Pre-Algebra Examples

$x = - 2 y^{2} - 8 y - 14$

Step 1

Find the properties of the given parabola.

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Step 1.1

Rewrite the equation in vertex form.

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Step 1.1.1

Complete the square for $- 2 y^{2} - 8 y - 14$ .

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Step 1.1.1.1

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

$a = - 2$

$b = - 8$

$c = - 14$

Step 1.1.1.2

Consider the vertex form of a parabola.

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$d = \frac{- 8}{2 \cdot - 2}$

Step 1.1.1.3.2

Simplify the right side.

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Step 1.1.1.3.2.1

Cancel the common factor of $- 8$ and $2$ .

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Step 1.1.1.3.2.1.1

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2 \cdot - 2}$

Step 1.1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.1.3.2.1.2.1

Factor $2$ out of $2 \cdot - 2$ .

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

Step 1.1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 \cdot - 2}$

Step 1.1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 4}{- 2}$

Step 1.1.1.3.2.2

Cancel the common factor of $- 4$ and $- 2$ .

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Step 1.1.1.3.2.2.1

Factor $- 2$ out of $- 4$ .

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

Step 1.1.1.3.2.2.2

Cancel the common factors.

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Step 1.1.1.3.2.2.2.1

Factor $- 2$ out of $- 2$ .

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

Step 1.1.1.3.2.2.2.2

Cancel the common factor.

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

Step 1.1.1.3.2.2.2.3

Rewrite the expression.

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

Step 1.1.1.3.2.2.2.4

Divide $2$ by $1$ .

$d = 2$

Step 1.1.1.4

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

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Step 1.1.1.4.1

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

$e = - 14 - \frac{{(- 8)}^{2}}{4 \cdot - 2}$

Step 1.1.1.4.2

Simplify the right side.

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Step 1.1.1.4.2.1

Simplify each term.

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Step 1.1.1.4.2.1.1

Raise $- 8$ to the power of $2$ .

$e = - 14 - \frac{64}{4 \cdot - 2}$

Step 1.1.1.4.2.1.2

Multiply $4$ by $- 2$ .

$e = - 14 - \frac{64}{- 8}$

Step 1.1.1.4.2.1.3

Divide $64$ by $- 8$ .

$e = - 14 - - 8$

Step 1.1.1.4.2.1.4

Multiply $- 1$ by $- 8$ .

$e = - 14 + 8$

Step 1.1.1.4.2.2

Add $- 14$ and $8$ .

$e = - 6$

Step 1.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(y + 2)}^{2} - 6$ .

$- 2 {(y + 2)}^{2} - 6$

Step 1.1.2

Set $x$ equal to the new right side.

$x = - 2 {(y + 2)}^{2} - 6$

Step 1.2

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

$a = - 2$

$h = - 6$

$k = - 2$

Step 1.3

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

Opens Left

Step 1.4

Find the vertex $(h, k)$ .

$(- 6, - 2)$

Step 1.5

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

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Step 1.5.1

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

$\frac{1}{4 a}$

Step 1.5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot - 2}$

Step 1.5.3

Simplify.

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Step 1.5.3.1

Multiply $4$ by $- 2$ .

$\frac{1}{- 8}$

Step 1.5.3.2

Move the negative in front of the fraction.

$- \frac{1}{8}$

Step 1.6

Find the focus.

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Step 1.6.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 1.6.2

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

$(- \frac{49}{8}, - 2)$

Step 1.7

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

$y = - 2$

Step 1.8

Find the directrix.

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Step 1.8.1

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 1.8.2

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

$x = - \frac{47}{8}$

Step 1.9

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

Direction: Opens Left

Vertex: $(- 6, - 2)$

Focus: $(- \frac{49}{8}, - 2)$

Axis of Symmetry: $y = - 2$

Directrix: $x = - \frac{47}{8}$

Direction: Opens Left

Vertex: $(- 6, - 2)$

Focus: $(- \frac{49}{8}, - 2)$

Axis of Symmetry: $y = - 2$

Directrix: $x = - \frac{47}{8}$

Step 2

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 2.1

Substitute the $x$ value $- 8$ into $f (x) = - \frac{4 + \sqrt{2 (- 6 - x)}}{2}$ . In this case, the point is $(- 8, - 3)$ .

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Step 2.1.1

Replace the variable $x$ with $- 8$ in the expression.

$f (- 8) = - \frac{4 + \sqrt{2 (- 6 - (- 8))}}{2}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the numerator.

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Step 2.1.2.1.1

Multiply $- 1$ by $- 8$ .

$f (- 8) = - \frac{4 + \sqrt{2 (- 6 + 8)}}{2}$

Step 2.1.2.1.2

Add $- 6$ and $8$ .

$f (- 8) = - \frac{4 + \sqrt{2 \cdot 2}}{2}$

Step 2.1.2.1.3

Multiply $2$ by $2$ .

$f (- 8) = - \frac{4 + \sqrt{4}}{2}$

Step 2.1.2.1.4

Rewrite $4$ as $2^{2}$ .

$f (- 8) = - \frac{4 + \sqrt{2^{2}}}{2}$

Step 2.1.2.1.5

Pull terms out from under the radical, assuming positive real numbers.

$f (- 8) = - \frac{4 + 2}{2}$

Step 2.1.2.1.6

Add $4$ and $2$ .

$f (- 8) = - \frac{6}{2}$

Step 2.1.2.2

Simplify the expression.

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Step 2.1.2.2.1

Divide $6$ by $2$ .

$f (- 8) = - 1 \cdot 3$

Step 2.1.2.2.2

Multiply $- 1$ by $3$ .

$f (- 8) = - 3$

Step 2.1.2.3

The final answer is $- 3$ .

$y = - 3$

Step 2.1.3

Convert $- 3$ to decimal.

$= - 3$

Step 2.2

Substitute the $x$ value $- 8$ into $f (x) = - \frac{4 - \sqrt{2 (- 6 - x)}}{2}$ . In this case, the point is $(- 8, - 1)$ .

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Step 2.2.1

Replace the variable $x$ with $- 8$ in the expression.

$f (- 8) = - \frac{4 - \sqrt{2 (- 6 - (- 8))}}{2}$

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

Simplify the numerator.

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Step 2.2.2.1.1

Multiply $- 1$ by $- 8$ .

$f (- 8) = - \frac{4 - \sqrt{2 (- 6 + 8)}}{2}$

Step 2.2.2.1.2

Add $- 6$ and $8$ .

$f (- 8) = - \frac{4 - \sqrt{2 \cdot 2}}{2}$

Step 2.2.2.1.3

Multiply $2$ by $2$ .

$f (- 8) = - \frac{4 - \sqrt{4}}{2}$

Step 2.2.2.1.4

Rewrite $4$ as $2^{2}$ .

$f (- 8) = - \frac{4 - \sqrt{2^{2}}}{2}$

Step 2.2.2.1.5

Pull terms out from under the radical, assuming positive real numbers.

$f (- 8) = - \frac{4 - 1 \cdot 2}{2}$

Step 2.2.2.1.6

Multiply $- 1$ by $2$ .

$f (- 8) = - \frac{4 - 2}{2}$

Step 2.2.2.1.7

Subtract $2$ from $4$ .

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

Step 2.2.2.2

Reduce the expression by cancelling the common factors.

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Step 2.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.2.1.2

Rewrite the expression.

$f (- 8) = - 1 \cdot 1$

Step 2.2.2.2.2

Multiply $- 1$ by $1$ .

$f (- 8) = - 1$

Step 2.2.2.3

The final answer is $- 1$ .

$y = - 1$

Step 2.2.3

Convert $- 1$ to decimal.

$= - 1$

Step 2.3

Substitute the $x$ value $- 7$ into $f (x) = - \frac{4 + \sqrt{2 (- 6 - x)}}{2}$ . In this case, the point is $(- 7, - 2.70710678)$ .

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Step 2.3.1

Replace the variable $x$ with $- 7$ in the expression.

$f (- 7) = - \frac{4 + \sqrt{2 (- 6 - (- 7))}}{2}$

Step 2.3.2

Simplify the result.

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Step 2.3.2.1

Simplify the numerator.

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Step 2.3.2.1.1

Multiply $- 1$ by $- 7$ .

$f (- 7) = - \frac{4 + \sqrt{2 (- 6 + 7)}}{2}$

Step 2.3.2.1.2

Add $- 6$ and $7$ .

$f (- 7) = - \frac{4 + \sqrt{2 \cdot 1}}{2}$

Step 2.3.2.1.3

Multiply $2$ by $1$ .

$f (- 7) = - \frac{4 + \sqrt{2}}{2}$

Step 2.3.2.2

The final answer is $- \frac{4 + \sqrt{2}}{2}$ .

$y = - \frac{4 + \sqrt{2}}{2}$

Step 2.3.3

Convert $- \frac{4 + \sqrt{2}}{2}$ to decimal.

$= - 2.70710678$

Step 2.4

Substitute the $x$ value $- 7$ into $f (x) = - \frac{4 - \sqrt{2 (- 6 - x)}}{2}$ . In this case, the point is $(- 7, - 1.29289321)$ .

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Step 2.4.1

Replace the variable $x$ with $- 7$ in the expression.

$f (- 7) = - \frac{4 - \sqrt{2 (- 6 - (- 7))}}{2}$

Step 2.4.2

Simplify the result.

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Step 2.4.2.1

Simplify the numerator.

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Step 2.4.2.1.1

Multiply $- 1$ by $- 7$ .

$f (- 7) = - \frac{4 - \sqrt{2 (- 6 + 7)}}{2}$

Step 2.4.2.1.2

Add $- 6$ and $7$ .

$f (- 7) = - \frac{4 - \sqrt{2 \cdot 1}}{2}$

Step 2.4.2.1.3

Multiply $2$ by $1$ .

$f (- 7) = - \frac{4 - \sqrt{2}}{2}$

Step 2.4.2.2

The final answer is $- \frac{4 - \sqrt{2}}{2}$ .

$y = - \frac{4 - \sqrt{2}}{2}$

Step 2.4.3

Convert $- \frac{4 - \sqrt{2}}{2}$ to decimal.

$= - 1.29289321$

Step 2.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 8 & - 3 \\ - 8 & - 1 \\ - 7 & - 2.71 \\ - 7 & - 1.29 \\ - 6 & - 2 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(- 6, - 2)$

Focus: $(- \frac{49}{8}, - 2)$

Axis of Symmetry: $y = - 2$

Directrix: $x = - \frac{47}{8}$

$\begin{array}{c} x & y \\ - 8 & - 3 \\ - 8 & - 1 \\ - 7 & - 2.71 \\ - 7 & - 1.29 \\ - 6 & - 2 \end{array}$

Step 4