To determine the correct mapping notation for reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis, we need to understand how this transformation affects the coordinates of the point.
1. Understanding Reflections Across the [tex]\(y\)[/tex]-axis:
- When a point [tex]\((x, y)\)[/tex] is reflected across the [tex]\(y\)[/tex]-axis, its [tex]\(y\)[/tex]-coordinate remains the same while its [tex]\(x\)[/tex]-coordinate changes sign. This means the point [tex]\((x, y)\)[/tex] will move to [tex]\((-x, y)\)[/tex].
2. Analyzing Options:
- Option 1: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]
- This matches our understanding of a reflection across the [tex]\(y\)[/tex]-axis. The [tex]\(x\)[/tex]-coordinate changes sign, becoming [tex]\(-x\)[/tex], while the [tex]\(y\)[/tex]-coordinate remains the same.
- Option 2: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]
- This would represent a reflection across the [tex]\(x\)[/tex]-axis, not the [tex]\(y\)[/tex]-axis. Here, the [tex]\(y\)[/tex]-coordinate changes sign, becoming [tex]\(-y\)[/tex], while the [tex]\(x\)[/tex]-coordinate remains the same.
- Option 3: [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]
- This would represent a rotation of 180 degrees around the origin, not a reflection across the [tex]\(y\)[/tex]-axis. Both coordinates change sign.
- Option 4: [tex]\((x, y) \rightarrow (y, x)\)[/tex]
- This would represent neither a reflection across the [tex]\(y\)[/tex]-axis nor any common reflection. It essentially swaps the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates.
3. Conclusion:
- Based on the definitions and transformations provided, the correct mapping notation for reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis is the first one: [tex]\((x, y) \rightarrow (-x, y)\)[/tex].
Hence, the correct abbreviated set of directions for reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis follows the notation:
[tex]\[
(x, y) \rightarrow (-x, y)
\][/tex]
Therefore, Marissa should choose Option 1: [tex]\((x, y) \rightarrow (-x, y)\)[/tex].
