Question

graph $overline{xy}$ with endpoints $x(-3,1)$ and $y(4,-5)$ and its image after the composition. rotation: $270^{circ}$ counterclockwise about the origin. reflection: in the y - axis. write a coordinate rule for the composition. $(x,y)\to( square,square)$

Explanation:

Step1: Apply rotation rule

The rule for a 270 - counterclockwise rotation about the origin is $(x,y)\to(y, - x)$.
For point $X(-3,1)$: $( - 3,1)\to(1,3)$.
For point $Y(4,-5)$: $(4,-5)\to(-5,-4)$.

Step2: Apply reflection rule

The rule for reflection in the $y$-axis is $(x,y)\to(-x,y)$.
For the rotated point $X'(1,3)$: $(1,3)\to(-1,3)$.
For the rotated point $Y'(-5,-4)$: $(-5,-4)\to(5,-4)$.

Step3: Find coordinate rule

First rotation: $(x,y)\to(y, - x)$. Then reflection: $(y,-x)\to(-y,-x)$. So the coordinate rule for the composition is $(x,y)\to(-y,-x)$.

Answer:

The new endpoints are $X''(-1,3)$ and $Y''(5,-4)$. The coordinate rule is $(x,y)\to(-y,-x)$.