Certainly! Let's walk through the problem step by step to determine the correct rule describing the translation of the rectangle.
1. Understanding Translation:
- Translation involves moving every point of a figure the same distance in a specified direction.
- Here, the translation is 5 units up and 3 units to the left.
2. Effect on Coordinates:
- When you translate a point [tex]\( (x, y) \)[/tex] on a coordinate plane:
- Moving 3 units to the left means subtracting 3 from the x-coordinate.
- Moving 5 units up means adding 5 to the y-coordinate.
3. Constructing the Rule:
- Given the point [tex]\( (x, y) \)[/tex]:
- After moving 3 units to the left, the new x-coordinate will be [tex]\( x - 3 \)[/tex].
- After moving 5 units up, the new y-coordinate will be [tex]\( y + 5 \)[/tex].
4. Formulating the Translation Rule:
- Combining these changes, the transformation rule that describes this translation is:
[tex]\[
(x, y) \rightarrow (x-3, y+5)
\][/tex]
5. Verifying Against Options:
- Let's compare our rule with the given options:
- [tex]\( (x, y) \rightarrow (x+5, y-3) \)[/tex]: This translates 5 units right and 3 units down.
- [tex]\( (x, y) \rightarrow (x+5, y+3) \)[/tex]: This translates 5 units right and 3 units up.
- [tex]\( (x, y) \rightarrow (x-3, y+5) \)[/tex]: This translates 3 units left and 5 units up.
- [tex]\( (x, y) \rightarrow (x+3, y+5) \)[/tex]: This translates 3 units right and 5 units up.
6. Choosing the Correct Option:
- The correct transformation rule is [tex]\((x, y) \rightarrow (x-3, y+5)\)[/tex].
Therefore, the rule that describes the translation of the rectangle 5 units up and 3 units to the left is:
[tex]\[
(x, y) \rightarrow (x-3, y+5)
\][/tex]
