To determine which of the given translations corresponds to the rule [tex]\( T_{-8,4}(x, y) \)[/tex], we need to understand the notation and how translations work in the coordinate plane.
The notation [tex]\( T_{-8,4}(x, y) \)[/tex] indicates a translation. Specifically:
- The [tex]\( -8 \)[/tex] indicates that each x-coordinate is decreased by 8.
- The [tex]\( +4 \)[/tex] indicates that each y-coordinate is increased by 4.
In other words, any point [tex]\((x, y)\)[/tex] on the plane will move to a new point [tex]\((x - 8, y + 4)\)[/tex].
Now, let's analyze each of the given translation rules:
1. [tex]\((x, y) \rightarrow (x+4, y-8)\)[/tex]:
- This rule means that the x-coordinate is increased by 4, and the y-coordinate is decreased by 8.
- This does not match the translation specified in [tex]\( T_{-8,4}(x, y) \)[/tex].
2. [tex]\((x, y) \rightarrow (x-4, y-8)\)[/tex]:
- This rule means that the x-coordinate is decreased by 4, and the y-coordinate is decreased by 8.
- This also does not match the translation specified in [tex]\( T_{-8,4}(x, y) \)[/tex].
3. [tex]\((x, y) \rightarrow (x-8, y+4)\)[/tex]:
- This rule means that the x-coordinate is decreased by 8, and the y-coordinate is increased by 4.
- This matches exactly with the translation specified in [tex]\( T_{-8,4}(x, y) \)[/tex].
4. [tex]\((x, y) \rightarrow (x+8, y-4)\)[/tex]:
- This rule means that the x-coordinate is increased by 8, and the y-coordinate is decreased by 4.
- This does not match the translation specified in [tex]\( T_{-8,4}(x, y) \)[/tex].
Therefore, the correct way to write the rule [tex]\( T_{-8,4}(x, y) \)[/tex] is [tex]\((x, y) \rightarrow (x-8, y+4)\)[/tex].
Hence, the answer is: [tex]\((x, y) \rightarrow (x-8, y+4)\)[/tex].
