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.

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 minus y negative 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−4,y)

Answer 1

To find a sequence of rigid transformations that maps rectangle ABCD to rectangle A"B"C"D", we first need to identify what transformations are required. Rigid transformations include translations and reflections.

1. If rectangle A B C D is reflected over the y-axis, the mapping can be expressed as \( (x, y) \to (-x, y) \).
2. Next, if we need to move the new rectangle (after reflection) down by 4 units, this can be expressed as \( (x, y) \to (x, y - 4) \).

The complete transformation sequence that maps ABCD to A"B"C"D" can be expressed as a composition of these transformations:

1. Reflect over the y-axis: \( (x, y) \to (-x, y) \)
2. Translate down by 4 units: \( (x, y) \to (x, y - 4) \)

In terms of composition, this can be written as:

1. First reflect: \( (x, y) \to (-x, y) \)
2. Then translate: \( (-x, y) \to (-x, y - 4) \)

Putting these operations together, the sequence of transformations is:

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

However, in terms of a single transformation:

The composition can be represented as: \[ (x, y) \to (-x, y - 4) \]

Therefore, the answer is: \( (x,y) \to (-x, y - 4) \)

So the final answer is: (x,y)→(−x,y−4).