Precalculus Examples

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

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

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.

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

Step 1.1.2.2.1.2

Divide $x + 1$ by $1$ .

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

Step 1.1.3

Subtract $1$ from both sides of the equation.

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

Step 1.1.4

Reorder terms.

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

Step 1.2

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

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

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

Step 1.2.1.1.2

Expand $(y - 2) (y - 2)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

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

Step 1.2.1.1.2.2

Apply the distributive property.

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

Step 1.2.1.1.2.3

Apply the distributive property.

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

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 $y$ .

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

Step 1.2.1.1.3.1.2

Move $- 2$ to the left of $y$ .

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

Step 1.2.1.1.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 1.2.1.1.3.2

Subtract $2 y$ from $- 2 y$ .

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

Step 1.2.1.1.4

Apply the distributive property.

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

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

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

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

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

Factor $4$ out of $12$ .

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

Step 1.2.1.1.5.2.2

Factor $4$ out of $- 4 y$ .

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

Step 1.2.1.1.5.2.3

Cancel the common factor.

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

Step 1.2.1.1.5.2.4

Rewrite the expression.

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

Step 1.2.1.1.5.3

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

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

Step 1.2.1.1.5.4

Cancel the common factor of $4$ .

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Step 1.2.1.1.5.4.1

Factor $4$ out of $12$ .

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

Step 1.2.1.1.5.4.2

Cancel the common factor.

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

Step 1.2.1.1.5.4.3

Rewrite the expression.

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

Step 1.2.1.2

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

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

Step 1.2.1.3

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

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

Step 1.2.1.4

Combine the numerators over the common denominator.

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

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Multiply $- 1$ by $3$ .

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

Step 1.2.1.5.2

Subtract $3$ from $1$ .

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

Step 1.2.1.6

Move the negative in front of the fraction.

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

Step 1.2.2

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

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

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

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

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

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.4.2.2

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

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

Step 1.2.4.2.3

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

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Step 1.2.4.2.3.1

Factor $2$ out of $2$ .

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

Step 1.2.4.2.3.2

Cancel the common factors.

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Step 1.2.4.2.3.2.1

Factor $2$ out of $12$ .

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

Step 1.2.4.2.3.2.2

Cancel the common factor.

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

Step 1.2.4.2.3.2.3

Rewrite the expression.

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

Step 1.2.4.2.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.4.2.5

Multiply $6$ by $1$ .

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

Step 1.2.4.2.6

Cancel the common factor of $3$ .

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Step 1.2.4.2.6.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Step 1.2.4.2.6.2

Factor $3$ out of $6$ .

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

Step 1.2.4.2.6.3

Cancel the common factor.

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

Step 1.2.4.2.6.4

Rewrite the expression.

$d = - 1 \cdot 2$

Step 1.2.4.2.7

Multiply $- 1$ by $2$ .

$d = - 2$

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

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

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

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

Step 1.2.5.2.1.1.2

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

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

Step 1.2.5.2.1.1.3

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

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

Step 1.2.5.2.1.1.4

One to any power is one.

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

Step 1.2.5.2.1.1.5

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

$e = - \frac{2}{3} - \frac{1 (\frac{1}{9})}{4 (\frac{1}{12})}$

Step 1.2.5.2.1.1.6

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

$e = - \frac{2}{3} - \frac{\frac{1}{9}}{4 (\frac{1}{12})}$

Step 1.2.5.2.1.2

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

$e = - \frac{2}{3} - \frac{\frac{1}{9}}{\frac{4}{12}}$

Step 1.2.5.2.1.3

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

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Step 1.2.5.2.1.3.1

Factor $4$ out of $4$ .

$e = - \frac{2}{3} - \frac{\frac{1}{9}}{\frac{4 (1)}{12}}$

Step 1.2.5.2.1.3.2

Cancel the common factors.

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Step 1.2.5.2.1.3.2.1

Factor $4$ out of $12$ .

$e = - \frac{2}{3} - \frac{\frac{1}{9}}{\frac{4 \cdot 1}{4 \cdot 3}}$

Step 1.2.5.2.1.3.2.2

Cancel the common factor.

$e = - \frac{2}{3} - \frac{\frac{1}{9}}{\frac{4 \cdot 1}{4 \cdot 3}}$

Step 1.2.5.2.1.3.2.3

Rewrite the expression.

$e = - \frac{2}{3} - \frac{\frac{1}{9}}{\frac{1}{3}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{2}{3} - (\frac{1}{9} \cdot 3)$

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 $9$ .

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

Step 1.2.5.2.1.5.2

Cancel the common factor.

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

Step 1.2.5.2.1.5.3

Rewrite the expression.

$e = - \frac{2}{3} - \frac{1}{3}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{- 2 - 1}{3}$

Step 1.2.5.2.3

Subtract $1$ from $- 2$ .

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

Step 1.2.5.2.4

Divide $- 3$ by $3$ .

$e = - 1$

Step 1.2.6

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

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

Step 1.3

Set $x$ equal to the new right side.

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

Step 2

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

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

$h = - 1$

$k = 2$

Step 3

Find the vertex $(h, k)$ .

$(- 1, 2)$

Step 4