Question

graph \\( \overline { r s } \\) with endpoints \\( r ( 2, - 1 ) \\) and \\( s ( 6, - 5 ) \\) and its image after the composition of a translation along \\( \langle - 2, - 2 \
angle \\) and a reflection in the \\( y \\)-axis.

Explanation:

Step1: Translation

For a translation along $\langle - 2,-2
angle$, the rule is $(x,y)\to(x - 2,y - 2)$.
For point $R(2,-1)$:
$x=2,y = - 1$
$x-2=2-2 = 0,y - 2=-1-2=-3$
So $R'=(0,-3)$
For point $S(6,-5)$:
$x = 6,y=-5$
$x-2=6 - 2=4,y - 2=-5-2=-7$
So $S'=(4,-7)$

Step2: Reflection in the $y$-axis

The rule for reflection in the $y$-axis is $(x,y)\to(-x,y)$
For point $R'(0,-3)$:
$x = 0,y=-3$
$-x=0$
So $R''=(0,-3)$
For point $S'(4,-7)$:
$x = 4,y=-7$
$-x=-4$
So $S''=(-4,-7)$

Answer:

The original segment $\overline{RS}$ has endpoints $R(2,-1)$ and $S(6,-5)$. After translation, the endpoints are $R'(0,-3)$ and $S'(4,-7)$. After reflection in the $y$-axis, the endpoints of the image are $R''(0,-3)$ and $S''(-4,-7)$. Graph the original segment with endpoints $(2,-1)$ and $(6,-5)$, then graph the translated - reflected segment with endpoints $(0,-3)$ and $(-4,-7)$.