Question

Am I suppose to substitute the variables with random numbers in order to answer these questions???

Translation A maps (x, y) to (x + n, y + 1). Translation B maps (x, y) to (x +s, y + m).
1. Translate a point using Translation A, followed by Translation B. Write an algebraic rule for the final image of the point after this composition.
2. Translate a point using Translation B, followed by Translation A. Write an algebraic rule for the final image of the point after this composition.
3. Compare the rules you wrote for parts (a) and (b). Does it matter which translation you do first? Explain your reasoning.

Answer 1

9514 1404 393

Answer:

1. (x, y) ⇒ (x +n +s, y +1 +m)
2. (x, y) ⇒ (x +s +n, y +m +1)
3. they are identical in effect; order does not matter

Explanation:

Substitute the expressions.

A then B

After the first translation, the value of x is (x+n). Put that as the value of x in the second translation.

x ⇒ x +s . . . . . . . . . the definition of the second translation

(x+n) ⇒ (x+n) +s . . . the result after both translations

The same thing goes for y. After the first translation, its new value is (y+1).

y ⇒ y +m . . . . . . . . the definition of the second translation

(y+1) ⇒ (y+1) +m . . . the result of both translations

Then the composition of A followed by B is (x, y) ⇒ (x +n +s, y +1 +m).

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B then A

The same reasoning applies. After the B translation, the x-coordinate is (x+s) and the y-coordinate is (y+m). Then the A translation changes these to ...

x ⇒ x +n . . . . . . . . . . definition of translation A

(x+s) ⇒ (x+s) +n . . . . translation A operating on point translated by B

y ⇒ y +1 . . . . . . . . . . . definition of translation A

(y+m) ⇒ (y+m) +1 . . . . translation A operating on point translated by B

The composition of B followed by A is (x, y) ⇒ (x + s + n, y + m + 1).

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You should recognize that the sums (x+n+s) and (x+s+n) are identical. The commutative and associative properties of addition let us rearrange the order of the terms with no effect on the outcome.

The two translations give the same result in either order.