To determine the rule for a translation that moves a point 6 units to the left and 4 units up, let's break down what each part of the translation means:
1. Moving 6 units to the left:
- When you move to the left on the x-axis, you are subtracting from the x-coordinate. Therefore, if a point [tex]\((x, y)\)[/tex] is translated 6 units to the left, its new x-coordinate will be [tex]\(x-6\)[/tex].
2. Moving 4 units up:
- When you move up on the y-axis, you are adding to the y-coordinate. Therefore, if a point [tex]\((x, y)\)[/tex] is translated 4 units up, its new y-coordinate will be [tex]\(y+4\)[/tex].
Putting both these steps together:
- The new x-coordinate will be [tex]\(x - 6\)[/tex].
- The new y-coordinate will be [tex]\(y + 4\)[/tex].
So, the rule describing this translation can be written as:
[tex]\[ (x, y) \rightarrow (x - 6, y + 4) \][/tex]
Reviewing the given options, we find that the correct rule is:
[tex]\[ (x, y) \rightarrow (x - 6, y + 4) \][/tex]
This corresponds to the fourth option in the list provided:
[tex]\[ (x, y) \rightarrow (x-6, y+4) \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
