A Mirrored Parabola

Algebra Level 3

A parabola generated by the equation y = a x 2 + b x + c y = ax^2 + bx + c has a "mirrored" parabola with the equation y = A x 2 + B x + C . y = Ax^2 + Bx + C. What are the "mirrored" coefficients A , B , A, B, and C C of the new parabola in terms of a , b , a, b, and c ? c?

A = − a , B = − b , C = c + b 2 2 a A = -a,\ B = -b,\ C = c + \frac{b^2}{2a} A = − a , B = b , C = c − b 2 4 a A = -a,\ B = b,\ C = c - \frac{b^2}{4a} A = a , B = b , C = c + b 2 4 a A = a,\ B = b,\ C = c + \frac{b^2}{4a} A = − a , B = − b , C = c − b 2 2 a A = -a,\ B = -b,\ C = c - \frac{b^2}{2a}

Tj Evert by TJ Evert , 53, USA

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1 solution

Tj Evert
Feb 23, 2018

The equation y = ax² + bx + c has a vertex at (-b/2a, c + b²/4a). For the "mirrored" parabola, y = Ax² + Bx + C:

A = -a (same curvature, different direction), B = -b (the only way the x-value of the vertex doesn't change when A = -a), B = -b, C = c - b²/2a , The y-value of the original vertex = c - b²/4a has to equal the y-value of the "mirrored" vertex = C - B²/4A, so, C - B²/4A = c - b²/4a. From there, C = c - b²/4a + B²/4A = c - b²/4a + (-b)²/4(-a) = c - b²/2a.

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