Let's break down the translation rule [tex]\( T_{-8,4}(x, y) \)[/tex]. This rule indicates that every point [tex]\((x, y)\)[/tex] on the triangle is moved according to the given translation. Specifically, the translation rule means that each [tex]\( x \)[/tex]-coordinate is decreased by 8 units and each [tex]\( y \)[/tex]-coordinate is increased by 4 units.
To formalize this, we can write the translation rule as:
[tex]\[ T_{-8, 4}(x, y) \rightarrow (x - 8, y + 4) \][/tex]
Now, let's examine each given option to see which one matches this rule:
1. [tex]\((x, y) \rightarrow (x + 4, y - 8)\)[/tex]
- This rule increases the [tex]\( x \)[/tex]-coordinate by 4 and decreases the [tex]\( y \)[/tex]-coordinate by 8. This does not match our translation rule.
2. [tex]\((x, y) \rightarrow (x - 4, y - 8)\)[/tex]
- This rule decreases the [tex]\( x \)[/tex]-coordinate by 4 and decreases the [tex]\( y \)[/tex]-coordinate by 8. This does not match our translation rule either.
3. [tex]\((x, y) \rightarrow (x - 8, y + 4)\)[/tex]
- This rule decreases the [tex]\( x \)[/tex]-coordinate by 8 and increases the [tex]\( y \)[/tex]-coordinate by 4. This perfectly matches our translation rule [tex]\( T_{-8, 4}(x, y) \)[/tex].
4. [tex]\((x, y) \rightarrow (x + 8, y - 4)\)[/tex]
- This rule increases the [tex]\( x \)[/tex]-coordinate by 8 and decreases the [tex]\( y \)[/tex]-coordinate by 4. This does not match our translation rule.
Based on the analysis above, the translation rule [tex]\( T_{-8, 4}(x, y) \)[/tex] is correctly represented by:
[tex]\[ (x, y) \rightarrow (x - 8, y + 4) \][/tex]
Thus, the correct option is:
[tex]\[ \boxed{(x, y) \rightarrow (x - 8, y + 4)} \][/tex]
