# For the parabola x-3 = 1/8 (y+5)^2, what are the coordinates of the vertex and focus and directrix?

Jun 24, 2018

Vertex: $\left(3 , - 5\right)$
Focus: $\left(5 , - 5\right)$
Directrix: $x = 1$

#### Explanation:

First off, we should recognize that this graph opens to the right. This will be important later. To find the vertex, find the $h$ and $k$ values:

$x - h = \frac{1}{4 p} {\left(y - k\right)}^{2}$
$x - 3 = \frac{1}{8} {\left(y + 5\right)}^{2}$
$h = 3 , k = - 5$

Vertex: $\left(3 , - 5\right)$

Next, let's find the focus and directrix. First, though, we should find $p$, which you might know as the distance from the vertex to both the focus and the directrix. How do we find that? You might have noticed $\frac{1}{4 p}$ in the first equation you saw. Let's set our scale factor, $\frac{1}{8}$, equal to that:

$\frac{1}{4 p} = \frac{1}{8}$
$4 p = 8$
$p = 2$

To find the focus, we need to start from the vertex. Since the focus is always on the axis of symmetry within the parabola, and since the parabola opens to the right, we need to move $2$ units right from the vertex:
$\left(3 + 2 , - 5\right) = \left(5 , - 5\right)$

Finally, the directrix will be $2$ units away from the vertex in the opposite direction. The graph opens to the side, so the directrix will start with $x =$. Our directrix, therefore, is $x = 1$.