Question

What are the vertex, focus and directrix of `y=3 -8x -4x^2`?

Answer 1

Vertex `(h, k)=(-1, 7)`

Focus `(h, k-p)=(-1, 7-1/16)=(-1, 111/16)`

Directrix is an equation a horizontal line

`y=k+p=7+1/16=113/16`
`y=113/16`

Explanation:

From the given equation `y=3-8x-4x^2`

Do a little rearrangement

`y=-4x^2-8x+3`

factor out -4

`y=-4(x^2+2x)+3`

Complete the square by adding 1 and subtracting 1 inside the parenthesis

`y=-4(x^2+2x+1-1)+3`

`y=-4(x+1)^2+4+3`

`y=-4(x+1)^2+7`

`y-7=-4(x+1)^2`

`(x--1)^2=-1/4(y-7)` The negative sign indicates that the parabola opens downward

`-4p=-1/4`

`p=1/16`

Vertex `(h, k)=(-1, 7)`

Focus `(h, k-p)=(-1, 7-1/16)=(-1, 111/16)`

Directrix is an equation a horizontal line

`y=k+p=7+1/16=113/16`
`y=113/16`

Kindly see the graph of `y=3-8x-4x^2`

graph{(y-3+8x+4x^2)(y-113/16)=0[-20,20,-10,10]}

God bless...I hope the explanation is useful.