Question

graph xy with endpoints x(-3, 1) and y(4, -5) and its image after the composition. reflection: in the line y = x rotation: 90° clockwise about the origin write a coordinate rule for the composition. (x, y)→(□,□)

Explanation:

Step1: Find reflection about y = x

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

Step2: Rotate the reflected points 90 - degree clockwise about the origin

The rule for rotating a point $(x,y)$ 90 - degree clockwise about the origin is $(x,y)\to(y,-x)$.
For $X_1(1,-3)$, the final image $X_2$ is $(-3,-1)$.
For $Y_1(-5,4)$, the final image $Y_2$ is $(4,5)$.
To find the coordinate - rule for the composition:
Let the original point be $(x,y)$.
After reflection about $y = x$, it becomes $(y,x)$.
After rotating $(y,x)$ 90 - degree clockwise about the origin, it becomes $(x,-y)$.

Answer:

$(x,y)\to(x,-y)$; The new endpoints are $X(-3,-1)$ and $Y(4,5)$