Find the Directrix (y+2)^2=-8(x+1)

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

Step 1

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

Rewrite the equation as $- 8 (x + 1) = {(y + 2)}^{2}$ .

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

Divide each term in $- 8 (x + 1) = {(y + 2)}^{2}$ by $- 8$ and simplify.

Divide each term in $- 8 (x + 1) = {(y + 2)}^{2}$ by $- 8$ .

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

Simplify the left side.

Cancel the common factor of $- 8$ .

Cancel the common factor.

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

Divide $x + 1$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Subtract $1$ from both sides of the equation.

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

Reorder terms.

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

Step 2

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

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

$h = - 1$

$k = - 2$

Step 3

Find the vertex $(h, k)$ .

$(- 1, - 2)$

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 vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

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

$x = 1$

Step 6

Find the Directrix (y+2)^2=-8(x+1)

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