Elliptic and Parabolic Curves

Algebra Level 4

Let \(f(x)\) be a quadratic function.

When

\[ y^2 = -x^3 + x^2 - x + 1 \text{ and } x^2 = - y^3 + y^2- y + 1 \]

are simultaneously graphed, there are three intersections. \(f(x)\) passes through all three of them, and there is one intersection where \(x=y=a\).

Find \(f(x)\) in terms of \(x\) and \(a\).

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