Precalculus Examples

$x^{2} = - 12 y$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

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

$- 12 y = x^{2}$

Step 1.1.2

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

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Step 1.1.2.1

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

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

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.

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

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

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

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.

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

Step 1.2

Complete the square for $- \frac{x^{2}}{12}$ .

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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 $c$ .

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

$b = 0$

$c = 0$

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

$d = \frac{0}{2 (- \frac{1}{12})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $0$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{1}{12})}$

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.

$d = \frac{2 \cdot 0}{2 (- \frac{1}{12})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{- \frac{1}{12}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- 1 \cdot 12)$

Step 1.2.3.2.3

Multiply $0 (- 1 \cdot 12)$ .

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Step 1.2.3.2.3.1

Multiply $- 1$ by $12$ .

$d = 0 \cdot - 12$

Step 1.2.3.2.3.2

Multiply $0$ by $- 12$ .

$d = 0$

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

$e = 0 - \frac{0^{2}}{4 (- \frac{1}{12})}$

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

Raising $0$ to any positive power yields $0$ .

$e = 0 - \frac{0}{4 (- \frac{1}{12})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 0 - \frac{0}{- 4 (\frac{1}{12})}$

Step 1.2.4.2.1.2.2

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

$e = 0 - \frac{0}{\frac{- 4}{12}}$

Step 1.2.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 1.2.4.2.1.3.1

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

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Step 1.2.4.2.1.3.1.1

Factor $4$ out of $- 4$ .

$e = 0 - \frac{0}{\frac{4 (- 1)}{12}}$

Step 1.2.4.2.1.3.1.2

Cancel the common factors.

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Step 1.2.4.2.1.3.1.2.1

Factor $4$ out of $12$ .

$e = 0 - \frac{0}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.2

Cancel the common factor.

$e = 0 - \frac{0}{\frac{4 \cdot - 1}{4 \cdot 3}}$

Step 1.2.4.2.1.3.1.2.3

Rewrite the expression.

$e = 0 - \frac{0}{\frac{- 1}{3}}$

Step 1.2.4.2.1.3.2

Move the negative in front of the fraction.

$e = 0 - \frac{0}{- \frac{1}{3}}$

Step 1.2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (0 (- 1 \cdot 3))$

Step 1.2.4.2.1.5

Multiply $- (0 (- 1 \cdot 3))$ .

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Step 1.2.4.2.1.5.1

Multiply $- 1$ by $3$ .

$e = 0 - (0 \cdot - 3)$

Step 1.2.4.2.1.5.2

Multiply $0$ by $- 3$ .

$e = 0 - 0$

Step 1.2.4.2.1.5.3

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Step 1.2.4.2.2

Add $0$ and $0$ .

$e = 0$

Step 1.2.5

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

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

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = 0$

$k = 0$

Step 3

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 4

Find the vertex $(h, k)$ .

$(0, 0)$

Step 5

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

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Step 5.1

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

$\frac{1}{4 a}$

Step 5.2

Substitute the value of $a$ into the formula.

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

Step 5.3

Simplify.

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Step 5.3.1

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

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Step 5.3.1.1

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

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

Step 5.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.2

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

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

Step 5.3.3

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

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Step 5.3.3.1

Factor $4$ out of $4$ .

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

Step 5.3.3.2

Cancel the common factors.

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Step 5.3.3.2.1

Factor $4$ out of $12$ .

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

Step 5.3.3.2.2

Cancel the common factor.

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

Step 5.3.3.2.3

Rewrite the expression.

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

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 3)$

Step 5.3.5

Multiply $- (1 \cdot 3)$ .

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Step 5.3.5.1

Multiply $3$ by $1$ .

$- 1 \cdot 3$

Step 5.3.5.2

Multiply $- 1$ by $3$ .

$- 3$

Step 6

Find the focus.

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Step 6.1

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

$(h, k + p)$

Step 6.2

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

$(0, - 3)$

Step 7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$x = 0$

Step 8

Find the directrix.

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Step 8.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 8.2

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

$y = 3$

Step 9

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Down

Vertex: $(0, 0)$

Focus: $(0, - 3)$

Axis of Symmetry: $x = 0$

Directrix: $y = 3$

Step 10