Certainly! Let's break down the problem step-by-step to find the pre-images of the given image vertices under the transformation rule \( r_{y \text{-axis}}(x, y) \rightarrow (-x, y) \).
Step 1: Understand the transformation rule
The given transformation rule states that for any point \((x, y)\) on the Cartesian plane, the image point after reflecting over the y-axis is \((-x, y)\). Essentially, this transformation changes the sign of the x-coordinate while keeping the y-coordinate the same.
Step 2: Identify the given image vertices
The given image vertices are:
1. \( A'(4, 2) \)
2. \( A'(2, -4) \)
3. \( A'(-2, 4) \)
4. \( A'(-4, -2) \)
Step 3: Determine the pre-images
To find the pre-image vertices, we reverse the transformation \( r_{y \text{-axis}}(x, y) \rightarrow (-x, y) \) which means we need to change the sign of the x-coordinate of each image vertex to find its corresponding pre-image.
1. For \( A'(4, 2) \):
- The x-coordinate \(x\) of the pre-image is \(-4\)
- The y-coordinate \(y\) of the pre-image remains \(2\)
- Thus, the pre-image is \( A(-4, 2) \)
2. For \( A'(2, -4) \):
- The x-coordinate \(x\) of the pre-image is \(-2\)
- The y-coordinate \(y\) of the pre-image remains \(-4\)
- Thus, the pre-image is \( A(-2, -4) \)
3. For \( A'(-2, 4) \):
- The x-coordinate \(x\) of the pre-image is \(2\)
- The y-coordinate \(y\) of the pre-image remains \(4\)
- Thus, the pre-image is \( A(2, 4) \)
4. For \( A'(-4, -2) \):
- The x-coordinate \(x\) of the pre-image is \(4\)
- The y-coordinate \(y\) of the pre-image remains \(-2\)
- Thus, the pre-image is \( A(4, -2) \)
Conclusion:
Here are the pre-image vertices corresponding to the given image vertices:
- \( A( -4, 2 ) \)
- \( A( -2, -4 ) \)
- \( A( 2, 4 ) \)
- \( A( 4, -2 ) \)
Each pre-image has been found by reversing the transformation [tex]\( r_{y \text{-axis}}(x, y) \)[/tex].
