Find the Directrix (y-3)^2=-12(x-1)

${(y - 3)}^{2} = - 12 (x - 1)$

Step 1

Rewrite the equation in vertex form.

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

Rewrite the equation as $- 12 (x - 1) = {(y - 3)}^{2}$ .

$- 12 (x - 1) = {(y - 3)}^{2}$

Divide each term in $- 12 (x - 1) = {(y - 3)}^{2}$ by $- 12$ and simplify.

Divide each term in $- 12 (x - 1) = {(y - 3)}^{2}$ by $- 12$ .

$\frac{- 12 (x - 1)}{- 12} = \frac{{(y - 3)}^{2}}{- 12}$

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

$\frac{- 12 (x - 1)}{- 12} = \frac{{(y - 3)}^{2}}{- 12}$

Divide $x - 1$ by $1$ .

$x - 1 = \frac{{(y - 3)}^{2}}{- 12}$

Simplify the right side.

Move the negative in front of the fraction.

$x - 1 = - \frac{{(y - 3)}^{2}}{12}$

Add $1$ to both sides of the equation.

$x = - \frac{{(y - 3)}^{2}}{12} + 1$

Reorder terms.

$x = - \frac{1}{12} \cdot {(y - 3)}^{2} + 1$

Complete the square for $- \frac{1}{12} \cdot {(y - 3)}^{2} + 1$ .

Simplify the expression.

Simplify each term.

Rewrite ${(y - 3)}^{2}$ as $(y - 3) (y - 3)$ .

$- \frac{1}{12} \cdot ((y - 3) (y - 3)) + 1$

Expand $(y - 3) (y - 3)$ using the FOIL Method.

Apply the distributive property.

$- \frac{1}{12} \cdot (y (y - 3) - 3 (y - 3)) + 1$

Apply the distributive property.

$- \frac{1}{12} \cdot (y \cdot y + y \cdot - 3 - 3 (y - 3)) + 1$

Apply the distributive property.

$- \frac{1}{12} \cdot (y \cdot y + y \cdot - 3 - 3 y - 3 \cdot - 3) + 1$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$- \frac{1}{12} \cdot (y^{2} + y \cdot - 3 - 3 y - 3 \cdot - 3) + 1$

Move $- 3$ to the left of $y$ .

$- \frac{1}{12} \cdot (y^{2} - 3 \cdot y - 3 y - 3 \cdot - 3) + 1$

Multiply $- 3$ by $- 3$ .

$- \frac{1}{12} \cdot (y^{2} - 3 y - 3 y + 9) + 1$

Subtract $3 y$ from $- 3 y$ .

$- \frac{1}{12} \cdot (y^{2} - 6 y + 9) + 1$

Apply the distributive property.

$- \frac{1}{12} y^{2} - \frac{1}{12} (- 6 y) - \frac{1}{12} \cdot 9 + 1$

Simplify.

Combine $y^{2}$ and $\frac{1}{12}$ .

$- \frac{y^{2}}{12} - \frac{1}{12} (- 6 y) - \frac{1}{12} \cdot 9 + 1$

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$- \frac{y^{2}}{12} + \frac{- 1}{12} (- 6 y) - \frac{1}{12} \cdot 9 + 1$

Factor $6$ out of $12$ .

$- \frac{y^{2}}{12} + \frac{- 1}{6 (2)} (- 6 y) - \frac{1}{12} \cdot 9 + 1$

Factor $6$ out of $- 6 y$ .

$- \frac{y^{2}}{12} + \frac{- 1}{6 (2)} (6 (- y)) - \frac{1}{12} \cdot 9 + 1$

Cancel the common factor.

$- \frac{y^{2}}{12} + \frac{- 1}{6 \cdot 2} (6 (- y)) - \frac{1}{12} \cdot 9 + 1$

Rewrite the expression.

$- \frac{y^{2}}{12} + \frac{- 1}{2} (- y) - \frac{1}{12} \cdot 9 + 1$

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

$- \frac{y^{2}}{12} - \frac{- y}{2} - \frac{1}{12} \cdot 9 + 1$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{12}$ into the numerator.

$- \frac{y^{2}}{12} - \frac{- y}{2} + \frac{- 1}{12} \cdot 9 + 1$

Factor $3$ out of $12$ .

$- \frac{y^{2}}{12} - \frac{- y}{2} + \frac{- 1}{3 (4)} \cdot 9 + 1$

Factor $3$ out of $9$ .

$- \frac{y^{2}}{12} - \frac{- y}{2} + \frac{- 1}{3 \cdot 4} \cdot (3 \cdot 3) + 1$

Cancel the common factor.

$- \frac{y^{2}}{12} - \frac{- y}{2} + \frac{- 1}{3 \cdot 4} \cdot (3 \cdot 3) + 1$

Rewrite the expression.

$- \frac{y^{2}}{12} - \frac{- y}{2} + \frac{- 1}{4} \cdot 3 + 1$

Combine $\frac{- 1}{4}$ and $3$ .

$- \frac{y^{2}}{12} - \frac{- y}{2} + \frac{- 1 \cdot 3}{4} + 1$

Multiply $- 1$ by $3$ .

$- \frac{y^{2}}{12} - \frac{- y}{2} + \frac{- 3}{4} + 1$

Simplify each term.

Move the negative in front of the fraction.

$- \frac{y^{2}}{12} - - \frac{y}{2} + \frac{- 3}{4} + 1$

Multiply $- - \frac{y}{2}$ .

Multiply $- 1$ by $- 1$ .

$- \frac{y^{2}}{12} + 1 \frac{y}{2} + \frac{- 3}{4} + 1$

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

$- \frac{y^{2}}{12} + \frac{y}{2} + \frac{- 3}{4} + 1$

Move the negative in front of the fraction.

$- \frac{y^{2}}{12} + \frac{y}{2} - \frac{3}{4} + 1$

Write $1$ as a fraction with a common denominator.

$- \frac{y^{2}}{12} + \frac{y}{2} - \frac{3}{4} + \frac{4}{4}$

Combine the numerators over the common denominator.

$- \frac{y^{2}}{12} + \frac{y}{2} + \frac{- 3 + 4}{4}$

Add $- 3$ and $4$ .

$- \frac{y^{2}}{12} + \frac{y}{2} + \frac{1}{4}$

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

$a = - \frac{1}{12}$

$b = \frac{1}{2}$

$c = \frac{1}{4}$

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 $d = \frac{b}{2 a}$ .

$d = \frac{\frac{1}{2}}{2 (- \frac{1}{12})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{2} \cdot \frac{1}{2 (- \frac{1}{12})}$

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

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

$d = \frac{1}{2} \cdot \frac{- 1 (- 1)}{2 (- \frac{1}{12})}$

Move the negative in front of the fraction.

$d = \frac{1}{2} (- \frac{1}{2 (\frac{1}{12})})$

Combine $2$ and $\frac{1}{12}$ .

$d = \frac{1}{2} (- \frac{1}{\frac{2}{12}})$

Cancel the common factor of $2$ and $12$ .

Factor $2$ out of $2$ .

$d = \frac{1}{2} (- \frac{1}{\frac{2 (1)}{12}})$

Cancel the common factors.

Factor $2$ out of $12$ .

$d = \frac{1}{2} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}})$

Cancel the common factor.

$d = \frac{1}{2} (- \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}})$

Rewrite the expression.

$d = \frac{1}{2} (- \frac{1}{\frac{1}{6}})$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{2} (- (1 \cdot 6))$

Multiply $- (1 \cdot 6)$ .

Multiply $6$ by $1$ .

$d = \frac{1}{2} (- 1 \cdot 6)$

Multiply $- 1$ by $6$ .

$d = \frac{1}{2} \cdot - 6$

Cancel the common factor of $2$ .

Factor $2$ out of $- 6$ .

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

Cancel the common factor.

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

Rewrite the expression.

$d = - 3$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{1}{4} - \frac{{(\frac{1}{2})}^{2}}{4 (- \frac{1}{12})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{1}{2}$ .

$e = \frac{1}{4} - \frac{\frac{1^{2}}{2^{2}}}{4 (- \frac{1}{12})}$

One to any power is one.

$e = \frac{1}{4} - \frac{\frac{1}{2^{2}}}{4 (- \frac{1}{12})}$

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

$e = \frac{1}{4} - \frac{\frac{1}{4}}{4 (- \frac{1}{12})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = \frac{1}{4} - \frac{\frac{1}{4}}{- 4 (\frac{1}{12})}$

Combine $- 4$ and $\frac{1}{12}$ .

$e = \frac{1}{4} - \frac{\frac{1}{4}}{\frac{- 4}{12}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4$ and $12$ .

Factor $4$ out of $- 4$ .

$e = \frac{1}{4} - \frac{\frac{1}{4}}{\frac{4 (- 1)}{12}}$

Cancel the common factors.

Factor $4$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

$e = \frac{1}{4} - \frac{\frac{1}{4}}{\frac{- 1}{3}}$

Move the negative in front of the fraction.

$e = \frac{1}{4} - \frac{\frac{1}{4}}{- \frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{1}{4} - (\frac{1}{4} (- 1 \cdot 3))$

Multiply $- 1$ by $3$ .

$e = \frac{1}{4} - (\frac{1}{4} \cdot - 3)$

Combine $\frac{1}{4}$ and $- 3$ .

$e = \frac{1}{4} - \frac{- 3}{4}$

Move the negative in front of the fraction.

$e = \frac{1}{4} - - \frac{3}{4}$

Multiply $- - \frac{3}{4}$ .

Multiply $- 1$ by $- 1$ .

$e = \frac{1}{4} + 1 (\frac{3}{4})$

Multiply $\frac{3}{4}$ by $1$ .

$e = \frac{1}{4} + \frac{3}{4}$

Combine the numerators over the common denominator.

$e = \frac{1 + 3}{4}$

Add $1$ and $3$ .

$e = \frac{4}{4}$

Divide $4$ by $4$ .

$e = 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{12} {(y - 3)}^{2} + 1$ .

$- \frac{1}{12} {(y - 3)}^{2} + 1$

Set $x$ equal to the new right side.

$x = - \frac{1}{12} \cdot {(y - 3)}^{2} + 1$

Step 2

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

$a = - \frac{1}{12}$

$h = 1$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(1, 3)$

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.

$\frac{1}{4 \cdot (- \frac{1}{12})}$

Simplify.

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

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{12})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{12})}$

Combine $4$ and $\frac{1}{12}$ .

$- \frac{1}{\frac{4}{12}}$

Cancel the common factor of $4$ and $12$ .

Factor $4$ out of $4$ .

$- \frac{1}{\frac{4 (1)}{12}}$

Cancel the common factors.

Factor $4$ out of $12$ .

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 3}}$

Cancel the common factor.

$- \frac{1}{\frac{4 \cdot 1}{4 \cdot 3}}$

Rewrite the expression.

$- \frac{1}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 3)$

Multiply $- (1 \cdot 3)$ .

Multiply $3$ by $1$ .

$- 1 \cdot 3$

Multiply $- 1$ by $3$ .

$- 3$

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.

$x = 4$

Step 6

Find the Directrix (y-3)^2=-12(x-1)

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