Question
Which composition transformation occurs in the following?
(X,y) (Y+5,-x)
(X,y) (Y+5,-x)
Answers
GPT-4o mini
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.
1. **Input Point:** Start with the point \((x, y)\).
2. **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:
1. Swapping \(x\) and \(y\) (but specifically \(y\) becomes the new \(x\)).
2. Adding 5 to the new \(x\) coordinate (which is originally \(y\)).
3. 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!
1. **Input Point:** Start with the point \((x, y)\).
2. **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:
1. Swapping \(x\) and \(y\) (but specifically \(y\) becomes the new \(x\)).
2. Adding 5 to the new \(x\) coordinate (which is originally \(y\)).
3. 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!