- The reflection \(r_{y=x}\) maps \((x, y)\) to \((y, x)\).
- Testing each composition, we find that \(R_{O, 270^\circ}\) maps \((x, y)\) to \((y, -x)\).
- Applying a reflection across the \(x\)-axis to this result, \(r_{x \text{-axis}}(y, -x)\), yields \((y, x)\).
- The correct composition is \(\boxed{r_{x \text{-axis}} \circ R_{O, 270^\circ}}\).
Explanation
- Analyze the target transformation and the given options.
The target transformation is a reflection across the line \(y=x\), which maps any point \((x, y)\) in the coordinate plane to the point \((y, x)\). Our objective is to identify which of the given composite transformations produces this same result. We will test each option step-by-step.
- Evaluate Option 2: \(R_{O, 270^\circ}\)
A rotation by \(270^\circ\) about the origin, denoted \(R_{O, 270^\circ}\), corresponds to a rotation by \(-90^\circ\). The rule for a \(270^\circ\) rotation is \((x, y) \to (x \cos 270^\circ - y \sin 270^\circ, x \sin 270^\circ + y \cos 270^\circ)\).
Given \(\cos 270^\circ = 0\) and \(\sin 270^\circ = -1\), the transformation is:
\((x, y) \to (0 - y(-1), x(-1) + 0) = (y, -x)\).
This does not result in \((y, x)\), so this option is incorrect.
- Evaluate Option 3: \(r_{x \text{-axis}} \circ R_{O, 270^\circ}\)
First, we apply the rotation \(R_{O, 270^\circ}\) to a point \((x, y)\), which results in \((y, -x)\).
Next, we apply the reflection across the \(x\)-axis, denoted \(r_{x \text{-axis}}\). The rule for reflection across the \(x\)-axis is \((x, y) \to (x, -y)\).
Applying this to the result of the rotation:
\(r_{x \text{-axis}}(y, -x) = (y, -(-x)) = (y, x)\).
Since this result matches the reflection across the line \(y=x\), this composition is correct.
- Conclusion and final verification.
We have verified that \(r_{x \text{-axis}} \circ R_{O, 270^\circ}\) maps \((x, y)\) to \((y, x)\), which is exactly what \(r_{y=x}\) does. The other options do not produce this mapping. The correct composition is \(r_{x \text{-axis}} \circ R_{O, 270^\circ}\).
Examples
Imagine you are working in graphic design software where you need to mirror an object across a diagonal axis. If the software only provides rotation and reflection tools relative to the standard coordinate axes, you can achieve a reflection across the line \(y=x\) by rotating your object by \(270^\circ\) and then reflecting it across the \(x\)-axis. Understanding these compositions allows designers and engineers to create complex symmetries using only a few basic transformation commands.