# How do you find the vertex, focus, and directrix of the parabola y^2-4y-4x=0?

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#### Explanation

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#### Explanation:

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7
Jan 4, 2017

The vertex is V $= \left(- 1 , 2\right)$
The focus is F $= \left(0 , 2\right)$
The directrix is $x = - 2$

#### Explanation:

Let's rearrange the equation and complete the squares

${y}^{2} - 4 y = 4 x$

${y}^{2} - 4 y + 4 = 4 x + 4$

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

Comparing this equation to

${\left(y - b\right)}^{2} = 2 p \left(x - a\right)$

$p = 2$

The vertex is V $= \left(a , b\right) = \left(- 1 , 2\right)$

The focus is F $= \left(a + \frac{p}{2} , b\right) = \left(0 , 2\right)$

The directrix is $x = a - \frac{p}{2}$

$x = - 1 - 1 = - 2$

graph{(y^2-4y-4x)(y-100x-200)=0 [-10, 10, -5, 5]}

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