To find the pre-image of a vertex [tex]\( A' \)[/tex] given that the rule used is [tex]\( r_{y \text{-axis}}(x, y) \rightarrow (-x, y) \)[/tex], we need to apply the inverse transformation to each image vertex.
The given image vertices [tex]\( A' \)[/tex] are:
1. [tex]\( A_1'(-4, 2) \)[/tex]
2. [tex]\( A_2'(-2, -4) \)[/tex]
3. [tex]\( A_3'(2, 4) \)[/tex]
4. [tex]\( A_4'(4, -2) \)[/tex]
The rule [tex]\( r_{y \text{-axis}}(x, y) \rightarrow (-x, y) \)[/tex] reflects a point across the y-axis by changing the sign of the x-coordinate. Now, we need to reverse this operation to find the original (pre-image) vertices.
1. For [tex]\( A_1'(-4, 2) \)[/tex]:
- Start with [tex]\((x, y) = (-4, 2)\)[/tex]
- To find the pre-image, we reverse the operation, so we take [tex]\((-x, y)\)[/tex] to [tex]\( (4, 2) \)[/tex]
2. For [tex]\( A_2'(-2, -4) \)[/tex]:
- Start with [tex]\((x, y) = (-2, -4)\)[/tex]
- To find the pre-image, we reverse the operation, so we take [tex]\((-x, y)\)[/tex] to [tex]\( (2, -4) \)[/tex]
3. For [tex]\( A_3'(2, 4) \)[/tex]:
- Start with [tex]\((x, y) = (2, 4)\)[/tex]
- To find the pre-image, we reverse the operation, so we take [tex]\((-x, y)\)[/tex] to [tex]\( (-2, 4) \)[/tex]
4. For [tex]\( A_4'(4, -2) \)[/tex]:
- Start with [tex]\((x, y) = (4, -2)\)[/tex]
- To find the pre-image, we reverse the operation, so we take [tex]\((-x, y)\)[/tex] to [tex]\( (-4, -2) \)[/tex]
Therefore, the pre-images of the vertices are:
1. [tex]\( A_1(4, 2) \)[/tex]
2. [tex]\( A_2(2, -4) \)[/tex]
3. [tex]\( A_3(-2, 4) \)[/tex]
4. [tex]\( A_4(-4, -2) \)[/tex]
These are the coordinates of the original vertices before the transformation was applied.
