How do you find the vertex, focus and directrix of #4x^2 + 6x -y + 2 = 0#?

1 Answer
Nov 9, 2016

The vertex #=(-3/4,-1/4)#
The focus is #=(-3/5,-3/16)#
The equation of the directrix is #y=-5/16#

Explanation:

Let's rewrite the equation by completing the squares
#4(x^2+3/2x+9/16)=y-2+9/4=y+1/4#
#(x+3/4)^2=1/4(y+1/4)#
This is a parabola
We compare this to the equation of the parabola #(x-a)^2=2p(y-b)#
#:. #The vertex is #(-3/4,-1/4)#
#p=1/8#
The focus is #(a,b+p/2)=(-3/4,-1/4+1/16)=(-3/4,-3/16)#

The equation of the directrix is #y=-1/4-1/16=-5/16#
graph{(4x^2+6x-y+2)(y+5/16)=0 [-3.459, 1.409, -1.378, 1.056]}