To find the line of reflection for the function \( f(x) = x^2 + 5x - 6 \) such that the reflected function \( f'(x) = -x^2 - 5x + 6 \), we can use the property of reflections:
When a point \( (a, b) \) is reflected over a line \( y = mx + b \), the relationship between the original point and the reflected point can be utilized. We know that the midpoint of the original point and its reflection lies on the line of reflection.
Given \( f(x) \) and \( f'(x) \), we can find the midpoint between \( (x, f(x)) \) and \( (x, f'(x)) \):
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Find the midpoint: \[ M(x) = \left( x, \frac{f(x) + f'(x)}{2} \right) \]
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Calculate \( f(x) + f'(x) \): \[ f(x) + f'(x) = (x^2 + 5x - 6) + (-x^2 - 5x + 6) \] \[ = 0 \]
Thus, we have: \[ f(x) + f'(x) = 0 \implies \frac{f(x) + f'(x)}{2} = 0 \]
So the y-coordinate of the midpoint \( M(x) \) is \( 0 \). Therefore, the midpoint \( M(x) \) is: \[ M(x) = \left( x, 0 \right) \]
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The line of reflection must be horizontal (since every point reflected over this line yields a vertical positioning at y = 0), thus the line of reflection is: \[ y = 0. \]
In conclusion, the equation of the line of reflection is:
\[ \boxed{y = 0}. \]
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