Algebra Examples

$(- 3, - 5)$ $x = - 7$

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.

Tap for more steps...

Step 2.1

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.

$(\frac{- 3 - 7}{2}, - 5)$

Step 2.2

Simplify the vertex.

Tap for more steps...

Step 2.2.1

Subtract $7$ from $- 3$ .

$(\frac{- 10}{2}, - 5)$

Step 2.2.2

Divide $- 10$ by $2$ .

$(- 5, - 5)$

Step 3

Find the distance from the focus to the vertex.

Tap for more steps...

Step 3.1

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 $p$ .

$p = - 3 + 5$

Step 3.2

Add $- 3$ and $5$ .

$p = 2$

Step 4

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

${(y + 5)}^{2} = 4 (2) (x + 5)$

Step 5

Simplify.

${(y + 5)}^{2} = 8 (x + 5)$

Step 6