To determine the pre-images of the given vertices \( A \) under the rule of reflection across the y-axis, which is given by the transformation \((x, y) \rightarrow (-x, y)\), we'll work through each vertex step by step.
1. For the vertex \( A(-4, 2) \):
- Given the transformation rule \((x, y) \rightarrow (-x, y)\), we need to find the original point that underwent this transformation to become \( (-4, 2) \).
- Let the original point be \( (x, y) \).
- Applying the rule: \(-x = -4\) and \( y = 2\).
- Solving for \( x \), we find \( x = 4 \).
- Therefore, the pre-image of \( A(-4, 2) \) is \( (4, 2) \).
2. For the vertex \( A(-2, -4) \):
- Let the original point be \( (x, y) \).
- Applying the rule: \(-x = -2\) and \( y = -4\).
- Solving for \( x \), we find \( x = 2 \).
- Therefore, the pre-image of \( A(-2, -4) \) is \( (2, -4) \).
3. For the vertex \( A(2, 4) \):
- Let the original point be \( (x, y) \).
- Applying the rule: \(-x = 2\) and \( y = 4\).
- Solving for \( x \), we find \( x = -2 \).
- Therefore, the pre-image of \( A(2, 4) \) is \( (-2, 4) \).
4. For the vertex \( A(4, -2) \):
- Let the original point be \( (x, y) \).
- Applying the rule: \(-x = 4\) and \( y = -2\).
- Solving for \( x \), we find \( x = -4 \).
- Therefore, the pre-image of \( A(4, -2) \) is \( (-4, -2) \).
In summary, the pre-images are:
- The pre-image of \( A(-4, 2) \) is \( (4, 2) \).
- The pre-image of \( A(-2, -4) \) is \( (2, -4) \).
- The pre-image of \( A(2, 4) \) is \( (-2, 4) \).
- The pre-image of \( A(4, -2) \) is \( (-4, -2) \).
Hence, the pre-images of the given vertices are [tex]\( (4, 2) \)[/tex], [tex]\( (2, -4) \)[/tex], [tex]\( (-2, 4) \)[/tex], and [tex]\( (-4, -2) \)[/tex].
