△XYZ has vertices X(0,−2), Y(1,4), and Z(5,3). Which of the following represents the translation of △XYZ along vector <3,−4> and its reflection across the x-axis?
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Answers:

X (0, −2) → X ′(3, −4) → X ″(−4, 3);
Y (1, 4) → Y ′(4, 0)→ Y ″(0, 4);
Z (5, 3)→ Z ′(8, 7)→ Z ″(−7, 8)

X (0, −2) → X ′(3, 2) → X ″(−2, 3);
Y (1, 4) → Y ′(4, 8) → Y ″(−8, 4);
Z (5, 3) → Z ′(8, 1) → Z ″(−8, 1)

X (0, −2) → X ′(3, −6) → X ″(3, 6);
Y (1, 4) → Y ′(4, 0) → Y ″(4, 0);
Z (5, 3)→ Z ′(8, −1)→ Z ″(8, 1)

X (0, −2) → X ′(0, 8) → X ″(0, −8);
Y (1, 4) → Y ′(3, −16) → Y ″(3, 16);
Z (5, 3) → Z ′(15, 12) → Z ″(15, −12)

X (0, −2) → X ′(3, −6) → X ″(3, 6);

Y (1, 4) → Y ′(4, 0) → Y ″(4, 0);

Z (5, 3)→ Z ′(8, −1)→ Z ″(8, 1)

Explanation:

use the rule for reflection: (x,y)→(x,−y)

X'(3,−6)→X''(3,6).

Y'(4,0)→Y''(4,0).

Z'(8,−1)→Z''(8,1).

X(0,−2)→X'(3,−6)→X''(3,6)

Y(1,4)→Y'(4,0)→Y''(4,0)

Z(5,3)→Z'(8,−1)→Z''(8,1)

X (0, −2) → X ′(3, −6) → X ″(3, 6);

Y (1, 4) → Y ′(4, 0) → Y ″(4, 0);

Z (5, 3)→ Z ′(8, −1)→ Z ″(8, 1)

Explanation:

Use the translation vector <3, −4>  to determine the rule for translation of the coordinates: (x, y) → (x + 3, y +(−4)).

Apply the rule to translate vertices X (0, −2), Y (1, 4), and Z (5, 3).

X (0, −2) → (0 + 3, −2 + (−4)) → X' (3, −6).

Y (1, 4) → (1 + 3, 4 + (−4)) → Y' (4, 0).

Z (5, 3) → (5 + 3, 3 + (−4)) → Z' (8, −1).

To apply the reflection across x-axis use the rule for reflection: (x, y) → (x, −y).

Apply the reflection rule to the vertices of △X'Y'Z'.

X ' (3, −6) → X '' (3, 6).

Y ' (4, 0) → Y '' (4, 0).

Z' (8, −1) → Z '' (8, 1).

Therefore,

X (0, −2) → X' (3, −6) → X'' (3, 6)

Y (1, 4) → Y' (4, 0) → Y'' (4, 0)

Z (5, 3) →Z' (8, −1) → Z'' (8, 1)

represents the translation of △XYZ along vector <3, −4>  and its reflection across the x-axis.