To determine which option correctly describes the given translation rule [tex]\( T_{-3,5}(x, y) \)[/tex], we need to understand what this translation entails.
The notation [tex]\( T_{-3,5}(x, y) \)[/tex] indicates a translation where:
- The [tex]\( x \)[/tex]-coordinate of any point is shifted by [tex]\(-3\)[/tex].
- The [tex]\( y \)[/tex]-coordinate of any point is shifted by [tex]\( +5 \)[/tex].
Let's break it down:
1. Translation of the [tex]\( x \)[/tex]-coordinate:
- Translating by [tex]\(-3\)[/tex] means that for any point [tex]\((x, y)\)[/tex], we subtract 3 from the [tex]\( x \)[/tex]-coordinate.
- Mathematically, this means the new [tex]\( x \)[/tex]-coordinate will be [tex]\( x - 3 \)[/tex].
2. Translation of the [tex]\( y \)[/tex]-coordinate:
- Translating by [tex]\( +5\)[/tex] means that for any point [tex]\((x, y)\)[/tex], we add 5 to the [tex]\( y \)[/tex]-coordinate.
- Mathematically, this means the new [tex]\( y \)[/tex]-coordinate will be [tex]\( y + 5 \)[/tex].
Combining these two, the rule [tex]\( T_{-3,5}(x, y) \)[/tex] can be written as:
[tex]\[ (x, y) \rightarrow (x - 3, y + 5) \][/tex]
We now compare this result to the provided options:
- [tex]\((x, y) \rightarrow (x - 3, y + 5)\)[/tex]
- [tex]\((x, y) \rightarrow (x - 3, y - 5)\)[/tex]
- [tex]\((x, y) \rightarrow (x + 3, y - 5)\)[/tex]
- [tex]\((x, y) \rightarrow (x + 3, y + 5)\)[/tex]
Clearly, the correct option that represents the rule [tex]\( T_{-3,5}(x, y) \)[/tex] is:
[tex]\[ (x, y) \rightarrow (x - 3, y + 5) \][/tex]
Therefore, the translation rule [tex]\( T_{-3,5}(x, y) \)[/tex] corresponds to the first option:
[tex]\[ (x, y) \rightarrow (x - 3, y + 5) \][/tex]
