Question

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

Answer 1

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]}