Find the Parabola with Focus (-4,-5/4) and Directrix y=27/4 (-4,-5/4) y=27/4

$(- 4, - \frac{5}{4})$ $y = \frac{27}{4}$

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.

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.

$(- 4, \frac{- \frac{5}{4} + \frac{27}{4}}{2})$

Simplify the vertex.

Simplify the numerator.

Combine the numerators over the common denominator.

$(- 4, \frac{\frac{- 5 + 27}{4}}{2})$

Add $- 5$ and $27$ .

$(- 4, \frac{\frac{22}{4}}{2})$

Cancel the common factor of $22$ and $4$ .

Factor $2$ out of $22$ .

$(- 4, \frac{\frac{2 (11)}{4}}{2})$

Cancel the common factors.

Factor $2$ out of $4$ .

$(- 4, \frac{\frac{2 \cdot 11}{2 \cdot 2}}{2})$

Cancel the common factor.

$(- 4, \frac{\frac{2 \cdot 11}{2 \cdot 2}}{2})$

Rewrite the expression.

$(- 4, \frac{\frac{11}{2}}{2})$

Multiply the numerator by the reciprocal of the denominator.

$(- 4, \frac{11}{2} \cdot \frac{1}{2})$

Multiply $\frac{11}{2} \cdot \frac{1}{2}$ .

Multiply $\frac{11}{2}$ by $\frac{1}{2}$ .

$(- 4, \frac{11}{2 \cdot 2})$

Multiply $2$ by $2$ .

$(- 4, \frac{11}{4})$

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 $y$ coordinate of the vertex from the $y$ coordinate of the focus to find $p$ .

$p = - \frac{5}{4} - \frac{11}{4}$

Simplify.

Combine the numerators over the common denominator.

$p = \frac{- 5 - 11}{4}$

Simplify the expression.

Subtract $11$ from $- 5$ .

$p = \frac{- 16}{4}$

Divide $- 16$ by $4$ .

$p = - 4$

Step 4

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

${(x + 4)}^{2} = 4 (- 4) (y - \frac{11}{4})$

Step 5

Simplify.

${(x + 4)}^{2} = - 16 (y - \frac{11}{4})$

Step 6

Find the Parabola with Focus (-4,-5/4) and Directrix y=27/4 (-4,-5/4) y=27/4

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