Course: Grade 8 (TX) > Unit 8
Lesson 9: Algebraic rules for transformations- Translations: graph to algebraic rule
- Translations: description to algebraic rule
- Rotations: graph to algebraic rule
- Rotations: description to algebraic rule
- Reflections: graph to algebraic rule
- Reflections: description to algebraic rule
- Algebraic rules for transformations
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Rotations: graph to algebraic rule
A worked example of going from a graph of a rotation to an algebraic rule. Created by Sal Khan.
Video transcript
- [Host] We're told that
Eduardo rotated triangle ABC by 90 degrees clockwise about the origin to create triangle A
prime, B prime, C prime. So what Eduardo did is took
this triangle right over here, rotated it 90 degrees
clockwise, so that's rotating at 90 degrees clockwise about the origin. So the rotation was like that, which does look right, write a rule that describes this transformation. So pause this video and have a go at that before we do this together. All right, so just to
remind ourselves what a rule for transformation looks like. It starts with each X or Y, coordinate in what
we could call the pre-image before we have transformed. And then it says, well, what will that coordinate become in the image? And so there's going to be
some new point that's going to have some X's or Y's here
and going to have some Xs or Ys there. Now, before we do that, let's just think about what's happening to the different points in
the pre-image when they get rotated onto that image. And to do that, I will set
up a little table here. So this is going to be the pre-image points, pre-image. So before we do the transformation, and then I'll try doing
this similar color, this will be the points
on the image in red. So let's write these points down. So point A right over here,
what are its coordinates before the rotation? A is that the coordinate X
equals four, Y equals five, so four comma five, what is point B in the pre-image? It is four comma three, four comma, three, point C in the pre-image
is seven comma five, seven comma five. Now what happens to these
once they get rotated? So the corresponding
point on the image to A, so I'll call that A prime. That is A prime right over there. That is at the 5 comma negative four, five comma negative four. B prime is at three, negative four, three, negative four. And then C prime is at
five, negative seven, five, negative seven. So can we see a pattern here? Well, we have a four for X here, and then we have a
negative four for X there. So it looks like what was the
X in A became the negative of that became the Y in A prime. So let me actually just write that down. I'll write that in purple
to show how tentative I am. So whatever the X here,
maybe the negative of that becomes the new Y and it looks like whatever was the old Y becomes the new X. At least that's what
I'm seeing for point A. So maybe we take the
old X in the pre-image, take the negative, and then
make that the images Y. And then what was the Y? We make that into the new X. Let's see if that holds up. So for point B, if we took this four,
took the negative of it and made that the new Y, that's
exactly what happened there. And then if we take the old Y and just put it in for X, that is exactly what happened there. So this is holding up and it also holds up with point C right over there. So there you have it. We have got our rule. This rotation, you can
view as if you're starting with an X, Y point. You can get the new coordinate
of where that point is, where its image is going to be after rotation by putting
the Y in place for the X, and then putting the negative of the X in, in place for the Y.