Find the equation of a parabola with given vertex and focus.

Question

vertex (-4,-2) and focus (-8,-2)

Answer 1

First one must determine whether the parabola opens upward, downward, left or right...

To do this look at the relative position of the vertex to the focus

With V = (-4,-2) and F = (-8,-2) we have the focus to left of the vertex so the parabola opens left

 

This means the standard form of the parabola will be given by

(y-k)² = 4p(x-h)

 

The vertex is (h,k) and p is the distance from vertex to focus

P is positive if parabola opens right and negative if parabola opens left....  In this case  p will be negative

The distance from Vertex to Focus is 4 ... from (-4,-2) to (-8,-2)... so p = -4 for this problem

 

Inputting this information into the standard form we get:

 

(y-(-2))² = 4(-4)(x-(-4))

(y+2)² = -16(x+4)