Any and all conics, parabolas included, take the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, with A, B, and C not all zero. The parabolas themselves have B2 - 4AC = 0.
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You need more than one tangent to find the equation of a parabola.
Look at the discriminant, B2 - 4AC, in the quadratic equation. As it goes from negative to positive, the parabola moves in the direction of its small end towards the X-axis. At zero, it touches the X-axis.
0,0
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Finding the vertex of the parabola is important because it tells you where the bottom (or the top, for a parabola that 'opens' downward), and thus where you can begin graphing.