To analyze the transformation given by the function \((x,y) \to (y + 5, -x)\), we can break it down into its components.
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First component: The new x-coordinate is \(y + 5\).
- This indicates a reliance on the original y-coordinate and a vertical shift upwards by 5 units.
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Second component: The new y-coordinate is \(-x\).
- This indicates a reflection across the x-axis (because the x-coordinate is negated).
Now, let's reshape this into a complete transformation description:
- The original point \((x, y)\) first gets reflected across the x-axis, changing the point to \((x, -y)\).
- Then, the y-coordinate is altered: the new x-coordinate becomes \(y + 5\) and the new y-coordinate becomes \(-x\).
So the transformation sequence is:
- Reflect across the x-axis, changing \((x, y)\) to \((x, -y)\).
- Change the x-coordinate to \(y + 5\), which involves a translation upwards by 5 units.
This matches with a reflection across the x-axis and a vertical translation up, not directly along the x or y-axis in the usual sense, but a reconfiguration of the coordinates based on their values.
Summarizing:
- Reflection across the x-axis.
- Translation vertically upwards by 5 units.
None of the given choices precisely frame this description, as they all involve reflections about the y-axis or translations along the x-axis. Hence, based on the transformations we examined, none of the responses accurately fits the transformation of \((x, y) \to (y + 5, -x)\).
However, if we purely consider negative x-translation or a typographical misalignment in answering formats with translations, the likely intent of the problem's formation would be leaning towards reflection across the x-axis and translation by 5 units up, albeit incorrectly provided in choices.
To choose the closest, you might want to ask for clarification or rephrase on what exact transformation was intended within the provided scope.
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