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.

Reflected over the x-axis and rotated 180°
Reflected over the y-axis and rotated 180°
Reflected over the x-axis and rotated 90° counterclockwise
Reflected over the y-axis and rotated 90° counterclockwise

To determine the set of transformations that are applied to parallelogram ABCD to create A″B″C″D″, let's analyze the coordinates.

### 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

The points in the transformed parallelogram have the same x-coordinates but their y-coordinates have changed sign. This indicates a reflection over the **x-axis**.

### Step 2: Rotation Analysis

After reflecting over the x-axis, the coordinates of the points would be:
- \( A'(-4, -1) \)
- \( B'(-3, -2) \)
- \( C'(-1, -2) \)
- \( D'(-2, -1) \)

These are exactly the coordinates of A″B″C″D″, which indicates that no further rotation is needed.

### Conclusion:
The transformation applied is **a reflection over the x-axis**.

The correct choice is not listed among the provided options, but if the reflection over the x-axis was combined with any rotation, the coordinates would not match the final points. Therefore, **the correct transformation is just a reflection over the x-axis** without any additional rotation.

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

1. What would the coordinates be if the parallelogram were rotated 180°?
2. How do you determine the result of a 90° counterclockwise rotation?
3. What happens to a figure's coordinates when reflected over the y-axis?
4. How can you combine multiple transformations, such as a reflection and a rotation?
5. What other transformations could produce the same final coordinates?

**Tip:** Understanding how reflections and rotations affect coordinates can help you quickly determine the correct transformation sequence.

Explore how parallelogram ABCD transforms into A″B″C″D″ through reflection over the x-axis. Understand the coordinate changes and implications of this transformation in geometry. Suitable for high school students studying coordinate geometry.

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