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How do you find a in parabola equations using just the vertex and the focus?

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2 Answers

If the axis of symmetry is vertical, as it is in this case based on the vertex and focus, one of the normal forms of a parabola is (x-h)2 = 4p(y-k)  where (h,k) is the vertex, and p is the distance from the vertex to the focus of the parabola.  
 
Vertex = (-1,-3),  Focus = (-1,0), the axis of symmetry is x = -1, and p = 0-(-3) = +3.  
 
So the equation of the parabola is (x-(-1))2 = 4*3*(y-(-3)), or (x+1)2 = 12(y+3)
 
Note:  A unique directrix is determined once you know the Vertex and Focus because it must be perpendicular to the axis of symmetry and the same distance from the vertex as the focus is.
The definition of a parabola is : the locus of points equidistant from a point(focus) and a line(directrix). What you have given is just two points. This is not enough information to determine a parabola because there are an infinite number of directrixes that would satisfy this requirement leaving an infinite amount of possibilities for parabolas. The significance of having the OTHER point you speak of (that one normally has) is that with it, you can determine the direction of the directrix. Without the other point, all that you leave possible to construct is a line. (Two points determine a line.) Therefore, you need more information to answer this question.

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