Question

which sequence of transformations maps quadrilateral pqrs onto quadrilateral pqrs? a rotation 180° counterclockwise about the origin followed by a reflection across the y - axis a reflection across the line y = 1 followed by a reflection across the y - axis

Explanation:

Step1: Analyze rotation rules

A $180^{\circ}$ counter - clockwise rotation about the origin changes the coordinates $(x,y)$ to $(-x,-y)$.

Step2: Analyze reflection rules

A reflection across the $y$ - axis changes the coordinates $(x,y)$ to $(-x,y)$. If we first rotate $180^{\circ}$ counter - clockwise about the origin and then reflect across the $y$ - axis, the combined transformation will not map the quadrilateral as required.

Step3: Analyze reflection across $y = 1$

The distance between a point $(x,y)$ and the line $y = 1$ is $d=\vert y - 1\vert$. After reflection across $y = 1$, the new $y$ - coordinate is $y'=1+(1 - y)=2 - y$ and the $x$ - coordinate remains the same, i.e., $(x,y)\to(x,2 - y)$.

Step4: Analyze second reflection across $y$ - axis

After reflecting the result of the first reflection across the $y$ - axis, the $x$ - coordinate changes sign, so $(x,2 - y)\to(-x,2 - y)$. This sequence of transformations maps quadrilateral $PQRS$ onto quadrilateral $P'Q'R'S'$.

Answer:

a reflection across the line $y = 1$ followed by a reflection across the $y$-axis