parabola conic form

Question

I how do I write the vertex form of a parabola given the vertex and the focus. 

Example: Vertex (-3,4) Focus (-23/8,4)

Answer 1

This is a "sideways" parabola because the focus is to the right of the vertex.  The usual roles of x and y are reversed.   Nevertheless the P value is the distance from the vertex to the focus.    In this case  P =  1/8  

 

Since the vertex is known, the equation of this sideways parabola can be written in vertex form as

 

x = a( y-4)² -3      for some value of a.

 

A standard result from conic section theory is that  a = 1/(4 P).   In this case   a =  2.    So

 

x = 2 ( y -4)² - 3