# What is the equation for a parabola with a vertex at (5,-1) and a focus at (5,-7)?

Jan 13, 2018

$y = - \frac{1}{24} {x}^{2} + \frac{5}{12} x - \frac{49}{24}$

#### Explanation:

$\text{note that the vertex and focus are on the vertical axis}$

$x = 5$

$\text{the focus and directrix are equidistant from the vertex}$

$\text{that is } - 1 - \left(- 7\right) = 6$

$\text{equation of directrix is } y = - 1 + 6 \to y = 5$

$\text{for any point "(x,y)" on the parabola}$

$\text{the distance to the focus and directrix are equidistant}$

$\text{using the "color(blue)"distance formula}$

$\sqrt{{\left(x - 5\right)}^{2} + {\left(y + 7\right)}^{2}} = | y - 5 |$

$\textcolor{b l u e}{\text{squaring both sides}}$

${\left(x - 5\right)}^{2} + {\left(y + 7\right)}^{2} = {\left(y - 5\right)}^{2}$

${\left(y + 7\right)}^{2} - {\left(y - 5\right)}^{2} = - {\left(x - 5\right)}^{2}$

$\cancel{{y}^{2}} + 14 y + 49 \cancel{- {y}^{2}} + 10 y - 25 = - {x}^{2} + 10 x - 25$

$\Rightarrow 24 y = - {x}^{2} + 10 x - 49$

$\Rightarrow y = - \frac{1}{24} {x}^{2} + \frac{5}{12} x - \frac{49}{24}$

Jan 13, 2018

$- 24 \left(y + 1\right) = {\left(x - 5\right)}^{2}$

#### Explanation:

Let's graph the points we have. The vertex and focus have the same x-coordinate. This tells us that this parabola doesn't open sideways, which means we'll be using the equation $4 p \left(y - k\right) = {\left(x - h\right)}^{2}$
(Here's a link if you want to see how this equation was derived!)

In the equation, $p$ is the distance from the vertex to the focus or directrix; $k$ is the y-coordinate of the vertex, and $h$ is the x-coordinate of the vertex. Since the vertex is already given, we can plug that in:
$4 p \left(y - k\right) = {\left(x - h\right)}^{2}$
$4 p \left(y + 1\right) = {\left(x - 5\right)}^{2}$

To find the p-value, simply subtract the y-value of the vertex from the y-value of the focus.
$- 7 + 1 = - 6$
The p-value is negative. This makes sense, because the focus of the parabola is below the vertex, making the parabola open downwards. Downward parabolas have negative p-values.

Now, just plug that into the equation:
$4 \left(- 6\right) \left(y + 1\right) = {\left(x - 5\right)}^{2}$
$- 24 \left(y + 1\right) = {\left(x - 5\right)}^{2}$ 