Question

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

Answer 1

Vertex=`(-2,0)`
Its directrix is `y=-1/4`
it's focus is `(-2,1/4)`

Explanation:

By completing the square

`y=color(green)((x+2)^2-4)+4`

`y=(x+2)^2`

the parabola is opened upwards

If a parabola is opened upwards then its equation will be

`color(blue)(y-k=4a(x-h)^2`

where `color(blue)((h,k)` are it's vertex

it's directrix is `color(blue)(y=k-a`

and its focus is `color(blue)((h,k+a)` `rarr` `"Where a is positive real number"`

so applying this for the following equation

`y=(x+2)^2`

`4a=1rarra=1/4`

it's vertex is `(-2,0)`

it's directrix is `y=0-1/4=-1/4`

it's focus is `(-2,0+1/4)=(-2,1/4)`