Find the equation of a parabola (in general form) Find the equation of the parabola with axis parallel to the $y$-axis, passing through $(1/2,-5/2),(3/2,-9/4)$ and $(-7/2,3/2)$.
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\begin{align}
{\rm y}\pars{x}&amp;=
A\pars{x - \half}\pars{x - {3 \over 2}}
+B\pars{x - {3 \over 2}}\pars{x + {7 \over 2}}
+C\pars{x - \half}\pars{x + {7 \over 2}}
\end{align}


  In order to determine $\ds{A, B , C}$ use 
  $$
{\rm y}\pars{-\,{7 \over 2}} = {3 \over 2}\,,\qquad
{\rm y}\pars{\half} = -\,{5 \over 2}\,,\qquad
{\rm y}\pars{3 \over 2} = -\,{9 \over 4}
$$
Standard form:   $(x+20)^2 + (y+24)^2 = (\dfrac{185}{2})^2$

General form: $2x^2 + 2y^2 - 20x -24y - 63$