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.



A. (x,y)→(x,−y−4)
B. (x,y)→(−x−4,y)
C. (x,y)→(x,y+4)
D.(x,y)→(−x,y−4)

Rectangle ABCD
Point A: (-4, 3)
Point B: (-2, 4)
Point C: (-1, 1)
Point D: (-3, 0)

Rectangle A"B"C"D"
Point A": (4, -1)
Point B": (2, 0)
Point C": (1, -3)
Point D": (3, -4)

Answer 1

To find the sequence of rigid transformations (which include translations, rotations, and reflections) that maps the rectangle ABCD to A"B"C"D", we will analyze the transformations one by one.

1. **Identify the changes in coordinates:**
- Point A: \( A(-4, 3) \) transforms to \( A"(4, -1) \).
- Point B: \( B(-2, 4) \) transforms to \( B"(2, 0) \).
- Point C: \( C(-1, 1) \) transforms to \( C"(1, -3) \).
- Point D: \( D(-3, 0) \) transforms to \( D"(3, -4) \).

2. **Determine the transformations needed:**
- The x-coordinates change from negative to positive, indicating a reflection over the y-axis.
- The y-coordinates decrease as the x-coordinates increase, suggesting a vertical shift downward.

3. **Testing the transformations:**
Let's analyze the given options to see which fit:

- **Option A:** \( (x,y) \to (x, -y-4) \)
- A (-4, 3) → (-4, -3-4) = (-4, -7) (not matching)

- **Option B:** \( (x,y) \to (-x-4, y) \)
- A (-4, 3) → (4-4, 3) = (0, 3) (not matching)

- **Option C:** \( (x,y) \to (x, y+4) \)
- A (-4, 3) → (-4, 3+4) = (-4, 7) (not matching)

- **Option D:** \( (x,y) \to (-x, y - 4) \)
- This performs a reflection across the y-axis and then moves down by 4 units. \
- A (-4, 3) → (4, 3-4) = (4, -1) → A"](4, -1) (this matches!)

Let's confirm for all points with **Option D**:
- B (-2, 4) → (2, 4 - 4) = (2, 0) → B"](2, 0) (this matches!)
- C (-1, 1) → (1, 1 - 4) = (1, -3) → C"](1, -3) (this matches!)
- D (-3, 0) → (3, 0 - 4) = (3, -4) → D"](3, -4) (this matches!)

### Conclusion:
The transformation that maps rectangle ABCD to rectangle A"B"C"D" is:
**Option D: (x, y) → (-x, y - 4)**

Thus, the composition of transformations is:
- Reflect over the y-axis, and
- Translate downward by 4 units.

In formal notation, it is:
\[ D \circ (x,y) \to (-x, y - 4) \]