Question

Use the image to answer the question. Identify a sequence of rigid transformations that would map ABCD→A"B"C"D" . Give your answer in the form of a composition transformation mapping. (1 point) Responses (x,y)→(−x,y−4) left parenthesis x comma y right parenthesis right arrow left parenthesis negative x comma y minus 4 right parenthesis (x,y)→(x,y+4) left parenthesis x comma y right parenthesis right arrow left parenthesis x comma y plus 4 right parenthesis (x,y)→(x,−y−4) left parenthesis x comma y right parenthesis right arrow left parenthesis x comma minus y negative 4 right parenthesis (x,y)→(−x−4,y)

Answer 1

To determine the sequence of rigid transformations that would map quadrilateral ABCD to A"B"C"D", we can analyze the potential transformations listed in your options.

1. Reflection over the y-axis: \( (x,y) \rightarrow (-x, y) \)
2. Translation: This will involve altering the y-coordinate:
   • Translating down: \( (x,y) \rightarrow (x, y - 4) \)
   • Translating up: \( (x,y) \rightarrow (x, y + 4) \)

Given the transformations provided:

• The option \( (x,y) \rightarrow (-x, y - 4) \) indicates a reflection across the y-axis followed by a downward translation of 4 units.
• The option \( (x,y) \rightarrow (x, y + 4) \) indicates a translation upward by 4 units and would not map ABCD to A"B"C"D" if we are looking to move down.
• The option \( (x,y) \rightarrow (x, -y - 4) \) suggests a reflection across the x-axis which does not align with the question.
• The option \( (x,y) \rightarrow (-x - 4, y) \) indicates a horizontal transformation that does not fit our need.

Based on standard mapping transformations for quadrilaterals and typically using the reflection followed by the translation that fits the new coordinates, the composition that maps ABCD to A"B"C"D" is:

\[ (x,y) \rightarrow (-x, y) \rightarrow (x, y - 4) \]

Thus, the sequence of transformations that maps ABCD to A"B"C"D" is given by the first option:

\[ (x,y) \rightarrow (-x, y - 4) \]