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To determine which rule describes a transformation across the [tex]\( y \)[/tex]-axis, let's break down the transformation process step-by-step:
1. Understand the transformation: A transformation across the [tex]\( y \)[/tex]-axis means that every point on a figure is reflected over the [tex]\( y \)[/tex]-axis. Essentially, for any point [tex]\((x, y)\)[/tex], this transformation changes the [tex]\( x \)[/tex]-coordinate to its opposite (i.e., negative of [tex]\( x \)[/tex]) while keeping the [tex]\( y \)[/tex]-coordinate the same.
2. Analyze the options:
- Option 1: [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]: This rule reflects the point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex]. This represents a reflection over the origin, not the [tex]\( y \)[/tex]-axis.
- Option 2: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]: This rule reflects the point [tex]\((x, y)\)[/tex] to [tex]\((-x, y)\)[/tex]. This change indicates that the [tex]\( x \)[/tex]-coordinate is negated while the [tex]\( y \)[/tex]-coordinate remains unchanged, consistent with the transformation across the [tex]\( y \)[/tex]-axis.
- Option 3: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]: This rule reflects the point [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex]. This describes a reflection across the [tex]\( x \)[/tex]-axis, not the [tex]\( y \)[/tex]-axis.
- Option 4: [tex]\((x, y) \rightarrow (y, x)\)[/tex]: This rule transposes the coordinates, which does not correlate with a reflection across the [tex]\( y \)[/tex]-axis.
3. Conclusion:
- The rule describing a transformation across the [tex]\( y \)[/tex]-axis is [tex]\((x, y) \rightarrow (-x, y)\)[/tex], as it correctly reflects the [tex]\( x \)[/tex]-coordinate across the [tex]\( y \)[/tex]-axis while keeping the [tex]\( y \)[/tex]-coordinate unchanged.
Thus, the correct answer is:
[tex]\[ (x, y) \rightarrow (-x, y) \][/tex]
1. Understand the transformation: A transformation across the [tex]\( y \)[/tex]-axis means that every point on a figure is reflected over the [tex]\( y \)[/tex]-axis. Essentially, for any point [tex]\((x, y)\)[/tex], this transformation changes the [tex]\( x \)[/tex]-coordinate to its opposite (i.e., negative of [tex]\( x \)[/tex]) while keeping the [tex]\( y \)[/tex]-coordinate the same.
2. Analyze the options:
- Option 1: [tex]\((x, y) \rightarrow (-x, -y)\)[/tex]: This rule reflects the point [tex]\((x, y)\)[/tex] to [tex]\((-x, -y)\)[/tex]. This represents a reflection over the origin, not the [tex]\( y \)[/tex]-axis.
- Option 2: [tex]\((x, y) \rightarrow (-x, y)\)[/tex]: This rule reflects the point [tex]\((x, y)\)[/tex] to [tex]\((-x, y)\)[/tex]. This change indicates that the [tex]\( x \)[/tex]-coordinate is negated while the [tex]\( y \)[/tex]-coordinate remains unchanged, consistent with the transformation across the [tex]\( y \)[/tex]-axis.
- Option 3: [tex]\((x, y) \rightarrow (x, -y)\)[/tex]: This rule reflects the point [tex]\((x, y)\)[/tex] to [tex]\((x, -y)\)[/tex]. This describes a reflection across the [tex]\( x \)[/tex]-axis, not the [tex]\( y \)[/tex]-axis.
- Option 4: [tex]\((x, y) \rightarrow (y, x)\)[/tex]: This rule transposes the coordinates, which does not correlate with a reflection across the [tex]\( y \)[/tex]-axis.
3. Conclusion:
- The rule describing a transformation across the [tex]\( y \)[/tex]-axis is [tex]\((x, y) \rightarrow (-x, y)\)[/tex], as it correctly reflects the [tex]\( x \)[/tex]-coordinate across the [tex]\( y \)[/tex]-axis while keeping the [tex]\( y \)[/tex]-coordinate unchanged.
Thus, the correct answer is:
[tex]\[ (x, y) \rightarrow (-x, y) \][/tex]
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