Precalculus Examples

$y^{2} - 4 x - 8 y - 12 = 0$

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

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1.1.1

Subtract $y^{2}$ from both sides of the equation.

$- 4 x - 8 y - 12 = - y^{2}$

Step 1.1.1.2

Add $8 y$ to both sides of the equation.

$- 4 x - 12 = - y^{2} + 8 y$

Step 1.1.1.3

Add $12$ to both sides of the equation.

$- 4 x = - y^{2} + 8 y + 12$

Step 1.1.2

Divide each term in $- 4 x = - y^{2} + 8 y + 12$ by $- 4$ and simplify.

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Step 1.1.2.1

Divide each term in $- 4 x = - y^{2} + 8 y + 12$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{- y^{2}}{- 4} + \frac{8 y}{- 4} + \frac{12}{- 4}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{- y^{2}}{- 4} + \frac{8 y}{- 4} + \frac{12}{- 4}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- y^{2}}{- 4} + \frac{8 y}{- 4} + \frac{12}{- 4}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Dividing two negative values results in a positive value.

$x = \frac{y^{2}}{4} + \frac{8 y}{- 4} + \frac{12}{- 4}$

Step 1.1.2.3.1.2

Cancel the common factor of $8$ and $- 4$ .

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Step 1.1.2.3.1.2.1

Factor $4$ out of $8 y$ .

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

Step 1.1.2.3.1.2.2

Move the negative one from the denominator of $\frac{2 y}{- 1}$ .

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

Step 1.1.2.3.1.3

Rewrite $- 1 \cdot (2 y)$ as $- (2 y)$ .

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

Step 1.1.2.3.1.4

Multiply $2$ by $- 1$ .

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

Step 1.1.2.3.1.5

Divide $12$ by $- 4$ .

$x = \frac{y^{2}}{4} - 2 y - 3$

Step 1.2

Complete the square for $\frac{y^{2}}{4} - 2 y - 3$ .

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Step 1.2.1

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

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

$b = - 2$

$c = - 3$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

$d = \frac{- 2}{2 (\frac{1}{4})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

Factor $2$ out of $- 2$ .

$d = \frac{2 \cdot - 1}{2 (\frac{1}{4})}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot - 1}{2 (\frac{1}{4})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

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

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 1 \cdot 4$

Step 1.2.3.2.3

Multiply $- 1$ by $4$ .

$d = - 4$

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

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

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

Step 1.2.4.2.1.2

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

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

Step 1.2.4.2.1.3

Divide $4$ by $4$ .

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

Step 1.2.4.2.1.4

Divide $4$ by $1$ .

$e = - 3 - 1 \cdot 4$

Step 1.2.4.2.1.5

Multiply $- 1$ by $4$ .

$e = - 3 - 4$

Step 1.2.4.2.2

Subtract $4$ from $- 3$ .

$e = - 7$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(y - 4)}^{2} - 7$ .

$\frac{1}{4} {(y - 4)}^{2} - 7$

Step 1.3

Set $x$ equal to the new right side.

$x = \frac{1}{4} \cdot {(y - 4)}^{2} - 7$

Step 2

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

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

$h = - 7$

$k = 4$

Step 3

Find the vertex $(h, k)$ .

$(- 7, 4)$

Step 4