Find the Focus (x+2)^2=-12(y-2)

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

Step 1

Rewrite the equation in vertex form.

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

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

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

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

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

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

Simplify the left side.

Cancel the common factor of $- 12$ .

Cancel the common factor.

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

Divide $y - 2$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Add $2$ to both sides of the equation.

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

Reorder terms.

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

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

Simplify the expression.

Simplify each term.

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$- \frac{1}{12} \cdot ((x + 2) (x + 2)) + 2$

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$- \frac{1}{12} \cdot (x (x + 2) + 2 (x + 2)) + 2$

Apply the distributive property.

$- \frac{1}{12} \cdot (x \cdot x + x \cdot 2 + 2 (x + 2)) + 2$

Apply the distributive property.

$- \frac{1}{12} \cdot (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) + 2$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $2$ to the left of $x$ .

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

Multiply $2$ by $2$ .

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

Add $2 x$ and $2 x$ .

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

Apply the distributive property.

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

Simplify.

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

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

Cancel the common factor of $4$ .

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

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

Factor $4$ out of $12$ .

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

Factor $4$ out of $4 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Cancel the common factor of $4$ .

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

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

Factor $4$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify each term.

Move the negative in front of the fraction.

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

Move the negative in front of the fraction.

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

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$- \frac{x^{2}}{12} - \frac{x}{3} + \frac{- 1 + 6}{3}$

Add $- 1$ and $6$ .

$- \frac{x^{2}}{12} - \frac{x}{3} + \frac{5}{3}$

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

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

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

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

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

$d = \frac{- \frac{1}{3}}{2 (- \frac{1}{12})}$

Simplify the right side.

Dividing two negative values results in a positive value.

$d = \frac{\frac{1}{3}}{2 (\frac{1}{12})}$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{3} \cdot \frac{1}{2 (\frac{1}{12})}$

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

$d = \frac{1}{3} \cdot \frac{1}{\frac{2}{12}}$

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

Factor $2$ out of $2$ .

$d = \frac{1}{3} \cdot \frac{1}{\frac{2 (1)}{12}}$

Cancel the common factors.

Factor $2$ out of $12$ .

$d = \frac{1}{3} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}}$

Cancel the common factor.

$d = \frac{1}{3} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 6}}$

Rewrite the expression.

$d = \frac{1}{3} \cdot \frac{1}{\frac{1}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{3} (1 \cdot 6)$

Multiply $6$ by $1$ .

$d = \frac{1}{3} \cdot 6$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

$d = 2$

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

$e = \frac{5}{3} - \frac{{(- \frac{1}{3})}^{2}}{4 (- \frac{1}{12})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{1}{3}$ .

$e = \frac{5}{3} - \frac{{(- 1)}^{2} {(\frac{1}{3})}^{2}}{4 (- \frac{1}{12})}$

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

$e = \frac{5}{3} - \frac{1 {(\frac{1}{3})}^{2}}{4 (- \frac{1}{12})}$

Apply the product rule to $\frac{1}{3}$ .

$e = \frac{5}{3} - \frac{1 \frac{1^{2}}{3^{2}}}{4 (- \frac{1}{12})}$

One to any power is one.

$e = \frac{5}{3} - \frac{1 \frac{1}{3^{2}}}{4 (- \frac{1}{12})}$

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

$e = \frac{5}{3} - \frac{1 (\frac{1}{9})}{4 (- \frac{1}{12})}$

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

$e = \frac{5}{3} - \frac{\frac{1}{9}}{4 (- \frac{1}{12})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = \frac{5}{3} - \frac{\frac{1}{9}}{- 4 (\frac{1}{12})}$

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

$e = \frac{5}{3} - \frac{\frac{1}{9}}{\frac{- 4}{12}}$

Reduce the expression by cancelling the common factors.

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

Factor $4$ out of $- 4$ .

$e = \frac{5}{3} - \frac{\frac{1}{9}}{\frac{4 (- 1)}{12}}$

Cancel the common factors.

Factor $4$ out of $12$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$e = \frac{5}{3} - (\frac{1}{3 (3)} (- 1 \cdot 3))$

Factor $3$ out of $- 1 \cdot 3$ .

$e = \frac{5}{3} - (\frac{1}{3 (3)} (3 \cdot - 1))$

Cancel the common factor.

$e = \frac{5}{3} - (\frac{1}{3 \cdot 3} (3 \cdot - 1))$

Rewrite the expression.

$e = \frac{5}{3} - (\frac{1}{3} \cdot - 1)$

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

$e = \frac{5}{3} - \frac{- 1}{3}$

Move the negative in front of the fraction.

$e = \frac{5}{3} - - \frac{1}{3}$

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

Multiply $- 1$ by $- 1$ .

$e = \frac{5}{3} + 1 (\frac{1}{3})$

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

$e = \frac{5}{3} + \frac{1}{3}$

Combine the numerators over the common denominator.

$e = \frac{5 + 1}{3}$

Add $5$ and $1$ .

$e = \frac{6}{3}$

Divide $6$ by $3$ .

$e = 2$

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

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

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = - 2$

$k = 2$

Step 3

Find the vertex $(h, k)$ .

$(- 2, 2)$

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.

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

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 focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

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

$(- 2, - 1)$

Step 6

Find the Focus (x+2)^2=-12(y-2)

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