Question

practice makes better
graph the image of a(0, -2), b(2, -2) and c(1, -5) & each transformation.

1. translation: (x + 2, y), then reflection: across the x - axis
2. reflection: across y - axis, then translation: (x - 4, y - 3)
3. translation: (x - 3, y + 2), then reflection: across x - axis

Explanation:

Step1: Translate original points

For $A(0,-2)$: $A_1 = (0+2, -2) = (2,-2)$
For $B(2,-2)$: $B_1 = (2+2, -2) = (4,-2)$
For $C(1,-5)$: $C_1 = (1+2, -5) = (3,-5)$

Step2: Reflect across x-axis

Reflection rule: $(x,y) \to (x,-y)$
$A_2 = (2, 2)$
$B_2 = (4, 2)$
$C_2 = (3, 5)$
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Step3: Reflect original points across y-axis

Reflection rule: $(x,y) \to (-x,y)$
$A_1 = (0,-2)$
$B_1 = (-2,-2)$
$C_1 = (-1,-5)$

Step4: Translate reflected points

Translation rule: $(x,y) \to (x-4,y-3)$
$A_2 = (0-4, -2-3) = (-4,-5)$
$B_2 = (-2-4, -2-3) = (-6,-5)$
$C_2 = (-1-4, -5-3) = (-5,-8)$

Answer:

1. Final transformed points: $A'(2,2)$, $B'(4,2)$, $C'(3,5)$
2. Final transformed points: $A'(-4,-5)$, $B'(-6,-5)$, $C'(-5,-8)$