To determine the correct function rule for transforming a geometric figure by reflecting it across the [tex]\(y\)[/tex]-axis and then moving it up 5 units, let's break down the transformations step-by-step.
1. Reflection across the [tex]\(y\)[/tex]-axis:
Reflecting a point [tex]\((x, y)\)[/tex] across the [tex]\(y\)[/tex]-axis changes its coordinates to [tex]\((-x, y)\)[/tex]. This is because reflecting across the [tex]\(y\)[/tex]-axis negates the [tex]\(x\)[/tex]-coordinate while leaving the [tex]\(y\)[/tex]-coordinate unaffected.
2. Translation up by 5 units:
Moving a point [tex]\((x, y)\)[/tex] up by 5 units changes its coordinates to [tex]\((x, y + 5)\)[/tex]. This is because adding 5 to the [tex]\(y\)[/tex]-coordinate raises the point vertically by 5 units.
Next, we need to combine these two operations:
- Start with the original coordinates [tex]\((x, y)\)[/tex].
- First, apply the reflection across the [tex]\(y\)[/tex]-axis: [tex]\((x, y) \rightarrow (-x, y)\)[/tex].
- Then, apply the translation up 5 units: [tex]\((-x, y) \rightarrow (-x, y + 5)\)[/tex].
So, the combined transformation is:
[tex]\[ f(x, y) = (-x, y + 5) \][/tex]
Therefore, the correct function rule is:
[tex]\[ f(x, y) = (-x, y + 5) \][/tex]
So the answer is:
[tex]\[ f(x, y) = (-x, y + 5) \][/tex]
