Question

3. What is the composition of the transformations written as

Answer 1

Answers:

a.   T(4, -3)

b.   T(-6, 5)

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Explanation:

The "T" stands for "translation" (not transformation; even though a translation is a type of transformation).

When we say something like T(3,-2), we mean "move 3 units to the right and 2 units down". The first coordinate handles left/right movement. Positive for right, negative for left. It's just like what you'd encounter on an xy grid system. The second coordinate handles the up and down motions (positive = up, negative = down).

The notation T(3,-2) is basically the same as writing the rule $(x,y) \to (x+3,y-2)$

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It's perfectly possible to chain multiple translations together. The goal is to get it as one translation instead of two separate ones.

As you can probably guess, the T(1,-1) will have the point move 1 unit to the right and one unit down.

If we add 1 unit to the right on top of the x+3, then we bump it up to x+4

If we subtract 1 unit from the y-2, then we get to y-3

So we would go from $(x,y) \to (x+3,y-2)$  to $(x,y) \to (x+4,y-3)$

This then condenses to the shorter notation T(4,-3)

It says "move 4 units to the right and 3 units down.

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An example is shown below involving the point (5,7).

The translation T(3,-2) moves point A to point B.

Then the translation T(1,-1) moves point B to point C.

We can directly move from A to C applying the translation T(4,-3).

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All of that above explained part (a). Part (b) is the same idea, but with different numbers. The order of translations doesn't matter.

Instead of repeating the same steps as above, I'll just talk about the shortcut.

If we had two translations T(a,b) and T(c,d) composited together, then they simplify to T(e,f) where e = a+c and f = b+d. It's as simple as adding the two corresponding coordinates.

So T(-4,0) and T(-2,5) combine to T(-6,5).