Find the Directrix (y-8)^2=24(x+1)

${(y - 8)}^{2} = 24 (x + 1)$

Step 1

Rewrite the equation in vertex form.

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

Rewrite the equation as $24 (x + 1) = {(y - 8)}^{2}$ .

$24 (x + 1) = {(y - 8)}^{2}$

Divide each term in $24 (x + 1) = {(y - 8)}^{2}$ by $24$ and simplify.

Divide each term in $24 (x + 1) = {(y - 8)}^{2}$ by $24$ .

$\frac{24 (x + 1)}{24} = \frac{{(y - 8)}^{2}}{24}$

Simplify the left side.

Cancel the common factor of $24$ .

Cancel the common factor.

$\frac{24 (x + 1)}{24} = \frac{{(y - 8)}^{2}}{24}$

Divide $x + 1$ by $1$ .

$x + 1 = \frac{{(y - 8)}^{2}}{24}$

Subtract $1$ from both sides of the equation.

$x = \frac{{(y - 8)}^{2}}{24} - 1$

Reorder terms.

$x = \frac{1}{24} \cdot {(y - 8)}^{2} - 1$

Complete the square for $\frac{1}{24} \cdot {(y - 8)}^{2} - 1$ .

Simplify the expression.

Simplify each term.

Rewrite ${(y - 8)}^{2}$ as $(y - 8) (y - 8)$ .

$\frac{1}{24} \cdot ((y - 8) (y - 8)) - 1$

Expand $(y - 8) (y - 8)$ using the FOIL Method.

Apply the distributive property.

$\frac{1}{24} \cdot (y (y - 8) - 8 (y - 8)) - 1$

Apply the distributive property.

$\frac{1}{24} \cdot (y \cdot y + y \cdot - 8 - 8 (y - 8)) - 1$

Apply the distributive property.

$\frac{1}{24} \cdot (y \cdot y + y \cdot - 8 - 8 y - 8 \cdot - 8) - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$\frac{1}{24} \cdot (y^{2} + y \cdot - 8 - 8 y - 8 \cdot - 8) - 1$

Move $- 8$ to the left of $y$ .

$\frac{1}{24} \cdot (y^{2} - 8 \cdot y - 8 y - 8 \cdot - 8) - 1$

Multiply $- 8$ by $- 8$ .

$\frac{1}{24} \cdot (y^{2} - 8 y - 8 y + 64) - 1$

Subtract $8 y$ from $- 8 y$ .

$\frac{1}{24} \cdot (y^{2} - 16 y + 64) - 1$

Apply the distributive property.

$\frac{1}{24} y^{2} + \frac{1}{24} (- 16 y) + \frac{1}{24} \cdot 64 - 1$

Simplify.

Combine $\frac{1}{24}$ and $y^{2}$ .

$\frac{y^{2}}{24} + \frac{1}{24} (- 16 y) + \frac{1}{24} \cdot 64 - 1$

Cancel the common factor of $8$ .

Factor $8$ out of $24$ .

$\frac{y^{2}}{24} + \frac{1}{8 (3)} (- 16 y) + \frac{1}{24} \cdot 64 - 1$

Factor $8$ out of $- 16 y$ .

$\frac{y^{2}}{24} + \frac{1}{8 (3)} (8 (- 2 y)) + \frac{1}{24} \cdot 64 - 1$

Cancel the common factor.

$\frac{y^{2}}{24} + \frac{1}{8 \cdot 3} (8 (- 2 y)) + \frac{1}{24} \cdot 64 - 1$

Rewrite the expression.

$\frac{y^{2}}{24} + \frac{1}{3} (- 2 y) + \frac{1}{24} \cdot 64 - 1$

Combine $- 2$ and $\frac{1}{3}$ .

$\frac{y^{2}}{24} + \frac{- 2}{3} y + \frac{1}{24} \cdot 64 - 1$

Combine $\frac{- 2}{3}$ and $y$ .

$\frac{y^{2}}{24} + \frac{- 2 y}{3} + \frac{1}{24} \cdot 64 - 1$

Cancel the common factor of $8$ .

Factor $8$ out of $24$ .

$\frac{y^{2}}{24} + \frac{- 2 y}{3} + \frac{1}{8 (3)} \cdot 64 - 1$

Factor $8$ out of $64$ .

$\frac{y^{2}}{24} + \frac{- 2 y}{3} + \frac{1}{8 \cdot 3} \cdot (8 \cdot 8) - 1$

Cancel the common factor.

$\frac{y^{2}}{24} + \frac{- 2 y}{3} + \frac{1}{8 \cdot 3} \cdot (8 \cdot 8) - 1$

Rewrite the expression.

$\frac{y^{2}}{24} + \frac{- 2 y}{3} + \frac{1}{3} \cdot 8 - 1$

Combine $\frac{1}{3}$ and $8$ .

$\frac{y^{2}}{24} + \frac{- 2 y}{3} + \frac{8}{3} - 1$

Move the negative in front of the fraction.

$\frac{y^{2}}{24} - \frac{2 y}{3} + \frac{8}{3} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{y^{2}}{24} - \frac{2 y}{3} + \frac{8}{3} - 1 \cdot \frac{3}{3}$

Combine $- 1$ and $\frac{3}{3}$ .

$\frac{y^{2}}{24} - \frac{2 y}{3} + \frac{8}{3} + \frac{- 1 \cdot 3}{3}$

Combine the numerators over the common denominator.

$\frac{y^{2}}{24} - \frac{2 y}{3} + \frac{8 - 1 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 1$ by $3$ .

$\frac{y^{2}}{24} - \frac{2 y}{3} + \frac{8 - 3}{3}$

Subtract $3$ from $8$ .

$\frac{y^{2}}{24} - \frac{2 y}{3} + \frac{5}{3}$

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = \frac{1}{24}$

$b = - \frac{2}{3}$

$c = \frac{5}{3}$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- \frac{2}{3}}{2 (\frac{1}{24})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{2}{3} \cdot \frac{1}{2 (\frac{1}{24})}$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$d = \frac{- 2}{3} \cdot \frac{1}{2 (\frac{1}{24})}$

Factor $2$ out of $- 2$ .

$d = \frac{2 (- 1)}{3} \cdot \frac{1}{2 (\frac{1}{24})}$

Cancel the common factor.

$d = \frac{2 \cdot - 1}{3} \cdot \frac{1}{2 (\frac{1}{24})}$

Rewrite the expression.

$d = \frac{- 1}{3} \cdot \frac{1}{\frac{1}{24}}$

Multiply $\frac{- 1}{3}$ by $\frac{1}{\frac{1}{24}}$ .

$d = \frac{- 1}{3 (\frac{1}{24})}$

Combine $3$ and $\frac{1}{24}$ .

$d = \frac{- 1}{\frac{3}{24}}$

Cancel the common factor of $3$ and $24$ .

Factor $3$ out of $3$ .

$d = \frac{- 1}{\frac{3 (1)}{24}}$

Cancel the common factors.

Factor $3$ out of $24$ .

$d = \frac{- 1}{\frac{3 \cdot 1}{3 \cdot 8}}$

Cancel the common factor.

$d = \frac{- 1}{\frac{3 \cdot 1}{3 \cdot 8}}$

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 8$

Multiply $- 1$ by $8$ .

$d = - 8$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{5}{3} - \frac{{(- \frac{2}{3})}^{2}}{4 (\frac{1}{24})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{2}{3}$ .

$e = \frac{5}{3} - \frac{{(- 1)}^{2} {(\frac{2}{3})}^{2}}{4 (\frac{1}{24})}$

Raise $- 1$ to the power of $2$ .

$e = \frac{5}{3} - \frac{1 {(\frac{2}{3})}^{2}}{4 (\frac{1}{24})}$

Apply the product rule to $\frac{2}{3}$ .

$e = \frac{5}{3} - \frac{1 \frac{2^{2}}{3^{2}}}{4 (\frac{1}{24})}$

Raise $2$ to the power of $2$ .

$e = \frac{5}{3} - \frac{1 \frac{4}{3^{2}}}{4 (\frac{1}{24})}$

Raise $3$ to the power of $2$ .

$e = \frac{5}{3} - \frac{1 (\frac{4}{9})}{4 (\frac{1}{24})}$

Multiply $\frac{4}{9}$ by $1$ .

$e = \frac{5}{3} - \frac{\frac{4}{9}}{4 (\frac{1}{24})}$

Combine $4$ and $\frac{1}{24}$ .

$e = \frac{5}{3} - \frac{\frac{4}{9}}{\frac{4}{24}}$

Cancel the common factor of $4$ and $24$ .

Factor $4$ out of $4$ .

$e = \frac{5}{3} - \frac{\frac{4}{9}}{\frac{4 (1)}{24}}$

Cancel the common factors.

Factor $4$ out of $24$ .

$e = \frac{5}{3} - \frac{\frac{4}{9}}{\frac{4 \cdot 1}{4 \cdot 6}}$

Cancel the common factor.

$e = \frac{5}{3} - \frac{\frac{4}{9}}{\frac{4 \cdot 1}{4 \cdot 6}}$

Rewrite the expression.

$e = \frac{5}{3} - \frac{\frac{4}{9}}{\frac{1}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{5}{3} - (\frac{4}{9} \cdot 6)$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$e = \frac{5}{3} - (\frac{4}{3 (3)} \cdot 6)$

Factor $3$ out of $6$ .

$e = \frac{5}{3} - (\frac{4}{3 \cdot 3} \cdot (3 \cdot 2))$

Cancel the common factor.

$e = \frac{5}{3} - (\frac{4}{3 \cdot 3} \cdot (3 \cdot 2))$

Rewrite the expression.

$e = \frac{5}{3} - (\frac{4}{3} \cdot 2)$

Combine $\frac{4}{3}$ and $2$ .

$e = \frac{5}{3} - \frac{4 \cdot 2}{3}$

Multiply $4$ by $2$ .

$e = \frac{5}{3} - \frac{8}{3}$

Combine the numerators over the common denominator.

$e = \frac{5 - 8}{3}$

Subtract $8$ from $5$ .

$e = \frac{- 3}{3}$

Divide $- 3$ by $3$ .

$e = - 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{24} {(y - 8)}^{2} - 1$ .

$\frac{1}{24} {(y - 8)}^{2} - 1$

Set $x$ equal to the new right side.

$x = \frac{1}{24} \cdot {(y - 8)}^{2} - 1$

Step 2

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

$a = \frac{1}{24}$

$h = - 1$

$k = 8$

Step 3

Find the vertex $(h, k)$ .

$(- 1, 8)$

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.

$\frac{1}{4 \cdot \frac{1}{24}}$

Simplify.

Combine $4$ and $\frac{1}{24}$ .

$\frac{1}{\frac{4}{24}}$

Cancel the common factor of $4$ and $24$ .

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{24}}$

Cancel the common factors.

Factor $4$ out of $24$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Rewrite the expression.

$\frac{1}{\frac{1}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 6$

Multiply $6$ by $1$ .

$6$

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.

$x = - 7$

Step 6

Find the Directrix (y-8)^2=24(x+1)

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