# What is the standard form of the parabola with a vertex at (4,0) and a focus at (4,-4)?

Jun 30, 2017

$y = - \frac{1}{16} {\left(x - 4\right)}^{2}$

#### Explanation:

The standard form of a parabola is

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

where $\left(h , k\right)$ is the vertex and $p$ is the distance from the vertex to the focus (or the distance from the vertex to the directrix).

Since we are given the vertex $\left(4 , 0\right)$, we can plug this into our parabola formula.

$y = \frac{1}{4 p} {\left(x - 4\right)}^{2} + 0$

$y = \frac{1}{4 p} {\left(x - 4\right)}^{2}$

To help visualize $p$, let's plot our given points on a graph.

$p$, or the distance from the vertex to the focus, is -4. Plug this value into the equation:

$y = \frac{1}{4 \left(- 4\right)} {\left(x - 4\right)}^{2}$
$y = - \frac{1}{16} {\left(x - 4\right)}^{2}$

That's your parabola in standard form!