To identify the sequence of transformations that carry the points A(-5,2), B(-2,4), and C(-3,0) to A"(-1,-2), B"(2,-4), and C"(1,0), we will analyze the transformations step by step.
Let's start with the points:
- A(-5,2) transforms to A"(-1,-2)
- B(-2,4) transforms to B"(2,-4)
- C(-3,0) transforms to C"(1,0)
Finding the transformation:
- Identify the changes in the coordinates:
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For point A: from (-5, 2) to (-1, -2)
- Change in x: -5 to -1 → +4
- Change in y: 2 to -2 → -4
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For point B: from (-2, 4) to (2, -4)
- Change in x: -2 to 2 → +4
- Change in y: 4 to -4 → -8
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For point C: from (-3, 0) to (1, 0)
- Change in x: -3 to 1 → +4
- Change in y: 0 to 0 → 0
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Notice the pattern in the x-coordinates: each x-coordinate increases by 4.
- Consider transformations available:
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Let's check option (a): \((x,y) \to (x, -y + 4)\)
- A(-5,2) → A(-5, -2 + 4) = A(-5, 2) (no change)
- This doesn’t work.
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Option (b): \((x,y) \to (x + 4, -y)\)
- A(-5,2) → A(-5 + 4, -2) = A(-1,-2)
- B(-2,4) → B(-2 + 4, -4) = B(2,-4)
- C(-3,0) → C(-3 + 4, 0) = C(1,0)
- This matches our required transformation.
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Option (c): \((x,y) \to (x, -y - 4)\)
- A(-5,2) → A(-5, -2 - 4) = A(-5, -6) (not matching)
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Option (d): \((x,y) \to (x - 4, y)\)
- A(-5,2) → A(-5 - 4, 2) = A(-9, 2) (not matching)
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Conclusion:
The required transformation to map A(-5,2), B(-2,4), C(-3,0) to A"(-1,-2), B"(2,-4), C"(1,0) is:
(x,y) \to (x + 4, -y)
Thus, the composition transformation that carries the points is: (x,y) → (x + 4, -y).
This matches option (b).
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