Precalculus Examples

${(y + 4)}^{2} = - 12 (x + 2)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

Rewrite the equation as $- 12 (x + 2) = {(y + 4)}^{2}$ .

$- 12 (x + 2) = {(y + 4)}^{2}$

Step 1.1.2

Divide each term in $- 12 (x + 2) = {(y + 4)}^{2}$ by $- 12$ and simplify.

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Step 1.1.2.1

Divide each term in $- 12 (x + 2) = {(y + 4)}^{2}$ by $- 12$ .

$\frac{- 12 (x + 2)}{- 12} = \frac{{(y + 4)}^{2}}{- 12}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $- 12$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{- 12 (x + 2)}{- 12} = \frac{{(y + 4)}^{2}}{- 12}$

Step 1.1.2.2.1.2

Divide $x + 2$ by $1$ .

$x + 2 = \frac{{(y + 4)}^{2}}{- 12}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Move the negative in front of the fraction.

$x + 2 = - \frac{{(y + 4)}^{2}}{12}$

Step 1.1.3

Subtract $2$ from both sides of the equation.

$x = - \frac{{(y + 4)}^{2}}{12} - 2$

Step 1.1.4

Reorder terms.

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

Step 1.2

Complete the square for $- \frac{1}{12} \cdot {(y + 4)}^{2} - 2$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(y + 4)}^{2}$ as $(y + 4) (y + 4)$ .

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

Step 1.2.1.1.2

Expand $(y + 4) (y + 4)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

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

Step 1.2.1.1.2.2

Apply the distributive property.

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

Step 1.2.1.1.2.3

Apply the distributive property.

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

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $y$ by $y$ .

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

Step 1.2.1.1.3.1.2

Move $4$ to the left of $y$ .

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

Step 1.2.1.1.3.1.3

Multiply $4$ by $4$ .

$- \frac{1}{12} \cdot (y^{2} + 4 y + 4 y + 16) - 2$

Step 1.2.1.1.3.2

Add $4 y$ and $4 y$ .

$- \frac{1}{12} \cdot (y^{2} + 8 y + 16) - 2$

Step 1.2.1.1.4

Apply the distributive property.

$- \frac{1}{12} y^{2} - \frac{1}{12} (8 y) - \frac{1}{12} \cdot 16 - 2$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

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

$- \frac{y^{2}}{12} - \frac{1}{12} (8 y) - \frac{1}{12} \cdot 16 - 2$

Step 1.2.1.1.5.2

Cancel the common factor of $4$ .

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Step 1.2.1.1.5.2.1

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

$- \frac{y^{2}}{12} + \frac{- 1}{12} (8 y) - \frac{1}{12} \cdot 16 - 2$

Step 1.2.1.1.5.2.2

Factor $4$ out of $12$ .

$- \frac{y^{2}}{12} + \frac{- 1}{4 (3)} (8 y) - \frac{1}{12} \cdot 16 - 2$

Step 1.2.1.1.5.2.3

Factor $4$ out of $8 y$ .

$- \frac{y^{2}}{12} + \frac{- 1}{4 (3)} (4 (2 y)) - \frac{1}{12} \cdot 16 - 2$

Step 1.2.1.1.5.2.4

Cancel the common factor.

$- \frac{y^{2}}{12} + \frac{- 1}{4 \cdot 3} (4 (2 y)) - \frac{1}{12} \cdot 16 - 2$

Step 1.2.1.1.5.2.5

Rewrite the expression.

$- \frac{y^{2}}{12} + \frac{- 1}{3} (2 y) - \frac{1}{12} \cdot 16 - 2$

Step 1.2.1.1.5.3

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

$- \frac{y^{2}}{12} + \frac{2 \cdot - 1}{3} y - \frac{1}{12} \cdot 16 - 2$

Step 1.2.1.1.5.4

Multiply $2$ by $- 1$ .

$- \frac{y^{2}}{12} + \frac{- 2}{3} y - \frac{1}{12} \cdot 16 - 2$

Step 1.2.1.1.5.5

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

$- \frac{y^{2}}{12} + \frac{- 2 y}{3} - \frac{1}{12} \cdot 16 - 2$

Step 1.2.1.1.5.6

Cancel the common factor of $4$ .

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Step 1.2.1.1.5.6.1

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

$- \frac{y^{2}}{12} + \frac{- 2 y}{3} + \frac{- 1}{12} \cdot 16 - 2$

Step 1.2.1.1.5.6.2

Factor $4$ out of $12$ .

$- \frac{y^{2}}{12} + \frac{- 2 y}{3} + \frac{- 1}{4 (3)} \cdot 16 - 2$

Step 1.2.1.1.5.6.3

Factor $4$ out of $16$ .

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

Step 1.2.1.1.5.6.4

Cancel the common factor.

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

Step 1.2.1.1.5.6.5

Rewrite the expression.

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

Step 1.2.1.1.5.7

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

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

Step 1.2.1.1.5.8

Multiply $- 1$ by $4$ .

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

Step 1.2.1.1.6

Simplify each term.

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Step 1.2.1.1.6.1

Move the negative in front of the fraction.

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

Step 1.2.1.1.6.2

Move the negative in front of the fraction.

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

Step 1.2.1.2

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

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

Step 1.2.1.3

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

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

Step 1.2.1.4

Combine the numerators over the common denominator.

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

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Multiply $- 2$ by $3$ .

$- \frac{y^{2}}{12} - \frac{2 y}{3} + \frac{- 4 - 6}{3}$

Step 1.2.1.5.2

Subtract $6$ from $- 4$ .

$- \frac{y^{2}}{12} - \frac{2 y}{3} + \frac{- 10}{3}$

Step 1.2.1.6

Move the negative in front of the fraction.

$- \frac{y^{2}}{12} - \frac{2 y}{3} - \frac{10}{3}$

Step 1.2.2

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

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

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

$c = - \frac{10}{3}$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.4.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.4.2.3

Cancel the common factor of $2$ .

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Step 1.2.4.2.3.1

Cancel the common factor.

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

Step 1.2.4.2.3.2

Rewrite the expression.

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

Step 1.2.4.2.4

Multiply $\frac{1}{3}$ by $\frac{1}{\frac{1}{12}}$ .

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

Step 1.2.4.2.5

Combine $3$ and $\frac{1}{12}$ .

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

Step 1.2.4.2.6

Cancel the common factor of $3$ and $12$ .

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Step 1.2.4.2.6.1

Factor $3$ out of $3$ .

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

Step 1.2.4.2.6.2

Cancel the common factors.

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Step 1.2.4.2.6.2.1

Factor $3$ out of $12$ .

$d = \frac{1}{\frac{3 \cdot 1}{3 \cdot 4}}$

Step 1.2.4.2.6.2.2

Cancel the common factor.

$d = \frac{1}{\frac{3 \cdot 1}{3 \cdot 4}}$

Step 1.2.4.2.6.2.3

Rewrite the expression.

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

Step 1.2.4.2.7

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 4$

Step 1.2.4.2.8

Multiply $4$ by $1$ .

$d = 4$

Step 1.2.5

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

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Step 1.2.5.1

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

$e = - \frac{10}{3} - \frac{{(- \frac{2}{3})}^{2}}{4 (- \frac{1}{12})}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Simplify the numerator.

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Step 1.2.5.2.1.1.1

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

$e = - \frac{10}{3} - \frac{{(- 1)}^{2} {(\frac{2}{3})}^{2}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.1.2

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

$e = - \frac{10}{3} - \frac{1 {(\frac{2}{3})}^{2}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.1.3

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

$e = - \frac{10}{3} - \frac{1 \frac{2^{2}}{3^{2}}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.1.4

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

$e = - \frac{10}{3} - \frac{1 \frac{4}{3^{2}}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.1.5

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

$e = - \frac{10}{3} - \frac{1 (\frac{4}{9})}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.1.6

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

$e = - \frac{10}{3} - \frac{\frac{4}{9}}{4 (- \frac{1}{12})}$

Step 1.2.5.2.1.2

Simplify the denominator.

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Step 1.2.5.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{10}{3} - \frac{\frac{4}{9}}{- 4 (\frac{1}{12})}$

Step 1.2.5.2.1.2.2

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

$e = - \frac{10}{3} - \frac{\frac{4}{9}}{\frac{- 4}{12}}$

Step 1.2.5.2.1.3

Reduce the expression by cancelling the common factors.

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Step 1.2.5.2.1.3.1

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

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Step 1.2.5.2.1.3.1.1

Factor $4$ out of $- 4$ .

$e = - \frac{10}{3} - \frac{\frac{4}{9}}{\frac{4 (- 1)}{12}}$

Step 1.2.5.2.1.3.1.2

Cancel the common factors.

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Step 1.2.5.2.1.3.1.2.1

Factor $4$ out of $12$ .

$e = - \frac{10}{3} - \frac{\frac{4}{9}}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 1.2.5.2.1.3.1.2.2

Cancel the common factor.

$e = - \frac{10}{3} - \frac{\frac{4}{9}}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 1.2.5.2.1.3.1.2.3

Rewrite the expression.

$e = - \frac{10}{3} - \frac{\frac{4}{9}}{\frac{- 1}{3}}$

Step 1.2.5.2.1.3.2

Move the negative in front of the fraction.

$e = - \frac{10}{3} - \frac{\frac{4}{9}}{- \frac{1}{3}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{10}{3} - (\frac{4}{9} (- 1 \cdot 3))$

Step 1.2.5.2.1.5

Cancel the common factor of $3$ .

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Step 1.2.5.2.1.5.1

Factor $3$ out of $9$ .

$e = - \frac{10}{3} - (\frac{4}{3 (3)} (- 1 \cdot 3))$

Step 1.2.5.2.1.5.2

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

$e = - \frac{10}{3} - (\frac{4}{3 (3)} (3 \cdot - 1))$

Step 1.2.5.2.1.5.3

Cancel the common factor.

$e = - \frac{10}{3} - (\frac{4}{3 \cdot 3} (3 \cdot - 1))$

Step 1.2.5.2.1.5.4

Rewrite the expression.

$e = - \frac{10}{3} - (\frac{4}{3} \cdot - 1)$

Step 1.2.5.2.1.6

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

$e = - \frac{10}{3} - \frac{4 \cdot - 1}{3}$

Step 1.2.5.2.1.7

Multiply $4$ by $- 1$ .

$e = - \frac{10}{3} - \frac{- 4}{3}$

Step 1.2.5.2.1.8

Move the negative in front of the fraction.

$e = - \frac{10}{3} - - \frac{4}{3}$

Step 1.2.5.2.1.9

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

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Step 1.2.5.2.1.9.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{10}{3} + 1 (\frac{4}{3})$

Step 1.2.5.2.1.9.2

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

$e = - \frac{10}{3} + \frac{4}{3}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{- 10 + 4}{3}$

Step 1.2.5.2.3

Add $- 10$ and $4$ .

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

Step 1.2.5.2.4

Divide $- 6$ by $3$ .

$e = - 2$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{12} {(y + 4)}^{2} - 2$ .

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

Step 1.3

Set $x$ equal to the new right side.

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

Step 2

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

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

$h = - 2$

$k = - 4$

Step 3

Find the vertex $(h, k)$ .

$(- 2, - 4)$

Step 4

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

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Step 4.1

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

$\frac{1}{4 a}$

Step 4.2

Substitute the value of $a$ into the formula.

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

Step 4.3

Simplify.

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Step 4.3.1

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

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Step 4.3.1.1

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

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

Step 4.3.1.2

Move the negative in front of the fraction.

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

Step 4.3.2

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

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

Step 4.3.3

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

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Step 4.3.3.1

Factor $4$ out of $4$ .

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

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

Factor $4$ out of $12$ .

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

Step 4.3.3.2.2

Cancel the common factor.

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

Step 4.3.3.2.3

Rewrite the expression.

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

Step 4.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 3)$

Step 4.3.5

Multiply $- (1 \cdot 3)$ .

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Step 4.3.5.1

Multiply $3$ by $1$ .

$- 1 \cdot 3$

Step 4.3.5.2

Multiply $- 1$ by $3$ .

$- 3$

Step 5

Find the directrix.

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Step 5.1

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.

$x = h - p$

Step 5.2

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

$x = 1$

Step 6