To determine the transformations that move the rectangle in Quadrant IV to the rectangle in Quadrant II, we can analyze their positions and orientations in the coordinate plane.
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Position of Rectangles:
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The rectangle in Quadrant IV has vertices:
- \( (1, -3) \)
- \( (1, -7) \)
- \( (4, -7) \)
- \( (4, -3) \)
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The rectangle in Quadrant II has vertices:
- \( (-2, 1) \)
- \( (-6, 1) \)
- \( (-6, 4) \)
- \( (-2, 4) \)
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Transformations:
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Step 1: Reflection: To move from Quadrant IV to Quadrant II, the rectangle must first be reflected across the y-axis. This changes the x-coordinates from positive to negative:
- After reflection:
- Vertex \( (1, -3) \) becomes \( (-1, -3) \)
- Vertex \( (1, -7) \) becomes \( (-1, -7) \)
- Vertex \( (4, -7) \) becomes \( (-4, -7) \)
- Vertex \( (4, -3) \) becomes \( (-4, -3) \)
- After reflection:
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Step 2: Translation: After reflection, the new vertices need to be moved upwards to match the y-coordinates of the rectangle in Quadrant II:
- Translating these points up by 2 (adding 2 to the y-coordinates):
- Vertex \( (-1, -3) \) becomes \( (-1, -1) \) (now matches y-range but shifted)
- and further translations yield vertices approaching the values of Quadrant II.
- Translating these points up by 2 (adding 2 to the y-coordinates):
However, instead of considering a simple translation after reflection, it's clear that these reflections reconfigure vertex alignments due to space adjustments leading to a form where translation is conceptualized by shifting these reflected points to match the Quadrant II placements.
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So the overall transformation sequence is a reflection followed by a translation. Hence, the correct answer is:
Reflection, then translation.