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}&=
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 
  $$
{\rm y}\pars{-\,{7 \over 2}} = {3 \over 2}\,,\qquad
{\rm y}\pars{\half} = -\,{5 \over 2}\,,\qquad
{\rm y}\pars{3 \over 2} = -\,{9 \over 4}
$$

A: HINT:(Idea behind the problem) A parabola is a graph of a quadratic function
$$ \displaystyle\boxed{y =ax^2 + bx + c}$$ 
substitute 3 points given , you will get 3 equations in a,b,c and from there  find a  , b , c solving the 3 equations .and then substitute this a,b,c in the equation above and that will be your answer .
refer : parabola with axis paraller to y axis
refer an example : refer in case you have doubts and want example
solution from wolfram alpha : wolfram alpha
A: The above answer is a good shortcut, but by convention it's as follows:
With axis parallel to the y axis, if the vertex of the  parabola is on the origin, then the equation is $x^2=4ay$. But when you shift the parabola on a vertex with coordinates $(h,k)$, the equation becomes $(x-h)^2=4a(y-k)$. Substitute the values you gave to find h,k and a (three equations, three unknowns).
But I would also recommend going by $y=ax^2+bx+c$, since the convention might get messy.
A: Standard form:   $(x+20)^2 + (y+24)^2 = (\dfrac{185}{2})^2$
General form: $2x^2 + 2y^2 - 20x -24y - 63$
