Precalculus Examples

$x^{2} + 6 x + 12 y + 9 = 0$

Step 1

Solve for $y$ .

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Step 1.1

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

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Step 1.1.1

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

$6 x + 12 y + 9 = - x^{2}$

Step 1.1.2

Subtract $6 x$ from both sides of the equation.

$12 y + 9 = - x^{2} - 6 x$

Step 1.1.3

Subtract $9$ from both sides of the equation.

$12 y = - x^{2} - 6 x - 9$

Step 1.2

Divide each term in $12 y = - x^{2} - 6 x - 9$ by $12$ and simplify.

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Step 1.2.1

Divide each term in $12 y = - x^{2} - 6 x - 9$ by $12$ .

$\frac{12 y}{12} = \frac{- x^{2}}{12} + \frac{- 6 x}{12} + \frac{- 9}{12}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $12$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{12 y}{12} = \frac{- x^{2}}{12} + \frac{- 6 x}{12} + \frac{- 9}{12}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{12} + \frac{- 6 x}{12} + \frac{- 9}{12}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{12} + \frac{- 6 x}{12} + \frac{- 9}{12}$

Step 1.2.3.1.2

Cancel the common factor of $- 6$ and $12$ .

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Step 1.2.3.1.2.1

Factor $6$ out of $- 6 x$ .

$y = - \frac{x^{2}}{12} + \frac{6 (- x)}{12} + \frac{- 9}{12}$

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $6$ out of $12$ .

$y = - \frac{x^{2}}{12} + \frac{6 (- x)}{6 (2)} + \frac{- 9}{12}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = - \frac{x^{2}}{12} + \frac{6 (- x)}{6 \cdot 2} + \frac{- 9}{12}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = - \frac{x^{2}}{12} + \frac{- x}{2} + \frac{- 9}{12}$

Step 1.2.3.1.3

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{12} - \frac{x}{2} + \frac{- 9}{12}$

Step 1.2.3.1.4

Cancel the common factor of $- 9$ and $12$ .

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Step 1.2.3.1.4.1

Factor $3$ out of $- 9$ .

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

Step 1.2.3.1.4.2

Cancel the common factors.

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Step 1.2.3.1.4.2.1

Factor $3$ out of $12$ .

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

Step 1.2.3.1.4.2.2

Cancel the common factor.

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

Step 1.2.3.1.4.2.3

Rewrite the expression.

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

Step 1.2.3.1.5

Move the negative in front of the fraction.

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

Step 2

Find the properties of the given parabola.

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Step 2.1

Rewrite the equation in vertex form.

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Step 2.1.1

Complete the square for $- \frac{x^{2}}{12} - \frac{x}{2} - \frac{3}{4}$ .

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Step 2.1.1.1

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

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

$b = - \frac{1}{2}$

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

Step 2.1.1.2

Consider the vertex form of a parabola.

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

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

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Dividing two negative values results in a positive value.

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

Step 2.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.1.3.2.3

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

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

Step 2.1.1.3.2.4

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

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Step 2.1.1.3.2.4.1

Factor $2$ out of $2$ .

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

Step 2.1.1.3.2.4.2

Cancel the common factors.

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Step 2.1.1.3.2.4.2.1

Factor $2$ out of $12$ .

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

Step 2.1.1.3.2.4.2.2

Cancel the common factor.

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

Step 2.1.1.3.2.4.2.3

Rewrite the expression.

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

Step 2.1.1.3.2.5

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{2} (1 \cdot 6)$

Step 2.1.1.3.2.6

Multiply $6$ by $1$ .

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

Step 2.1.1.3.2.7

Cancel the common factor of $2$ .

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Step 2.1.1.3.2.7.1

Factor $2$ out of $6$ .

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

Step 2.1.1.3.2.7.2

Cancel the common factor.

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

Step 2.1.1.3.2.7.3

Rewrite the expression.

$d = 3$

Step 2.1.1.4

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

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Step 2.1.1.4.1

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

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

Step 2.1.1.4.2

Simplify the right side.

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Step 2.1.1.4.2.1

Simplify each term.

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Step 2.1.1.4.2.1.1

Simplify the numerator.

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Step 2.1.1.4.2.1.1.1

Apply the product rule to $- \frac{1}{2}$ .

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

Step 2.1.1.4.2.1.1.2

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

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

Step 2.1.1.4.2.1.1.3

Apply the product rule to $\frac{1}{2}$ .

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

Step 2.1.1.4.2.1.1.4

One to any power is one.

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

Step 2.1.1.4.2.1.1.5

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

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

Step 2.1.1.4.2.1.1.6

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

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

Step 2.1.1.4.2.1.2

Simplify the denominator.

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Step 2.1.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

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

Step 2.1.1.4.2.1.2.2

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

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

Step 2.1.1.4.2.1.3

Reduce the expression by cancelling the common factors.

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Step 2.1.1.4.2.1.3.1

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

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Step 2.1.1.4.2.1.3.1.1

Factor $4$ out of $- 4$ .

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

Step 2.1.1.4.2.1.3.1.2

Cancel the common factors.

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Step 2.1.1.4.2.1.3.1.2.1

Factor $4$ out of $12$ .

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

Step 2.1.1.4.2.1.3.1.2.2

Cancel the common factor.

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

Step 2.1.1.4.2.1.3.1.2.3

Rewrite the expression.

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

Step 2.1.1.4.2.1.3.2

Move the negative in front of the fraction.

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

Step 2.1.1.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.1.4.2.1.5

Multiply $- 1$ by $3$ .

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

Step 2.1.1.4.2.1.6

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

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

Step 2.1.1.4.2.1.7

Move the negative in front of the fraction.

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

Step 2.1.1.4.2.1.8

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

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Step 2.1.1.4.2.1.8.1

Multiply $- 1$ by $- 1$ .

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

Step 2.1.1.4.2.1.8.2

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

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

Step 2.1.1.4.2.2

Combine the numerators over the common denominator.

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

Step 2.1.1.4.2.3

Add $- 3$ and $3$ .

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

Step 2.1.1.4.2.4

Divide $0$ by $4$ .

$e = 0$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{12} {(x + 3)}^{2} + 0$ .

$- \frac{1}{12} {(x + 3)}^{2} + 0$

Step 2.1.2

Set $y$ equal to the new right side.

$y = - \frac{1}{12} \cdot {(x + 3)}^{2} + 0$

Step 2.2

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

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

$h = - 3$

$k = 0$

Step 2.3

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

Opens Down

Step 2.4

Find the vertex $(h, k)$ .

$(- 3, 0)$

Step 2.5

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

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Step 2.5.1

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

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of $a$ into the formula.

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

Step 2.5.3

Simplify.

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Step 2.5.3.1

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

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Step 2.5.3.1.1

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

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

Step 2.5.3.1.2

Move the negative in front of the fraction.

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

Step 2.5.3.2

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

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

Step 2.5.3.3

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

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Step 2.5.3.3.1

Factor $4$ out of $4$ .

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

Step 2.5.3.3.2

Cancel the common factors.

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Step 2.5.3.3.2.1

Factor $4$ out of $12$ .

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

Step 2.5.3.3.2.2

Cancel the common factor.

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

Step 2.5.3.3.2.3

Rewrite the expression.

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

Step 2.5.3.4

Multiply the numerator by the reciprocal of the denominator.

$- (1 \cdot 3)$

Step 2.5.3.5

Multiply $- (1 \cdot 3)$ .

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Step 2.5.3.5.1

Multiply $3$ by $1$ .

$- 1 \cdot 3$

Step 2.5.3.5.2

Multiply $- 1$ by $3$ .

$- 3$

Step 2.6

Find the focus.

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Step 2.6.1

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 2.6.2

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

$(- 3, - 3)$

Step 2.7

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

$x = - 3$

Step 2.8

Find the directrix.

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Step 2.8.1

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 2.8.2

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

$y = 3$

Step 2.9

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

Direction: Opens Down

Vertex: $(- 3, 0)$

Focus: $(- 3, - 3)$

Axis of Symmetry: $x = - 3$

Directrix: $y = 3$

Direction: Opens Down

Vertex: $(- 3, 0)$

Focus: $(- 3, - 3)$

Axis of Symmetry: $x = - 3$

Directrix: $y = 3$

Step 3

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Step 3.1

Replace the variable $x$ with $- 4$ in the expression.

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

Step 3.2

Simplify the result.

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Step 3.2.1

Find the common denominator.

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Step 3.2.1.1

Multiply $\frac{- 4}{2}$ by $\frac{6}{6}$ .

$f (- 4) = - \frac{{(- 4)}^{2}}{12} - (\frac{- 4}{2} \cdot \frac{6}{6}) - \frac{3}{4}$

Step 3.2.1.2

Multiply $\frac{- 4}{2}$ by $\frac{6}{6}$ .

$f (- 4) = - \frac{{(- 4)}^{2}}{12} - \frac{- 4 \cdot 6}{2 \cdot 6} - \frac{3}{4}$

Step 3.2.1.3

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

$f (- 4) = - \frac{{(- 4)}^{2}}{12} - \frac{- 4 \cdot 6}{2 \cdot 6} - (\frac{3}{4} \cdot \frac{3}{3})$

Step 3.2.1.4

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

$f (- 4) = - \frac{{(- 4)}^{2}}{12} - \frac{- 4 \cdot 6}{2 \cdot 6} - \frac{3 \cdot 3}{4 \cdot 3}$

Step 3.2.1.5

Multiply $2$ by $6$ .

$f (- 4) = - \frac{{(- 4)}^{2}}{12} - \frac{- 4 \cdot 6}{12} - \frac{3 \cdot 3}{4 \cdot 3}$

Step 3.2.1.6

Reorder the factors of $4 \cdot 3$ .

$f (- 4) = - \frac{{(- 4)}^{2}}{12} - \frac{- 4 \cdot 6}{12} - \frac{3 \cdot 3}{3 \cdot 4}$

Step 3.2.1.7

Multiply $3$ by $4$ .

$f (- 4) = - \frac{{(- 4)}^{2}}{12} - \frac{- 4 \cdot 6}{12} - \frac{3 \cdot 3}{12}$

Step 3.2.2

Combine the numerators over the common denominator.

$f (- 4) = \frac{- {(- 4)}^{2} + 4 \cdot 6 - 3 \cdot 3}{12}$

Step 3.2.3

Simplify each term.

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Step 3.2.3.1

Raise $- 4$ to the power of $2$ .

$f (- 4) = \frac{- 1 \cdot 16 + 4 \cdot 6 - 3 \cdot 3}{12}$

Step 3.2.3.2

Multiply $- 1$ by $16$ .

$f (- 4) = \frac{- 16 + 4 \cdot 6 - 3 \cdot 3}{12}$

Step 3.2.3.3

Multiply $4$ by $6$ .

$f (- 4) = \frac{- 16 + 24 - 3 \cdot 3}{12}$

Step 3.2.3.4

Multiply $- 3$ by $3$ .

$f (- 4) = \frac{- 16 + 24 - 9}{12}$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $- 16$ and $24$ .

$f (- 4) = \frac{8 - 9}{12}$

Step 3.2.4.2

Subtract $9$ from $8$ .

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

Step 3.2.4.3

Move the negative in front of the fraction.

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

Step 3.2.5

The final answer is $- \frac{1}{12}$ .

$- \frac{1}{12}$

Step 3.3

The $y$ value at $x = - 4$ is $- \frac{1}{12}$ .

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

Step 3.4

Replace the variable $x$ with $- 5$ in the expression.

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

Step 3.5

Simplify the result.

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Step 3.5.1

Find the common denominator.

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Step 3.5.1.1

Multiply $\frac{- 5}{2}$ by $\frac{6}{6}$ .

$f (- 5) = - \frac{{(- 5)}^{2}}{12} - (\frac{- 5}{2} \cdot \frac{6}{6}) - \frac{3}{4}$

Step 3.5.1.2

Multiply $\frac{- 5}{2}$ by $\frac{6}{6}$ .

$f (- 5) = - \frac{{(- 5)}^{2}}{12} - \frac{- 5 \cdot 6}{2 \cdot 6} - \frac{3}{4}$

Step 3.5.1.3

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

$f (- 5) = - \frac{{(- 5)}^{2}}{12} - \frac{- 5 \cdot 6}{2 \cdot 6} - (\frac{3}{4} \cdot \frac{3}{3})$

Step 3.5.1.4

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

$f (- 5) = - \frac{{(- 5)}^{2}}{12} - \frac{- 5 \cdot 6}{2 \cdot 6} - \frac{3 \cdot 3}{4 \cdot 3}$

Step 3.5.1.5

Multiply $2$ by $6$ .

$f (- 5) = - \frac{{(- 5)}^{2}}{12} - \frac{- 5 \cdot 6}{12} - \frac{3 \cdot 3}{4 \cdot 3}$

Step 3.5.1.6

Reorder the factors of $4 \cdot 3$ .

$f (- 5) = - \frac{{(- 5)}^{2}}{12} - \frac{- 5 \cdot 6}{12} - \frac{3 \cdot 3}{3 \cdot 4}$

Step 3.5.1.7

Multiply $3$ by $4$ .

$f (- 5) = - \frac{{(- 5)}^{2}}{12} - \frac{- 5 \cdot 6}{12} - \frac{3 \cdot 3}{12}$

Step 3.5.2

Combine the numerators over the common denominator.

$f (- 5) = \frac{- {(- 5)}^{2} + 5 \cdot 6 - 3 \cdot 3}{12}$

Step 3.5.3

Simplify each term.

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Step 3.5.3.1

Raise $- 5$ to the power of $2$ .

$f (- 5) = \frac{- 1 \cdot 25 + 5 \cdot 6 - 3 \cdot 3}{12}$

Step 3.5.3.2

Multiply $- 1$ by $25$ .

$f (- 5) = \frac{- 25 + 5 \cdot 6 - 3 \cdot 3}{12}$

Step 3.5.3.3

Multiply $5$ by $6$ .

$f (- 5) = \frac{- 25 + 30 - 3 \cdot 3}{12}$

Step 3.5.3.4

Multiply $- 3$ by $3$ .

$f (- 5) = \frac{- 25 + 30 - 9}{12}$

Step 3.5.4

Reduce the expression by cancelling the common factors.

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Step 3.5.4.1

Add $- 25$ and $30$ .

$f (- 5) = \frac{5 - 9}{12}$

Step 3.5.4.2

Subtract $9$ from $5$ .

$f (- 5) = \frac{- 4}{12}$

Step 3.5.4.3

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

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Step 3.5.4.3.1

Factor $4$ out of $- 4$ .

$f (- 5) = \frac{4 (- 1)}{12}$

Step 3.5.4.3.2

Cancel the common factors.

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Step 3.5.4.3.2.1

Factor $4$ out of $12$ .

$f (- 5) = \frac{4 \cdot - 1}{4 \cdot 3}$

Step 3.5.4.3.2.2

Cancel the common factor.

$f (- 5) = \frac{4 \cdot - 1}{4 \cdot 3}$

Step 3.5.4.3.2.3

Rewrite the expression.

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

Step 3.5.4.4

Move the negative in front of the fraction.

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

Step 3.5.5

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

Step 3.6

The $y$ value at $x = - 5$ is $- \frac{1}{3}$ .

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

Step 3.7

Replace the variable $x$ with $- 2$ in the expression.

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

Step 3.8

Simplify the result.

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Step 3.8.1

Find the common denominator.

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Step 3.8.1.1

Multiply $\frac{- 2}{2}$ by $\frac{6}{6}$ .

$f (- 2) = - \frac{{(- 2)}^{2}}{12} - (\frac{- 2}{2} \cdot \frac{6}{6}) - \frac{3}{4}$

Step 3.8.1.2

Multiply $\frac{- 2}{2}$ by $\frac{6}{6}$ .

$f (- 2) = - \frac{{(- 2)}^{2}}{12} - \frac{- 2 \cdot 6}{2 \cdot 6} - \frac{3}{4}$

Step 3.8.1.3

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

$f (- 2) = - \frac{{(- 2)}^{2}}{12} - \frac{- 2 \cdot 6}{2 \cdot 6} - (\frac{3}{4} \cdot \frac{3}{3})$

Step 3.8.1.4

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

$f (- 2) = - \frac{{(- 2)}^{2}}{12} - \frac{- 2 \cdot 6}{2 \cdot 6} - \frac{3 \cdot 3}{4 \cdot 3}$

Step 3.8.1.5

Multiply $2$ by $6$ .

$f (- 2) = - \frac{{(- 2)}^{2}}{12} - \frac{- 2 \cdot 6}{12} - \frac{3 \cdot 3}{4 \cdot 3}$

Step 3.8.1.6

Reorder the factors of $4 \cdot 3$ .

$f (- 2) = - \frac{{(- 2)}^{2}}{12} - \frac{- 2 \cdot 6}{12} - \frac{3 \cdot 3}{3 \cdot 4}$

Step 3.8.1.7

Multiply $3$ by $4$ .

$f (- 2) = - \frac{{(- 2)}^{2}}{12} - \frac{- 2 \cdot 6}{12} - \frac{3 \cdot 3}{12}$

Step 3.8.2

Combine the numerators over the common denominator.

$f (- 2) = \frac{- {(- 2)}^{2} + 2 \cdot 6 - 3 \cdot 3}{12}$

Step 3.8.3

Simplify each term.

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Step 3.8.3.1

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

$f (- 2) = \frac{- 1 \cdot 4 + 2 \cdot 6 - 3 \cdot 3}{12}$

Step 3.8.3.2

Multiply $- 1$ by $4$ .

$f (- 2) = \frac{- 4 + 2 \cdot 6 - 3 \cdot 3}{12}$

Step 3.8.3.3

Multiply $2$ by $6$ .

$f (- 2) = \frac{- 4 + 12 - 3 \cdot 3}{12}$

Step 3.8.3.4

Multiply $- 3$ by $3$ .

$f (- 2) = \frac{- 4 + 12 - 9}{12}$

Step 3.8.4

Simplify the expression.

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Step 3.8.4.1

Add $- 4$ and $12$ .

$f (- 2) = \frac{8 - 9}{12}$

Step 3.8.4.2

Subtract $9$ from $8$ .

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

Step 3.8.4.3

Move the negative in front of the fraction.

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

Step 3.8.5

The final answer is $- \frac{1}{12}$ .

$- \frac{1}{12}$

Step 3.9

The $y$ value at $x = - 2$ is $- \frac{1}{12}$ .

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

Step 3.10

Replace the variable $x$ with $- 1$ in the expression.

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

Step 3.11

Simplify the result.

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Step 3.11.1

Find the common denominator.

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Step 3.11.1.1

Multiply $\frac{- 1}{2}$ by $\frac{6}{6}$ .

$f (- 1) = - \frac{{(- 1)}^{2}}{12} - (\frac{- 1}{2} \cdot \frac{6}{6}) - \frac{3}{4}$

Step 3.11.1.2

Multiply $\frac{- 1}{2}$ by $\frac{6}{6}$ .

$f (- 1) = - \frac{{(- 1)}^{2}}{12} - \frac{- 1 \cdot 6}{2 \cdot 6} - \frac{3}{4}$

Step 3.11.1.3

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

$f (- 1) = - \frac{{(- 1)}^{2}}{12} - \frac{- 1 \cdot 6}{2 \cdot 6} - (\frac{3}{4} \cdot \frac{3}{3})$

Step 3.11.1.4

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

$f (- 1) = - \frac{{(- 1)}^{2}}{12} - \frac{- 1 \cdot 6}{2 \cdot 6} - \frac{3 \cdot 3}{4 \cdot 3}$

Step 3.11.1.5

Multiply $2$ by $6$ .

$f (- 1) = - \frac{{(- 1)}^{2}}{12} - \frac{- 1 \cdot 6}{12} - \frac{3 \cdot 3}{4 \cdot 3}$

Step 3.11.1.6

Reorder the factors of $4 \cdot 3$ .

$f (- 1) = - \frac{{(- 1)}^{2}}{12} - \frac{- 1 \cdot 6}{12} - \frac{3 \cdot 3}{3 \cdot 4}$

Step 3.11.1.7

Multiply $3$ by $4$ .

$f (- 1) = - \frac{{(- 1)}^{2}}{12} - \frac{- 1 \cdot 6}{12} - \frac{3 \cdot 3}{12}$

Step 3.11.2

Combine the numerators over the common denominator.

$f (- 1) = \frac{- {(- 1)}^{2} + 1 \cdot 6 - 3 \cdot 3}{12}$

Step 3.11.3

Simplify each term.

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Step 3.11.3.1

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.11.3.1.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 3.11.3.1.1.1

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

$f (- 1) = \frac{(- 1) {(- 1)}^{2} + 1 \cdot 6 - 3 \cdot 3}{12}$

Step 3.11.3.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- 1) = \frac{{(- 1)}^{1 + 2} + 1 \cdot 6 - 3 \cdot 3}{12}$

Step 3.11.3.1.2

Add $1$ and $2$ .

$f (- 1) = \frac{{(- 1)}^{3} + 1 \cdot 6 - 3 \cdot 3}{12}$

Step 3.11.3.2

Raise $- 1$ to the power of $3$ .

$f (- 1) = \frac{- 1 + 1 \cdot 6 - 3 \cdot 3}{12}$

Step 3.11.3.3

Multiply $6$ by $1$ .

$f (- 1) = \frac{- 1 + 6 - 3 \cdot 3}{12}$

Step 3.11.3.4

Multiply $- 3$ by $3$ .

$f (- 1) = \frac{- 1 + 6 - 9}{12}$

Step 3.11.4

Reduce the expression by cancelling the common factors.

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Step 3.11.4.1

Add $- 1$ and $6$ .

$f (- 1) = \frac{5 - 9}{12}$

Step 3.11.4.2

Subtract $9$ from $5$ .

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

Step 3.11.4.3

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

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Step 3.11.4.3.1

Factor $4$ out of $- 4$ .

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

Step 3.11.4.3.2

Cancel the common factors.

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Step 3.11.4.3.2.1

Factor $4$ out of $12$ .

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

Step 3.11.4.3.2.2

Cancel the common factor.

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

Step 3.11.4.3.2.3

Rewrite the expression.

$f (- 1) = \frac{- 1}{3}$

Step 3.11.4.4

Move the negative in front of the fraction.

$f (- 1) = - \frac{1}{3}$

Step 3.11.5

The final answer is $- \frac{1}{3}$ .

$- \frac{1}{3}$

Step 3.12

The $y$ value at $x = - 1$ is $- \frac{1}{3}$ .

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

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 5 & - \frac{1}{3} \\ - 4 & - \frac{1}{12} \\ - 3 & 0 \\ - 2 & - \frac{1}{12} \\ - 1 & - \frac{1}{3} \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(- 3, 0)$

Focus: $(- 3, - 3)$

Axis of Symmetry: $x = - 3$

Directrix: $y = 3$

$\begin{array}{c} x & y \\ - 5 & - \frac{1}{3} \\ - 4 & - \frac{1}{12} \\ - 3 & 0 \\ - 2 & - \frac{1}{12} \\ - 1 & - \frac{1}{3} \end{array}$

Step 5