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Compositions of Transformations - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

A composition of transformations is a combination of two or more transformations, each performed on the previous image. A composition of reflections over parallel lines has the same effect as a translation (twice the distance between the parallel lines). A composition of reflections over intersecting lines is the same as a rotation (twice the measure of the angle formed by the lines).

We said there are 3 types of isometries, translations, reflections and rotations. When you put 2 or more of those together what you have is the composition of transformations, so basically what you're saying is you could translate something and then reflect it and that would a composition of transformations. So let's look at what translating point a according to these two order pair rules would actually be equivalent to, so our first one we have xy maps onto x-2 and y-7, so to get an idea let's just pick a point a anywhere on our order plane. So let's say point a is right here and that is at 1, 2, 3, 4 up 1, 2, 3, 4 so point a is going to be at 4, 4 so when I perform my first translation which I know because we're adding and subtracting from y and x, x-2 well 2-4 is going, excuse me 4-2 is 2 so we know that our a prime which I'm going to write right here, a prime will be at 2 and from 4 I need to subtract 7 so that's going to be at -3 so I'm going to go over 2 and then down 1, 2, 3 and that will be my a prime we performed our first translation.
Our second one says to add 1 to our x so a double prime, if I add 1 to 2 that's going to be 3 and if I add 3 to my y we're going to be at 0 because -3+3 is 0 so a double prime is that 1, 2, 3 0 so I'm going to write a double prime right there. So notice what we did our first translation went down and then our next one, went kind of so our firstdown onto the left next went up into the right. But couldn't they have just been the same as moving from a to a double prime? Could we have just said that in one single translation? And the answer is yes so the way that you would do this is if we come back here add up what we're doing to our x's so we're going to say that -2+1 is -1 so the equivalent translation was doing x-1 but let's go back and make sure that that was correct we started at 4 and a double prime is at 3, 0 then yes 4-1 is 3 so it checks out. If we go back here we see that we have y-7 and y+3, -7+3 is -4 so according to our overall translation we should have from where we started dropped down 4 and if we started at y=4 and if we've ended up at 0 then yes that is correct. So whenever you have more than one translations, all you need to do is look at your x coordinates add up those up whatever you're doing to them and do the same thing for the y's and you'll have your overall translation.

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