Question 1054626
Some of the description should be changed.  Okay to let vertex be (h,k), but for some other given point on the graph, say this is some "given" point of something like  (p,v).  You do not want to confuse general point (x,y) with any specific point.


That now taken care of, start with Standard Form.
{{{y=m(x-h)^2+k}}}, for some real number m so that {{{m<>0}}}.


Your graph for the equation passes through some given point, (p,v).  
{{{m(x-h)^2=y-k}}}

{{{m=(y-k)/(x-h)^2}}}


{{{m=(v-k)/(p-h)^2}}}-------Now you would "know" value for m.


Put everything back into the standard form.
{{{y=((v-k)/(p-h)^2)(x-h)^2+k}}}
Obviously this is not yet in GENERAL form, but still in fairly standard form.  Just do all the needed "simplifications" or multiplications and arrange into general form.