# What are the vertex, focus, and directrix of the parabola described by (x − 5)^2 = −4(y + 2)?

Aug 7, 2018

$\left(5 , - 2\right) , \left(5 , - 3\right) , y = - 1$

#### Explanation:

$\text{the standard form of a vertically opening parabola is}$

•color(white)(x)(x-h)^2=4a(y-k)

$\text{where "(h,k)" are the coordinates of the vertex and a}$
$\text{is the distance from the vertex to the focus and}$
$\text{directrix}$

${\left(x - 5\right)}^{2} = - 4 \left(y + 2\right) \text{ is in this form}$

$\text{with vertex } = \left(5 , - 2\right)$

$\text{ and } 4 a = - 4 \Rightarrow a = - 1$

$\text{Focus } = \left(h , a + k\right) = \left(5 , - 1 - 2\right) = \left(5 , - 3\right)$

$\text{directrix is } y = - a + k = 1 - 2 = - 1$
graph{(x-5)^2=-4(y+2) [-10, 10, -5, 5]}