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Question

A coordinate plane shows two rectangles A B C D and A double dash B double dash C double dash D double dash. Both the axes range from negative 5 to 5 in unit increments.

Identify a sequence of rigid transformations that would map A (-4,3)B (-2,4) C (-1,1)D (-3,0)→A" (4,-1)B" (2,0) C" (1,-3) D" (3,-4) . Give your answer in the form of a composition transformation mapping.

(1 point)
Responses choose the correct answer carefully and tell me why

(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,y−4)
left parenthesis x comma y right parenthesis right arrow left parenthesis negative x comma y minus 4 right parenthesis

(x,y)→(−x−4,y)
left parenthesis x comma y right parenthesis right arrow left parenthesis negative x minus 4 comma y 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
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Answer 1

To determine the correct sequence of rigid transformations that maps rectangle A B C D with vertices A (-4, 3), B (-2, 4), C (-1, 1), and D (-3, 0) to the new rectangle A" (4, -1), B" (2, 0), C" (1, -3), and D" (3, -4), we analyze the changes in the coordinates.

The transformations involve reflecting and translating the points from the original rectangle to the target rectangle.

1. Analyze the transformation involved:

   • The x-coordinates of the new rectangle are the negatives of the x-coordinates of the original rectangle:

     • A: (-4) → 4
     • B: (-2) → 2
     • C: (-1) → 1
     • D: (-3) → 3

     This suggests a reflection over the y-axis, which can be described with the transformation \((x, y) \rightarrow (-x, y)\).

2. Next, look at the y-coordinates:

   • The y-coordinates of the new rectangle appear to have undergone a downward shift:

     • A: 3 → -1 (shift of -4)
     • B: 4 → 0 (shift of -4)
     • C: 1 → -3 (shift of -4)
     • D: 0 → -4 (shift of -4)

     This suggests a downward translation by 4 units, which can be described with the transformation \((x, y) \rightarrow (x, y - 4)\).

3. Combine these transformations: Based on the analysis, the sequence of transformations can thus be expressed as:

• First, reflect over the y-axis: \((x, y) \rightarrow (-x, y)\).
• Then, translate downward: \((x, y) \rightarrow (x, y - 4)\).

The combined transformations can be written as:

• Start with \((x, y)\) → reflect over the y-axis: \((−x, y)\) → then translate downward: \((-x, y) \rightarrow (-x, y - 4)\).

This can be expressed in function order as:

1. Reflection: \((x, y) \rightarrow (-x, y)\)
2. Translation: \((x, y) \rightarrow (x, y - 4)\)

Thus, the sequence of transformations in composition form is: \((x,y) → (-x,y)\) followed by \((x,y) → (x, y - 4)\).

Now checking the options provided:

• The option \((x,y)→(−x,y−4)\) corresponds to first reflecting over the y-axis and then translating downward by 4.

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

Reason: This captures the necessary transformation of reflecting over the y-axis followed by a downward translation by 4 units.