Algebra Examples

$(0, - 5)$ $y = - 3$

Step 1

Since the directrix is vertical, use the equation of a parabola that opens up or down.

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

Step 2

Find the vertex.

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Step 2.1

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

$(0, \frac{- 5 - 3}{2})$

Step 2.2

Simplify the vertex.

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Step 2.2.1

Subtract $3$ from $- 5$ .

$(0, \frac{- 8}{2})$

Step 2.2.2

Divide $- 8$ by $2$ .

$(0, - 4)$

Step 3

Find the distance from the focus to the vertex.

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Step 3.1

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

$p = - 5 + 4$

Step 3.2

Add $- 5$ and $4$ .

$p = - 1$

Step 4

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

${(x - 0)}^{2} = 4 (- 1) (y + 4)$

Step 5

Simplify.

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

Step 6