Find the Parabola with Focus (-6,0) and Directrix x=-2 (-6,0) x=-2

$(- 6, 0)$ $x = - 2$

Since the directrix is horizontal, use the equation of a parabola that opens left or right.

${(y - k)}^{2} = 4 p (x - h)$

Find the vertex.

The vertex $(h, k)$ is halfway between the directrix and focus. Find the $x$ coordinate of the vertex using the formula $x = \frac{x coordinate of focus + directrix}{2}$ . The $y$ coordinate will be the same as the $y$ coordinate of the focus.

$(\frac{- 6 - 2}{2}, 0)$

Simplify the vertex.

Cancel the common factor of $- 6 - 2$ and $2$ .

Factor $2$ out of $- 6$ .

$(\frac{2 \cdot - 3 - 2}{2}, 0)$

Factor $2$ out of $- 2$ .

$(\frac{2 \cdot - 3 + 2 \cdot - 1}{2}, 0)$

Factor $2$ out of $2 \cdot - 3 + 2 \cdot - 1$ .

$(\frac{2 \cdot (- 3 - 1)}{2}, 0)$

Cancel the common factors.

Factor $2$ out of $2$ .

$(\frac{2 \cdot (- 3 - 1)}{2 (1)}, 0)$

Cancel the common factor.

$(\frac{2 \cdot (- 3 - 1)}{2 \cdot 1}, 0)$

Rewrite the expression.

$(\frac{- 3 - 1}{1}, 0)$

Divide $- 3 - 1$ by $1$ .

$(- 3 - 1, 0)$

Subtract $1$ from $- 3$ .

$(- 4, 0)$

Find the distance from the focus to the vertex.

The distance from the focus to the vertex and from the vertex to the directrix is $| p |$ . Subtract the $x$ coordinate of the vertex from the $x$ coordinate of the focus to find $p$ .

$p = - 6 + 4$

Add $- 6$ and $4$ .

$p = - 2$

Substitute in the known values for the variables into the equation ${(y - k)}^{2} = 4 p (x - h)$ .

${(y - 0)}^{2} = 4 (- 2) (x + 4)$

Simplify.

$y^{2} = - 8 (x + 4)$

Find the Parabola with Focus (-6,0) and Directrix x=-2 (-6,0) x=-2

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