To find the location of [tex]\( X^ \)[/tex] starting with the point [tex]\( X \)[/tex] at coordinates [tex]\((8, 4)\)[/tex], we need to follow two transformations: a translation and a reflection. Here are the steps:
1. Translation: The translation rule given is [tex]\((x, y) \rightarrow (x + 4, y - 1)\)[/tex].
- For the point [tex]\( X \)[/tex] at coordinates [tex]\((8, 4)\)[/tex]:
- The new [tex]\( x \)[/tex]-coordinate after translation: [tex]\( 8 + 4 = 12 \)[/tex].
- The new [tex]\( y \)[/tex]-coordinate after translation: [tex]\( 4 - 1 = 3 \)[/tex].
- So, after translation, the coordinates are [tex]\((12, 3)\)[/tex].
2. Reflection across the [tex]\( y \)[/tex]-axis: Reflecting a point across the [tex]\( y \)[/tex]-axis changes the sign of the [tex]\( x \)[/tex]-coordinate while keeping the [tex]\( y \)[/tex]-coordinate the same.
- For the translated point at coordinates [tex]\((12, 3)\)[/tex]:
- The new [tex]\( x \)[/tex]-coordinate after reflection: [tex]\(-12\)[/tex].
- The [tex]\( y \)[/tex]-coordinate remains the same: [tex]\( 3 \)[/tex].
- So, after reflection, the coordinates are [tex]\((-12, 3)\)[/tex].
Hence, the location of [tex]\( X^ \)[/tex] after performing both transformations is [tex]\((-12, 3)\)[/tex].
