What set of transformations are applied to parallelogram ABCD to create A″B″C″D″?

Parallelogram formed by ordered pairs A at negative 4, 1, B at negative 3, 2, C at negative 1, 2, D at negative 2, 1. Second parallelogram transformed formed by ordered pairs A double prime at negative 4, negative 1, B double prime at negative 3, negative 2, C double prime at negative 1, negative 2, D double prime at negative 2, negative 1.

A. Reflected over the x-axis and rotated 180°
B. Reflected over the y-axis and rotated 180°
C. Reflected over the x-axis and rotated 90° counterclockwise
D. Reflected over the y-axis and rotated 90° counterclockwise

Let's analyze the transformation of the parallelogram step by step:

### Original Parallelogram ABCD:
- \( A(-4, 1) \)
- \( B(-3, 2) \)
- \( C(-1, 2) \)
- \( D(-2, 1) \)

### Transformed Parallelogram A″B″C″D″:
- \( A''(-4, -1) \)
- \( B''(-3, -2) \)
- \( C''(-1, -2) \)
- \( D''(-2, -1) \)

### Step 1: Reflection Analysis

When reflecting over the x-axis, the y-coordinates of each point change sign, while the x-coordinates remain the same. Applying this transformation to the original points:

- \( A(-4, 1) \) becomes \( A'(-4, -1) \)
- \( B(-3, 2) \) becomes \( B'(-3, -2) \)
- \( C(-1, 2) \) becomes \( C'(-1, -2) \)
- \( D(-2, 1) \) becomes \( D'(-2, -1) \)

After reflecting over the x-axis, we get the points:
- \( A'(-4, -1) \)
- \( B'(-3, -2) \)
- \( C'(-1, -2) \)
- \( D'(-2, -1) \)

These are exactly the same as the points in A″B″C″D″.

### Step 2: Rotation Analysis

Next, we check if any rotation was applied after the reflection. Since the coordinates match exactly after reflecting over the x-axis, no further rotation is necessary.

### Conclusion:
The correct transformation is **a reflection over the x-axis**, and no additional rotation is needed.

However, based on the provided options:
- **A. Reflected over the x-axis and rotated 180°** would result in the original points being rotated further, which would not match A″B″C″D″.
- **B. Reflected over the y-axis and rotated 180°** would not produce the required coordinates.
- **C. Reflected over the x-axis and rotated 90° counterclockwise** would also not match the final coordinates.
- **D. Reflected over the y-axis and rotated 90° counterclockwise** would not match either.

Since the reflection over the x-axis alone produces the correct result, the best option to choose is **A**, even though the rotation is not necessary.

Would you like more details or have any questions? Here are some related questions:

1. How does a 180° rotation affect coordinates?
2. What happens when a figure is rotated 90° counterclockwise?
3. What is the effect of reflecting a figure over the y-axis?
4. Can you combine transformations like reflections and rotations to achieve specific results?
5. What other transformations could be applied to get the same result?

**Tip:** When analyzing transformations, always check if the simplest transformation alone accounts for the change before considering combinations.

Explore how to transform parallelogram ABCD to A″B″C″D″ using reflections and rotations. Detailed step-by-step analysis and explanation of the transformation process. Suitable for high school students studying geometry.

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