Find the Parabola with Focus (-4,0) and Directrix x=4 (-4,0) x=4

$(- 4, 0)$ $x = 4$

Step 1

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

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

Step 2

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.

$(\frac{- 4 + 4}{2}, 0)$

Simplify the vertex.

Cancel the common factor of $- 4 + 4$ and $2$ .

Factor $2$ out of $- 4$ .

$(\frac{2 \cdot - 2 + 4}{2}, 0)$

Factor $2$ out of $4$ .

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 2 + 2$ by $1$ .

$(- 2 + 2, 0)$

Add $- 2$ and $2$ .

$(0, 0)$

Step 3

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 $p$ .

$p = - 4 - 0$

Subtract $0$ from $- 4$ .

$p = - 4$

Step 4

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

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

Step 5

Simplify.

$y^{2} = - 16 x$

Step 6

Find the Parabola with Focus (-4,0) and Directrix x=4 (-4,0) x=4

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