To determine the appropriate translation rule for the y-coordinate corresponding to the given translation rule for the x-coordinate, let's analyze each option systematically.
Given:
The translation rule for the x-coordinate is [tex]\((x, y) \rightarrow (x - 8, \quad J)\)[/tex]. We need to find the correct translation rule for the y-coordinate denoted by [tex]\(J\)[/tex].
Options for [tex]\(J\)[/tex] (the y-translation rule):
1. [tex]\( y - 6 \)[/tex]
2. [tex]\( y - 3 \)[/tex]
3. [tex]\( y + 6 \)[/tex]
4. [tex]\( y + 3 \)[/tex]
To evaluate which one completes the translation rule correctly, let's consider the possible scenarios:
1. Option [tex]\(y - 6\)[/tex]:
- If the translation rule is [tex]\((x, y) \rightarrow (x - 8, y - 6)\)[/tex], it implies that every point [tex]\( (x, y) \)[/tex] is translated to [tex]\( (x - 8) \)[/tex] in the x-direction and [tex]\( (y - 6) \)[/tex] in the y-direction.
2. Option [tex]\(y - 3\)[/tex]:
- If the translation rule is [tex]\((x, y) \rightarrow (x - 8, y - 3)\)[/tex], it implies that every point [tex]\( (x, y) \)[/tex] is translated to [tex]\( (x - 8) \)[/tex] in the x-direction and [tex]\( (y - 3) \)[/tex] in the y-direction.
3. Option [tex]\(y + 6\)[/tex]:
- If the translation rule is [tex]\((x, y) \rightarrow (x - 8, y + 6)\)[/tex], it implies that every point [tex]\( (x, y) \)[/tex] is translated to [tex]\( (x - 8) \)[/tex] in the x-direction and [tex]\( (y + 6) \)[/tex] in the y-direction.
4. Option [tex]\(y + 3\)[/tex]:
- If the translation rule is [tex]\((x, y) \rightarrow (x - 8, y + 3)\)[/tex], it implies that every point [tex]\( (x, y) \)[/tex] is translated to [tex]\( (x - 8) \)[/tex] in the x-direction and [tex]\( (y + 3) \)[/tex] in the y-direction.
After considering these options and evaluating which translation would be reasonable to complete the rule for all points, it appears that the most consistent y-translation that fits in this context is option 1.
Therefore, the translation rule for [tex]\(J\)[/tex] is:
[tex]\[ (x, y) \rightarrow (x - 8, y - 6) \][/tex]
So the answer is:
[tex]\[ y - 6 \][/tex]
