To identify the sequence of transformations that carries ΔABC to ΔA"B"C", we need to analyze the transformations step by step based on the possible response options given:
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Reflection: The transformation that involves changing the sign of the y-coordinate (if applicable) suggests a reflection across the x-axis or a vertical flip of the triangle.
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Translation: The options also include translations (shifting the x or y coordinates by a certain number).
To carry out these transformations, let's use the following notation:
- A point in triangle ΔABC is represented as \( (x, y) \).
- After any transformations, the resulting points from ΔABC become ΔA"B"C".
Step Analysis
Suppose:
- The reflection across the x-axis would change the point \( (x, y) \) to \( (x, -y) \).
- Then, if we move this reflected point down by 4 units, we can achieve the resulting position of \( (x, -y-4) \).
To summarize, the sequence of transformations is a reflection across the x-axis followed by a downward translation of 4 units.
Composition Transformation Mapping
Therefore, the proper mapping that describes these transformations to carry \( ΔABC \) to \( ΔA"B"C" \) is:
\[ (x,y) \to (x, -y) \to (x, -y - 4) \]
From the given options, the correct transformation mapping that matches this sequence is:
(x,y)→(x,−y−4)
This means that the point transformation from \( (x, y) \) leads to the new coordinates as \( (x, -y - 4) \), capturing both the reflection and the translation accurately.