Certainly! To find the pre-image of the given vertices after the rule [tex]\(r_{y\text{-axis}}\)[/tex] which maps [tex]\((x, y) \to (-x, y)\)[/tex], we need to reverse the transformation. This means we will negate the x-coordinates again to return to the original points.
Here is the step-by-step process for each vertex:
1. For vertex [tex]\(A'(-4, 2)\)[/tex]:
- The original point would have a negated x-coordinate.
- Reverse the rule: Negate the x-coordinate to get [tex]\(-( -4) = 4\)[/tex].
- The pre-image is [tex]\(A(4, 2)\)[/tex].
2. For vertex [tex]\(A'(-2, -4)\)[/tex]:
- The original point would have a negated x-coordinate.
- Reverse the rule: Negate the x-coordinate to get [tex]\(-( -2) = 2\)[/tex].
- The pre-image is [tex]\(A(2, -4)\)[/tex].
3. For vertex [tex]\(A'(2, 4)\)[/tex]:
- The original point would have a negated x-coordinate.
- Reverse the rule: Negate the x-coordinate to get [tex]\(-(2) = -2\)[/tex].
- The pre-image is [tex]\(A(-2, 4)\)[/tex].
4. For vertex [tex]\(A'(4, -2)\)[/tex]:
- The original point would have a negated x-coordinate.
- Reverse the rule: Negate the x-coordinate to get [tex]\(-(4) = -4\)[/tex].
- The pre-image is [tex]\(A(-4, -2)\)[/tex].
Thus, the pre-image coordinates of the given vertices after reversing the rule [tex]\(r_{y\text{-axis}}\)[/tex] are:
- [tex]\(A(4, 2)\)[/tex]
- [tex]\(A(2, -4)\)[/tex]
- [tex]\(A(-2, 4)\)[/tex]
- [tex]\(A(-4, -2)\)[/tex]
