Algebra Examples

$(1, - 2)$ $x = - 3$

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.

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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.

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

Simplify the vertex.

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Subtract $3$ from $1$ .

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

Divide $- 2$ by $2$ .

$(- 1, - 2)$

Find the distance from the focus to the vertex.

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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 $p$ .

$p = 1 + 1$

Add $1$ and $1$ .

$p = 2$

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

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

Simplify.

${(y + 2)}^{2} = 8 (x + 1)$