Graphing a non similar pre image algebraically
About this video
Triangle DEF is the image of triangle ABC after the transformation (x, y) -> (-x, 2y), and the task is to graph the pre-image ABC. Sal undoes the transformation with the reverse rule (x, y) -> (-x, y/2), mapping F(-8, 8), D(5, 6) and E(1, -4) to (8, 4), (-5, 3) and (-1, -2), then checks one point. He notes this transformation distorts angles, so it does not preserve similarity.
You'll be able to
- tell the difference between finding an image and finding a pre-image.
- write the reverse of a coordinate transformation.
- find pre-image coordinates and check them with the original rule.
- explain why stretching in only one direction does not preserve similarity.
Key words
- pre-image
- The original figure before a transformation is applied.
- image
- The figure you get after applying a transformation.
- composed transformation
- Two or more transformations applied one after another, written as a single rule.
- reverse transformation
- A rule that undoes a transformation, taking the image back to the pre-image.
- similarity
- Two figures are similar if they have the same shape: equal angles and proportional sides.
Transcript
- [Presenter] We're told triangle DEF. They just had to write one triangle there. So triangle DEF is the image of triangle ABC after the following composed transformation, all right. Graph the pre-image of triangle ABC. So pause this video, and have a go at it before we do this together.
Okay. So the key here is that they want the pre-image. Some of you might've just said, "Oh, I just have to perform this transformation." But that's not what they're saying. They're not saying apply this transformation to this triangle. They said, "We started with some triangle ABC.
"Then we performed this transformation, "and then we got triangle DEF." So what we really want to do is go in reverse. So you might just want to think about what's happening here. The original transformation took the negative of the x value to get our new x value, and then it doubled the y value.
So if you wanted to go and unwind this, what you would need to do, if you wanted to go back to the... Let me do it this way. If you wanted to go back to the original, you would take your x, and then you would multiply that times -1, and then you would take your y, and you would multiply that times 1/2.
Or another way to think about it is, if we performed the following transformation, which you could almost view as the reverse transformation, where you take the negative of the x and you take half of the y, and we perform this on DEF, we should go back to ABC.
And we can verify that. Once we think we have ABC, we can do the original transformation and see what happens. So let's just do that. So this F right over here is at the point, this is at the point -8,8. So if we apply this transformation, this negative 8 becomes a positive 8.
And then this other 8, you take half of that and you get a 4. So it becomes positive 8 and 4. So it goes over there. Let me just write that. That is 8,4. This point D right over here is at the point 5,6. Well, you take the negative of that 5, you get to -5, and you take half of that 6.
So -5,3. So it takes us right there. So we write that's -5,3. And then last, but not least, this point right over here, E, is at negative, oh, sorry, it's at positive 2 and negative 4. So you take the opposite of that positive. Oh, positive 1. Let me... What am I doing?
1,-4. I don't want to make a mistake here. So you take the negative of that 1, you get to -1, and then you take half of that -4, you get to -2. You get right over here. So you go to -1 and then -2. And so this is what I am saying. I would argue that this is our ABC triangle, and, if we want, we can validate that.
We can actually apply this original transformation. For example, if I started with -5,3. If I took the negative of that x coordinate, it would become positive 5, and if I were to double the y coordinate, I'd go to 6. And that's exactly what happened. This got mapped to that.
And you can verify the other points would map just like we said they would. Now, one thing to take note of is that the angles in this transformation do get distorted. So it does not preserve similarity. Think about why that is.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.