To find the corresponding coordinate on the left parabola given the point (3, 1) on the right parabola, we need to analyze the equations of the parabolas.
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Identify the right parabola: The right parabola is given by \( f(x) = (x - 2)^2 \).
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Evaluate \( f(3) \): \[ f(3) = (3 - 2)^2 = (1)^2 = 1 \] The point (3, 1) is indeed on the right parabola.
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Identify the left parabola: The left parabola is given by \( y = (-x - 2)^2 \), which can be rewritten as \( y = (-(x + 2))^2 \) or \( y = (-1)^2 \cdot (x + 2)^2 \).
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Determine the relation between the parabolas: The left parabola is a reflection of the right parabola across the line \( x = -2 \). To find the corresponding x-coordinate for the point (3, 1):
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Calculate the reflection: The line \( x = -2 \) is the axis of reflection. The distance between the point 3 and this line is calculated as follows: \[ d = 3 - (-2) = 3 + 2 = 5 \] To find the reflected x-coordinate, subtract this distance from -2: \[ x' = -2 - 5 = -7 \]
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Determine the y-coordinate on the left parabola \( y = (-x - 2)^2 \): Use \( x' = -7 \) to find the corresponding y-coordinate: \[ y' = (-(-7) - 2)^2 = (7 - 2)^2 = (5)^2 = 25 \]
Thus, the coordinate on the left parabola that corresponds to the point (3, 1) on the right parabola is:
\[ (-7, 25) \]
Therefore, the answer is: \[ (-7, 25) \]