Question

tangle defg with vertices d(-2, 7), e(2, 3), f(0, 1), and g(-4, 5): translation along the rule (x, y)→(x + 6, y - 8) reflection in the y - axis

Explanation:

Step1: Apply translation rule

For point $D(-2,7)$: $x=-2,y = 7$, after translation $(x,y)\to(x + 6,y-8)$, we have $x=-2 + 6=4,y=7-8=-1$, so $D'=(4,-1)$.
For point $E(2,3)$: $x = 2,y=3$, after translation $x=2 + 6=8,y=3-8=-5$, so $E'=(8,-5)$.
For point $F(0,1)$: $x = 0,y=1$, after translation $x=0 + 6=6,y=1-8=-7$, so $F'=(6,-7)$.
For point $G(-4,5)$: $x=-4,y = 5$, after translation $x=-4+6 = 2,y=5-8=-3$, so $G'=(2,-3)$.

Step2: Apply reflection in y - axis rule

The rule for reflection in the $y$-axis is $(x,y)\to(-x,y)$.
For $D'(4,-1)$: After reflection, $D''=(-4,-1)$.
For $E'(8,-5)$: After reflection, $E''=(-8,-5)$.
For $F'(6,-7)$: After reflection, $F''=(-6,-7)$.
For $G'(2,-3)$: After reflection, $G''=(-2,-3)$.

Answer:

$D''(-4,-1),E''(-8,-5),F''(-6,-7),G''(-2,-3)$