What is the equation of the parabola that has a vertex at # (6, 2) # and passes through point # (3,20) #?

Given:
#color(white)("XXX")#Vertex at #(color(red)6,color(blue)2)#, and
#color(white)("XXX")#Additional point at #(3,20)#

If we assume the desired parabola has a vertical axis,
then the vertex form of any such parabola is
#color(white)("XXX")y=color(green)m(x-color(red)a)^2+color(blue)b# with vertex at #(color(red)a,color(blue)b)#

Therefore our desired parabola must have the vertex form
#color(white)("XXX")y=color(green)m(x-color(red)6)^2+color(blue)2#

Furthermore we know that the "additional point" #(x,y)=(color(magenta)3,color(teal)20)#

Therefore
#color(white)("XXX")color(teal)20=color(green)m(color(magenta)3-color(red)6)^2+color(blue)2#

#color(white)("XXX")rArr 18=9color(green)m#

#color(white)("XXX")rArr color(green)m=2#

Plugging this value back into our earier version of the desired parabola, we get
#color(white)("XXX")y=color(green)2(x-color(red)6)^2+color(blue)2#

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If the axis of symmetry is not vertical:
[1] if it is vertical a similar process can be used working with the general form #x=m(y-b)^2+a#
[2] if it is neither vertical nor horizontal, the process becomes more involved (ask as a separate question if this is the case; in general you will need to know the angle of the axis of symmetry in order to develop an answer).