To solve the problem of identifying the correct translation rule, we need to understand the given rule [tex]\( T_{-8,4}(x, y) \)[/tex]. This rule tells us how to translate a point [tex]\((x, y)\)[/tex] on a coordinate plane.
Let's break down what this translation rule does:
- The notation [tex]\( T_{-8,4} \)[/tex] means that we shift the point by [tex]\(-8\)[/tex] units in the x-direction and [tex]\(+4\)[/tex] units in the y-direction.
This specific transformation can be represented by the following rule:
[tex]\[ (x, y) \rightarrow (x - 8, y + 4) \][/tex]
Now, let's analyze the given options to determine which one accurately represents this translation:
1. [tex]\((x, y) \rightarrow (x+4, y-8)\)[/tex]:
- This translates a point by adding 4 to the x-coordinate and subtracting 8 from the y-coordinate. It does not match our rule.
2. [tex]\((x, y) \rightarrow (x-4, y-8)\)[/tex]:
- This translates a point by subtracting 4 from the x-coordinate and subtracting 8 from the y-coordinate. It does not match our rule.
3. [tex]\((x, y) \rightarrow (x-8, y+4)\)[/tex]:
- This translates a point by subtracting 8 from the x-coordinate and adding 4 to the y-coordinate. This matches our rule exactly.
4. [tex]\( (x, y) \rightarrow (x+8, y-4)\)[/tex]:
- This translates a point by adding 8 to the x-coordinate and subtracting 4 from the y-coordinate. It does not match our rule.
Thus, the correct alternative way to write the translation rule [tex]\( T_{-8,4}(x, y) \)[/tex] is:
[tex]\[ (x, y) \rightarrow (x-8, y+4) \][/tex]
The answer is:
[tex]\[ \boxed{(x, y) \rightarrow (x-8, y+4)} \][/tex]
