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Use the image to answer the question. WHICH ANSWER IS CORRECT AND WHY

A coordinate plane shows two triangles A B C and A double dash B double dash C double dash. Both the axes range from negative 6 to 6 in unit increments.

Identify the sequence of transformations that will carry ΔA (-5,2) B (-2,4) C (-3,0) →ΔA" (-1,-2) B" (2,-4) C" (1,0). Give your answer in the form of a composition transformation mapping.

(1 point)
Responses

(x,y)→(x+4,−y)
left parenthesis x comma y right parenthesis right arrow left parenthesis x plus 4 comma negative y right parenthesis

(x,y)→(x,−y−4)
open paren x comma y close paren rightarrow open paren x comma negative y minus 4 close paren

1 answer

To determine the correct sequence of transformations that maps triangle \( \Delta A(-5,2) B(-2,4) C(-3,0) \) to triangle \( \Delta A''(-1,-2) B''(2,-4) C''(1,0) \), we can analyze the transformations required to go from the original coordinates to the new coordinates.

  1. Identify the transformations:

    • The first triangle points are:
      • \( A(-5, 2) \) transforms to \( A''(-1,-2) \)
      • \( B(-2, 4) \) transforms to \( B''(2,-4) \)
      • \( C(-3, 0) \) transforms to \( C''(1, 0) \)

  2. Translation and Reflection Analysis:

    • Observing how \( A(-5, 2) \) becomes \( A''(-1, -2) \):
      • The x-coordinate \( -5 \) to \( -1 \) indicates a translation of \( +4 \).
      • The y-coordinate changes from \( 2 \) to \( -2 \), which is a reflection over the x-axis and then a downward translation of \( -4 \).

  3. Determine the transformation.

    • Reflection over the x-axis: This transformation is represented by \( (x, y) \to (x, -y) \). This step will change \( y \) values to their negatives.

    • After reflecting, we add \( 4 \) to the x-coordinate. This will transform \( (x, -y) \) to \( (x + 4, -y) \).

    • The combined transformation can be written as: \[ (x,y) \to (x + 4, -y) \]

Thus, \( (x,y) \to (x + 4, -y) \) correctly represents the sequence of transformations required to map triangle \( \Delta A \) to triangle \( \Delta A'' \).

Therefore, the correct answer is: (x,y)→(x+4,−y).