Question

graph $overline{xy}$ with endpoints $x(-3,1)$ and $y(4, - 5)$ and its image after the composition.
reflection: in the line $y = x$
rotation: $90^{circ}$ clockwise about the origin
write a coordinate rule for the composition.
$(x,y)\to(\square,\square)$

Explanation:

Step1: Find coordinates after reflection

The rule for reflection in the line $y = x$ is $(x,y)\to(y,x)$. For point $X(-3,1)$, after reflection, $X_1=(1, - 3)$. For point $Y(4,-5)$, after reflection, $Y_1=(-5,4)$.

Step2: Find coordinates after rotation

The rule for a $90^{\circ}$ clock - wise rotation about the origin is $(x,y)\to(y,-x)$. For $X_1=(1,-3)$, after rotation, $X_2=(-3,-1)$. For $Y_1=(-5,4)$, after rotation, $Y_2=(4,5)$.

Step3: Find the coordinate rule for the composition

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

Answer:

The coordinates of the endpoints of the image are $X_2(-3,-1)$ and $Y_2=(4,5)$, and the coordinate rule for the composition is $(x,y)\to(y,-x)$