Let's start by applying the transformation rules step by step to find the coordinates of points A", B", and C" after the transformations.
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Initial Points:
- A(-8, 4)
- B(-5, 8)
- C(-3, 3)
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First Transformation (to get A', B', C'): The transformation rule is given by \( (x, y) \to (x + 4, y + 6) \).
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For A: \[ A' = (-8 + 4, 4 + 6) = (-4, 10) \]
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For B: \[ B' = (-5 + 4, 8 + 6) = (-1, 14) \]
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For C: \[ C' = (-3 + 4, 3 + 6) = (1, 9) \]
So after the first transformation, we have:
- A'(-4, 10)
- B'(-1, 14)
- C'(1, 9)
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Second Transformation (to get A'', B'', C''): The transformation rule is given by \( (x, y) \to (x + 5, y + 3) \).
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For A': \[ A'' = (-4 + 5, 10 + 3) = (1, 13) \]
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For B': \[ B'' = (-1 + 5, 14 + 3) = (4, 17) \]
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For C': \[ C'' = (1 + 5, 9 + 3) = (6, 12) \]
So after the second transformation, we have:
- A''(1, 13)
- B''(4, 17)
- C''(6, 12)
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Chain Rule (Composition of Transformations): The composed transformation rule from the original points is calculated by combining the two transformations:
Starting from \( (x, y) \), we first apply \( (x + 4, y + 6) \) followed by \( (x + 5, y + 3) \):
- First transformation: \[(x, y) \to (x + 4, y + 6)\]
- Second transformation applied to the result of the first: \[(x + 4, y + 6) \to ((x + 4) + 5, (y + 6) + 3)\]
- This simplifies to: \[ (x + 9, y + 9) \]
Therefore, the overall transformation rule is: \[ (x, y) \to (x + 9, y + 9) \]
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Predicted Quadrant: Now let's consider the final coordinates of A'', B'', and C'':
- A''(1, 13) is in Quadrant I (both x and y are positive).
- B''(4, 17) is also in Quadrant I.
- C''(6, 12) is in Quadrant I as well.
Since all the transformed points A'', B'', and C'' have positive x and y coordinates, we can predict that the image will be in Quadrant I.