Question

How do you find the vertex, focus and directrix of `y+12x-2x^2=15`?

Answer 1

The vertex is `V=(3,-3)`
The focus is `F=(3,-23/8)`
The directrix is `y=-25/8`

Explanation:

Let's rewrite the equation and complete the squares

`y+12x-2x^2=15`

`y-15=2x^2-12x`

`y-15=2(x^2-6x)`

`y-15+18=2(x^2-6x+9)`

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

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

We compare this equation to

`(x-a)^2=2p(y-b)`

The vertex is `V=(a,b)=(3,-3)`

`2p=1/2`, `=>`,`p=1/4`

The focus is `F=(a,b+p/2)=(3,-23/8)`

The directrix is

`y=b-p/2=-3-1/8=-25/8`

graph{(y-2x^2+12x-15)(y+25/8)=0 [-4.264, 13.514, -4.284, 4.605]}