Sure, let's understand the translation rule [tex]\( T_{-3,5}(x, y) \)[/tex] step-by-step.
1. Translation Rule: A translation [tex]\( T_{a,b} \)[/tex] means shifting all points in the plane by [tex]\( a \)[/tex] units horizontally and [tex]\( b \)[/tex] units vertically.
2. Given Translation: The rule [tex]\( T_{-3, 5}(x, y) \)[/tex] tells us how to move each point [tex]\((x, y)\)[/tex] of the triangle.
- The [tex]\( -3 \)[/tex] indicates a shift left by 3 units (horizontally).
- The [tex]\( 5 \)[/tex] indicates a shift up by 5 units (vertically).
3. Writing the Translation:
- If you start with a point [tex]\((x, y)\)[/tex], applying the translation [tex]\( T_{-3, 5} \)[/tex] will change the point to [tex]\((x - 3, y + 5)\)[/tex].
4. Options Analysis:
- [tex]\((x, y) \rightarrow (x - 3, y - 5)\)[/tex]: This translation moves [tex]\( x \)[/tex] to the left by 3 units and [tex]\( y \)[/tex] down by 5 units. This is not our rule.
- [tex]\((x, y) \rightarrow (x + 3, y - 5)\)[/tex]: This translation moves [tex]\( x \)[/tex] to the right by 3 units and [tex]\( y \)[/tex] down by 5 units. This is also not our rule.
- [tex]\((x, y) - (x + 3, y + 5)\)[/tex]: This seems to be an incorrect notation and does not represent any useful transformation.
- [tex]\((x, y) \rightarrow (x - 3, y + 5)\)[/tex]: This translation moves [tex]\( x \)[/tex] to the left by 3 units and [tex]\( y \)[/tex] up by 5 units, which matches our given rule [tex]\( T_{-3, 5}(x, y)\)[/tex].
Therefore, the correct way to write the translation rule [tex]\( T_{-3, 5}(x, y) \)[/tex] is:
[tex]\[
(x, y) \rightarrow (x - 3, y + 5)
\][/tex]
So, the correct answer is:
[tex]\[
(x, y) \rightarrow (x - 3, y + 5)
\][/tex]
