A polygon is graphed in the first quadrant of a coordinate plane, then transformed according to the rule $r_{y=x}$. Which composition of transformations will produce the same result?

A$r_{y\text{-axis}} \circ R_{0, 300^\circ}$
B$R_{0, 200^\circ} \circ r_{y\text{-axis}}$
C$r_{y\text{-axis}} \circ R_{0, 270^\circ}$

The target transformation is a reflection across the line $y = x$, which maps a point $(x, y)$ to $(y, x)$. We need to determine which of the provided compositions of transformations yields this result. Let's analyze each component transformation:

Reflection across the x-axis: $r_{x\text{-axis}}(x, y) = (x, -y)$.
Reflection across the y-axis: $r_{y\text{-axis}}(x, y) = (-x, y)$.
Rotation about the origin by $\theta$: $R_{0, \theta}(x, y) = (x \cos\theta - y \sin\theta, x \sin\theta + y \cos\theta)$.

The rotation by $270^\circ$ is defined by $\cos 270^\circ = 0$ and $\sin 270^\circ = -1$. Thus, $R_{0, 270^\circ}(x, y) = (x(0) - y(-1), x(-1) + y(0)) = (y, -x)$.

Test the given compositions.

Now we test each composition against the target mapping $(x, y) \to (y, x)$:

$r_{y\text{-axis}} \circ R_{0, 300^\circ}$: Since $\cos 300^\circ = 0.5$ and $\sin 300^\circ = -\frac{\sqrt{3}}{2}$, $R_{0, 300^\circ}(x, y) = (0.5x + y \frac{\sqrt{3}}{2}, -x \frac{\sqrt{3}}{2} + 0.5y)$. Applying $r_{y\text{-axis}}$ yields $(-0.5x - y \frac{\sqrt{3}}{2}, -x \frac{\sqrt{3}}{2} + 0.5y)$, which is not $(y, x)$.
$R_{0, 270^\circ}$: As calculated previously, this is $(y, -x)$, which is not $(y, x)$.
$r_{x\text{-axis}} \circ R_{0, 270^\circ}$: Starting with $R_{0, 270^\circ}(x, y) = (y, -x)$, we then apply $r_{x\text{-axis}}(y, -x) = (y, -(-x)) = (y, x)$. This matches our target!
$r_{y\text{-axis}} \circ R_{0, 270^\circ}$: Starting with $R_{0, 270^\circ}(x, y) = (y, -x)$, we then apply $r_{y\text{-axis}}(y, -x) = (-y, -x)$, which is not $(y, x)$.

Identify the correct composition.

The composition $r_{x\text{-axis}} \circ R_{0, 270^\circ}$ maps $(x, y)$ to $(y, x)$, which is exactly the result of reflecting a point across the line $y=x$. Therefore, this is the correct composition.
The final answer is $\boxed{r_{x\text{-axis}} \circ R_{0, 270^\circ}}$.

Examples

In computer graphics and image processing, transformations are fundamental for manipulating objects on a screen. For example, if you want to flip a digital image or a shape across a diagonal to create a mirror effect, you can combine basic rotations and reflections. Recognizing that a reflection across a diagonal (like $y=x$) can be achieved by a combination of a $270^\circ$ rotation followed by a vertical flip (reflection across the x-axis) allows developers to build efficient geometric transformation libraries.