# What are the vertex, focus, and directrix of  y=8 - (x + 2) ^2?

The vertex is at $\left(h , k\right) = \left(- 2 , 8\right)$
Focus is at $\left(- 2 , 7\right)$
Directrix: $y = 9$

#### Explanation:

The given equation is $y = 8 - {\left(x + 2\right)}^{2}$

The equation is almost presented in the vertex form

$y = 8 - {\left(x + 2\right)}^{2}$

$y - 8 = - {\left(x + 2\right)}^{2}$

$- \left(y - 8\right) = {\left(x + 2\right)}^{2}$

${\left(x - - 2\right)}^{2} = - \left(y - 8\right)$

The vertex is at $\left(h , k\right) = \left(- 2 , 8\right)$

$a = \frac{1}{4 p}$ and $4 p = - 1$

$p = - \frac{1}{4}$

$a = \frac{1}{4 \cdot \left(- \frac{1}{4}\right)}$

$a = - 1$

Focus is at $\left(h , k - \left\mid a \right\mid\right) = \left(- 2 , 8 - 1\right) = \left(- 2 , 7\right)$

Directrix is the horizontal line equation

$y = k + \left\mid a \right\mid = 8 + 1 = 9$

$y = 9$

Kindly see the graph of $y = 8 - {\left(x + 2\right)}^{2}$ and the directrix $y = 9$

graph{(y-8+(x+2)^2)(y-9)=0[-25,25,-15,15]}

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