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

Jun 29, 2018

Vertex=$\left(- 2 , 0\right)$
Its directrix is $y = - \frac{1}{4}$
it's focus is $\left(- 2 , \frac{1}{4}\right)$

#### Explanation:

$y = \textcolor{g r e e n}{{\left(x + 2\right)}^{2} - 4} + 4$

$y = {\left(x + 2\right)}^{2}$

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)$\rightarrow$$\text{Where a is positive real number}$

so applying this for the following equation

$y = {\left(x + 2\right)}^{2}$

$4 a = 1 \rightarrow a = \frac{1}{4}$

it's vertex is $\left(- 2 , 0\right)$

it's directrix is $y = 0 - \frac{1}{4} = - \frac{1}{4}$

it's focus is $\left(- 2 , 0 + \frac{1}{4}\right) = \left(- 2 , \frac{1}{4}\right)$