The transformation you provided seems to describe a composition of two transformations applied to a point \((x, y)\). Let's analyze the transformation step by step.
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Input Point: Start with the point \((x, y)\).
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First Transformation: The transformation \((X, y) \to (Y + 5, -x)\) suggests that:
- The new \(x\)-coordinate is \(Y + 5\).
- The new \(y\)-coordinate is \(-x\).
In terms of the original coordinates \((x, y)\):
- Replacing \(Y\) with \(y\), we get: \[ (x, y) \to (y + 5, -x) \]
Now we can summarize the transformation as follows:
- The new \(x\)-coordinate is \(y + 5\).
- The new \(y\)-coordinate is \(-x\).
Therefore, the overall transformation can be expressed as: \[ (x, y) \to (y + 5, -x) \]
To summarize, the composition transformation involves:
- Swapping \(x\) and \(y\) (but specifically \(y\) becomes the new \(x\)).
- Adding 5 to the new \(x\) coordinate (which is originally \(y\)).
- Negating the original \(x\) coordinate to find the new \(y\) coordinate.
Thus, the transformation involves both a reflection and a translation:
- Reflection across the line \(y = -x\) (which changes the sign and swaps coordinates).
- Translation by 5 units in the \(x\) direction.
If you'd like further clarification or a deeper exploration into the transformation, just let me know!
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