Question

3 translate 2 units right and 4 units down. rotate 180° around the origin.

Explanation:

Step1: Apply translation rule

The translation rule for moving 2 units right and 4 units down is $(x,y)\to(x + 2,y-4)$.
Let the coordinates of the vertices of the polygon be: $H(-6,4)$, $I(-2,4)$, $J(-2,2)$, $K(-4,2)$, $L(-4,4)$.
After translation:
$H'(-6 + 2,4-4)=(-4,0)$
$I'(-2 + 2,4-4)=(0,0)$
$J'(-2 + 2,2-4)=(0,-2)$
$K'(-4 + 2,2-4)=(-2,-2)$
$L'(-4 + 2,4-4)=(-2,0)$

Step2: Apply rotation rule

The rule for rotating a point $(x,y)$ 180° around the origin is $(x,y)\to(-x,-y)$.
For $H'(-4,0)$, the new point $H''(4,0)$
For $I'(0,0)$, the new point $I''(0,0)$
For $J'(0,-2)$, the new point $J''(0,2)$
For $K'(-2,-2)$, the new point $K''(2,2)$
For $L'(-2,0)$, the new point $L''(2,0)$

Answer:

The new - coordinates of the vertices of the polygon after translation and rotation are $H''(4,0)$, $I''(0,0)$, $J''(0,2)$, $K''(2,2)$, $L''(2,0)$