Answer :
To determine the image of the point [tex]\((11, -5)\)[/tex] after a translation [tex]\(T_{(11, -5)}\)[/tex] followed by a reflection across the y-axis [tex]\(R_{x=0}\)[/tex], we follow these steps:
1. Translation [tex]\(T_{(11, -5)}\)[/tex]:
- Start with the initial point [tex]\((11, -5)\)[/tex].
- Translate the point by [tex]\( (11, -5) \)[/tex].
Translation can be done by adding the translation vector to the original point:
[tex]\[ \begin{align*} x' &= x + \Delta x = 11 + 11 = 22, \\ y' &= y + \Delta y = -5 - 5 = -10. \end{align*} \][/tex]
So, the translated point is [tex]\((22, -10)\)[/tex].
2. Reflection across the y-axis [tex]\(R_{x=0}\)[/tex]:
- After translation, we have the point [tex]\((22, -10)\)[/tex].
- Reflect this point across the y-axis.
Reflection across the y-axis inverts the x-coordinate:
[tex]\[ \begin{align*} x'' &= -x' = -22, \\ y'' &= y' = -10. \end{align*} \][/tex]
So, the reflected point is [tex]\((-22, -10)\)[/tex].
Therefore, the image of [tex]\((11, -5)\)[/tex] after the transformation [tex]\(R_{x=0} \circ T_{(11, -5)}\)[/tex] is [tex]\((-22, -10)\)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{(-22, -10)} \][/tex]
1. Translation [tex]\(T_{(11, -5)}\)[/tex]:
- Start with the initial point [tex]\((11, -5)\)[/tex].
- Translate the point by [tex]\( (11, -5) \)[/tex].
Translation can be done by adding the translation vector to the original point:
[tex]\[ \begin{align*} x' &= x + \Delta x = 11 + 11 = 22, \\ y' &= y + \Delta y = -5 - 5 = -10. \end{align*} \][/tex]
So, the translated point is [tex]\((22, -10)\)[/tex].
2. Reflection across the y-axis [tex]\(R_{x=0}\)[/tex]:
- After translation, we have the point [tex]\((22, -10)\)[/tex].
- Reflect this point across the y-axis.
Reflection across the y-axis inverts the x-coordinate:
[tex]\[ \begin{align*} x'' &= -x' = -22, \\ y'' &= y' = -10. \end{align*} \][/tex]
So, the reflected point is [tex]\((-22, -10)\)[/tex].
Therefore, the image of [tex]\((11, -5)\)[/tex] after the transformation [tex]\(R_{x=0} \circ T_{(11, -5)}\)[/tex] is [tex]\((-22, -10)\)[/tex].
Hence, the correct answer is:
[tex]\[ \boxed{(-22, -10)} \][/tex]
