Question

find a translation that has the same effect as the composition of translations below. t_(1,3)(x,y) followed by t_(-5,7)(x,y)
choose the correct answer below.
a. (x,y)→(x + 6,y + 10)
b. (x,y)→(x - 4,y + 10)
c. (x,y)→(x - 4,y - 4)
d. (x,y)→(x + 6,y - 4)

Explanation:

Step1: Analyze first translation

For $T_{(1,3)}(x,y)$, the transformation is $(x,y)\to(x + 1,y+3)$.

Step2: Analyze second translation

For $T_{(-5,7)}(x,y)$, starting from the result of the first - translation $(x + 1,y + 3)$, the new transformation is $(x+1,y + 3)\to((x + 1)-5,(y + 3)+7)$.

Step3: Simplify the combined transformation

Simplify $((x + 1)-5,(y + 3)+7)$:
\[

$$\begin{align*} (x + 1-5,y+3 + 7)&=(x-4,y + 10) \end{align*}$$

\]

Answer:

B. $(x,y)\to(x - 4,y + 10)$