To solve the problem, let's break it down step by step:
1. Translation:
- We start with the point [tex]\( (11, -5) \)[/tex].
- We need to apply a translation by vector [tex]\( (11, -5) \)[/tex].
When you translate a point [tex]\( (x, y) \)[/tex] by a vector [tex]\( (a, b) \)[/tex], the new coordinates are:
[tex]\[
(x + a, y + b)
\][/tex]
Therefore, translating [tex]\( (11, -5) \)[/tex] by [tex]\( (11, -5) \)[/tex] gives us:
[tex]\[
(11 + 11, -5 + (-5)) = (22, -10)
\][/tex]
2. Reflection:
- Next, we need to reflect the translated point [tex]\( (22, -10) \)[/tex] over the line [tex]\( x = 0 \)[/tex] (the y-axis).
When you reflect a point [tex]\( (x, y) \)[/tex] over the y-axis, the x-coordinate changes sign, while the y-coordinate remains the same:
[tex]\[
(-x, y)
\][/tex]
Therefore, reflecting [tex]\( (22, -10) \)[/tex] over the y-axis gives us:
[tex]\[
(-22, -10)
\][/tex]
Thus, the image of the point [tex]\( (11, -5) \)[/tex] after applying the composition of translation by [tex]\( (11, -5) \)[/tex] followed by reflection over the line [tex]\( x = 0 \)[/tex] is:
[tex]\[
(-22, -10)
\][/tex]
Hence, the correct answer is [tex]\((-22, -10)\)[/tex].
