Find the Directrix -8(y-3)=(x+4)^2

$- 8 (y - 3) = {(x + 4)}^{2}$

Step 1

Isolate $y$ to the left side of the equation.

Divide each term in $- 8 (y - 3) = {(x + 4)}^{2}$ by $- 8$ and simplify.

Divide each term in $- 8 (y - 3) = {(x + 4)}^{2}$ by $- 8$ .

$\frac{- 8 (y - 3)}{- 8} = \frac{{(x + 4)}^{2}}{- 8}$

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

$\frac{- 8 (y - 3)}{- 8} = \frac{{(x + 4)}^{2}}{- 8}$

Divide $y - 3$ by $1$ .

$y - 3 = \frac{{(x + 4)}^{2}}{- 8}$

Simplify the right side.

Move the negative in front of the fraction.

$y - 3 = - \frac{{(x + 4)}^{2}}{8}$

Add $3$ to both sides of the equation.

$y = - \frac{{(x + 4)}^{2}}{8} + 3$

Reorder terms.

$y = - \frac{1}{8} \cdot {(x + 4)}^{2} + 3$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - \frac{1}{8}$

$h = - 4$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(- 4, 3)$

Step 4

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{1}{8})}$

Simplify.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{8})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{8})}$

Combine $4$ and $\frac{1}{8}$ .

$- \frac{1}{\frac{4}{8}}$

Cancel the common factor of $4$ and $8$ .

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{8}}$

Cancel the common factors.

Factor $4$ out of $8$ .

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 2}}$

Rewrite the expression.

$- \frac{1}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 2)$

Multiply $- (1 \cdot 2)$ .

Multiply $2$ by $1$ .

$- 1 \cdot 2$

Multiply $- 1$ by $2$ .

$- 2$

Step 5

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Substitute the known values of $p$ and $k$ into the formula and simplify.

$y = 5$

Step 6

Find the Directrix -8(y-3)=(x+4)^2

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