Question

What is the composition of the translations (T<−3, 4> ∘ T<8, −7>) (x, y) as one translation?

(T<−3, 4> ∘ T<8, −7>)(x, y) = T<m, n>(x, y) where m =
and n =
.

Answer 1

To find the composition of the translations \( T_{<-3, 4>} \) and \( T_{<8, -7>} \), we need to understand how translations work.

A translation \( T_{<a, b>} \) transforms a point \( (x, y) \) to the point \( (x + a, y + b) \).

For the two translations given:

1. \( T_{<-3, 4>} \) transforms \( (x, y) \) to \( (x - 3, y + 4) \).
2. \( T_{<8, -7>} \) transforms \( (x, y) \) to \( (x + 8, y - 7) \).

Now, we compose \( T_{<-3, 4>} \) and \( T_{<8, -7>} \):

1. First apply \( T_{<8, -7>} \) to \( (x, y) \):
\[
(x, y) \rightarrow (x + 8, y - 7)
\]

2. Now apply \( T_{<-3, 4>} \) to the result from step 1:
\[
(x + 8, y - 7) \rightarrow ((x + 8) - 3, (y - 7) + 4) = (x + 5, y - 3)
\]

Thus, the composition \( (T_{<-3, 4>} \circ T_{<8, -7>})(x, y) \) gives us the single translation to point \( (x + 5, y - 3) \).

This implies that we can express this as a single translation \( T_{<m, n>} \) where:

- \( m = 5 \)
- \( n = -3 \)

Therefore, we have:
- \( m = 5 \)
- \( n = -3 \)