Question

Question

Use the image to answer the question.

A coordinate plane shows two rectangles A (-4,3) B (-2,4) C (-1,1) D (-3,0) and A double dash (4,-1) B double dash (2,0) C double dash (1,-3) D double dash (3,-4). Both the axes range from negative 5 to 5 in unit increments.

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 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 find the transformation that maps rectangle ABCD to rectangle A"B"C"D", we analyze the coordinates of the rectangles:

1. Rectangle ABCD has vertices:

   • A(-4, 3)
   • B(-2, 4)
   • C(-1, 1)
   • D(-3, 0)

2. Rectangle A"B"C"D" has vertices:

   • A"(4, -1)
   • B"(2, 0)
   • C"(1, -3)
   • D"(3, -4)

Step 1: Analyze the transformations

It appears that to transition from ABCD to A"B"C"D", we first need to reflect the coordinates across the origin (which switches the signs of both x and y) and then apply a vertical translation.

Transformation Process

1. Reflection across the origin:

   • This changes (x, y) to (-x, -y).

2. Vertical translation downward:

   • After reflection, we notice the y-coordinates of A"B"C"D" are 4 units lower than the reflected coordinates. Thus, we will then translate down by 4 units, or (x, y) becomes (x, y - 4).

Combined Transformation

The first transformation is reflecting across the origin:

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

Then, translate down by 4 units:

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

Final Composition Transformation

Put together, the complete transformation from ABCD to A"B"C"D" can be expressed as:

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

Conclusion

The correct response that corresponds to this transformation is:

\((x,y)→(−x,y−4)\)

In words: Reflecting across the y-axis and then translating down by 4 units.