Translations: graph to algebraic rule
About this video
The video finds the algebraic rule for a translation shown on a graph. By counting how far point C moves (5 down and 6 left) and checking with point B, it writes the rule (x, y) -> (x - 6, y - 5).
You'll be able to
- tell a translation apart from a dilation by comparing sizes.
- count horizontal and vertical shifts between a point and its image.
- write a translation rule in the form (x, y) -> (x + a, y + b) and check it with another point.
Key words
- translation
- A transformation that slides every point the same distance in the same direction.
- dilation
- A transformation that makes a shape bigger or smaller.
- image
- The new shape after a transformation, labelled with primes like A'B'C'.
- rule
- An instruction like (x, y) -> (x - 6, y - 5) that says where each point goes.
Transcript
- [Instructor] We're told Lucas translated triangle ABC to create triangle A'B'C'. So we went from this blue one or blue-green one to this burgundy one or this red one. Write a rule that describes this transformation. So pause this video and try to figure this out on your own before we do this together.
Alright, so when we look at what's going on here, and actually they tell us what type of transformation it is, this is a translation and we can tell it hasn't been dilated, that the triangles look like they're the same size. For example, if we go from A' to B', it's down two, and to the right one.
And if we look at going from A to B, it's also down two, and to the right one. So they're the same size, they haven't been scaled, but they definitely have been translated, or A, B, C has been translated to get us to A'B'C'. And to think about what that translation is, we just have to say, okay, for each of these points, what happened to 'em?
How did they get moved? Well, it looks like, for example, point C right over here, to go from C to C', we had to go, it looks like, how many units down is that? One, two, three, four, five. So we had to go five units down. And then we had to go, it looks like one, two, three, four, five, six units to the left.
So just based off of that, and it looks like the other ones have done the same, but I can double check that. The rule I would say is for every X and Y for A, B and C, we have now mapped it where in the X direction we have been shifted to the left by six. So whatever our original X was, we are now going to subtract by six, and whatever our original Y was, we are now going to be five lower, so we're going to subtract by five.
So that's the rule. And we could double check it with one of these other points. For example, we could look at, let's say point B. If we go down five, we will end up right over here. And then if we go to the left six, we are going to end up right over there at B'.
So this rule definitely describes this translation.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.