Algebra Examples

$(- 4, \frac{17}{8})$ $y = \frac{15}{8}$

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.

$(- 4, \frac{\frac{17}{8} + \frac{15}{8}}{2})$

Step 2.2

Simplify the vertex.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Combine the numerators over the common denominator.

$(- 4, \frac{\frac{17 + 15}{8}}{2})$

Step 2.2.1.2

Add $17$ and $15$ .

$(- 4, \frac{\frac{32}{8}}{2})$

Step 2.2.1.3

Divide $32$ by $8$ .

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

Step 2.2.2

Divide $4$ by $2$ .

$(- 4, 2)$

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 $p$ .

$p = \frac{17}{8} - 2$

Step 3.2

Simplify.

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Step 3.2.1

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$p = \frac{17}{8} - 2 \cdot \frac{8}{8}$

Step 3.2.2

Combine $- 2$ and $\frac{8}{8}$ .

$p = \frac{17}{8} + \frac{- 2 \cdot 8}{8}$

Step 3.2.3

Combine the numerators over the common denominator.

$p = \frac{17 - 2 \cdot 8}{8}$

Step 3.2.4

Simplify the numerator.

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Step 3.2.4.1

Multiply $- 2$ by $8$ .

$p = \frac{17 - 16}{8}$

Step 3.2.4.2

Subtract $16$ from $17$ .

$p = \frac{1}{8}$

Step 4

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

${(x + 4)}^{2} = 4 (\frac{1}{8}) (y - 2)$

Step 5

Simplify.

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

Step 6