Pre-Algebra Examples

$x^{2} + 18 x - 9 y - 9 = 0$

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.

$18 x - 9 y - 9 = - x^{2}$

Step 1.1.2

Subtract $18 x$ from both sides of the equation.

$- 9 y - 9 = - x^{2} - 18 x$

Step 1.1.3

Add $9$ to both sides of the equation.

$- 9 y = - x^{2} - 18 x + 9$

Step 1.2

Divide each term in $- 9 y = - x^{2} - 18 x + 9$ by $- 9$ and simplify.

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Step 1.2.1

Divide each term in $- 9 y = - x^{2} - 18 x + 9$ by $- 9$ .

$\frac{- 9 y}{- 9} = \frac{- x^{2}}{- 9} + \frac{- 18 x}{- 9} + \frac{9}{- 9}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 9 y}{- 9} = \frac{- x^{2}}{- 9} + \frac{- 18 x}{- 9} + \frac{9}{- 9}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{- 9} + \frac{- 18 x}{- 9} + \frac{9}{- 9}$

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

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{9} + \frac{- 18 x}{- 9} + \frac{9}{- 9}$

Step 1.2.3.1.2

Cancel the common factor of $- 18$ and $- 9$ .

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Step 1.2.3.1.2.1

Factor $- 9$ out of $- 18 x$ .

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

Step 1.2.3.1.2.2

Cancel the common factors.

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Step 1.2.3.1.2.2.1

Factor $- 9$ out of $- 9$ .

$y = \frac{x^{2}}{9} + \frac{- 9 (2 x)}{- 9 (1)} + \frac{9}{- 9}$

Step 1.2.3.1.2.2.2

Cancel the common factor.

$y = \frac{x^{2}}{9} + \frac{- 9 (2 x)}{- 9 \cdot 1} + \frac{9}{- 9}$

Step 1.2.3.1.2.2.3

Rewrite the expression.

$y = \frac{x^{2}}{9} + \frac{2 x}{1} + \frac{9}{- 9}$

Step 1.2.3.1.2.2.4

Divide $2 x$ by $1$ .

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

Step 1.2.3.1.3

Divide $9$ by $- 9$ .

$y = \frac{x^{2}}{9} + 2 x - 1$

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 $\frac{x^{2}}{9} + 2 x - 1$ .

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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 $c$ .

$a = \frac{1}{9}$

$b = 2$

$c = - 1$

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

$d = \frac{2}{2 (\frac{1}{9})}$

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

Cancel the common factor of $2$ .

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Step 2.1.1.3.2.1.1

Cancel the common factor.

$d = \frac{2}{2 (\frac{1}{9})}$

Step 2.1.1.3.2.1.2

Rewrite the expression.

$d = \frac{1}{\frac{1}{9}}$

Step 2.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 9$

Step 2.1.1.3.2.3

Multiply $9$ by $1$ .

$d = 9$

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

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

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

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

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

Step 2.1.1.4.2.1.2

Combine $4$ and $\frac{1}{9}$ .

$e = - 1 - \frac{4}{\frac{4}{9}}$

Step 2.1.1.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.1.4.2.1.4

Cancel the common factor of $4$ .

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Step 2.1.1.4.2.1.4.1

Cancel the common factor.

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

Step 2.1.1.4.2.1.4.2

Rewrite the expression.

$e = - 1 - 1 \cdot 9$

Step 2.1.1.4.2.1.5

Multiply $- 1$ by $9$ .

$e = - 1 - 9$

Step 2.1.1.4.2.2

Subtract $9$ from $- 1$ .

$e = - 10$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{9} {(x + 9)}^{2} - 10$ .

$\frac{1}{9} {(x + 9)}^{2} - 10$

Step 2.1.2

Set $y$ equal to the new right side.

$y = \frac{1}{9} \cdot {(x + 9)}^{2} - 10$

Step 2.2

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

$a = \frac{1}{9}$

$h = - 9$

$k = - 10$

Step 2.3

Since the value of $a$ is positive, the parabola opens up.

Opens Up

Step 2.4

Find the vertex $(h, k)$ .

$(- 9, - 10)$

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.

$\frac{1}{4 \cdot \frac{1}{9}}$

Step 2.5.3

Simplify.

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Step 2.5.3.1

Combine $4$ and $\frac{1}{9}$ .

$\frac{1}{\frac{4}{9}}$

Step 2.5.3.2

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{9}{4})$

Step 2.5.3.3

Multiply $\frac{9}{4}$ by $1$ .

$\frac{9}{4}$

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.

$(- 9, - \frac{31}{4})$

Step 2.7

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

$x = - 9$

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.

$y = - \frac{49}{4}$

Step 2.9

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

Direction: Opens Up

Vertex: $(- 9, - 10)$

Focus: $(- 9, - \frac{31}{4})$

Axis of Symmetry: $x = - 9$

Directrix: $y = - \frac{49}{4}$

Direction: Opens Up

Vertex: $(- 9, - 10)$

Focus: $(- 9, - \frac{31}{4})$

Axis of Symmetry: $x = - 9$

Directrix: $y = - \frac{49}{4}$

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 $- 10$ in the expression.

$f (- 10) = \frac{{(- 10)}^{2}}{9} + 2 (- 10) - 1$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raise $- 10$ to the power of $2$ .

$f (- 10) = \frac{100}{9} + 2 (- 10) - 1$

Step 3.2.1.2

Multiply $2$ by $- 10$ .

$f (- 10) = \frac{100}{9} - 20 - 1$

Step 3.2.2

Find the common denominator.

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Step 3.2.2.1

Write $- 20$ as a fraction with denominator $1$ .

$f (- 10) = \frac{100}{9} + \frac{- 20}{1} - 1$

Step 3.2.2.2

Multiply $\frac{- 20}{1}$ by $\frac{9}{9}$ .

$f (- 10) = \frac{100}{9} + \frac{- 20}{1} \cdot \frac{9}{9} - 1$

Step 3.2.2.3

Multiply $\frac{- 20}{1}$ by $\frac{9}{9}$ .

$f (- 10) = \frac{100}{9} + \frac{- 20 \cdot 9}{9} - 1$

Step 3.2.2.4

Write $- 1$ as a fraction with denominator $1$ .

$f (- 10) = \frac{100}{9} + \frac{- 20 \cdot 9}{9} + \frac{- 1}{1}$

Step 3.2.2.5

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

$f (- 10) = \frac{100}{9} + \frac{- 20 \cdot 9}{9} + \frac{- 1}{1} \cdot \frac{9}{9}$

Step 3.2.2.6

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

$f (- 10) = \frac{100}{9} + \frac{- 20 \cdot 9}{9} + \frac{- 1 \cdot 9}{9}$

Step 3.2.3

Combine the numerators over the common denominator.

$f (- 10) = \frac{100 - 20 \cdot 9 - 1 \cdot 9}{9}$

Step 3.2.4

Simplify each term.

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Step 3.2.4.1

Multiply $- 20$ by $9$ .

$f (- 10) = \frac{100 - 180 - 1 \cdot 9}{9}$

Step 3.2.4.2

Multiply $- 1$ by $9$ .

$f (- 10) = \frac{100 - 180 - 9}{9}$

Step 3.2.5

Simplify the expression.

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Step 3.2.5.1

Subtract $180$ from $100$ .

$f (- 10) = \frac{- 80 - 9}{9}$

Step 3.2.5.2

Subtract $9$ from $- 80$ .

$f (- 10) = \frac{- 89}{9}$

Step 3.2.5.3

Move the negative in front of the fraction.

$f (- 10) = - \frac{89}{9}$

Step 3.2.6

The final answer is $- \frac{89}{9}$ .

$- \frac{89}{9}$

Step 3.3

The $y$ value at $x = - 10$ is $- \frac{89}{9}$ .

$y = - \frac{89}{9}$

Step 3.4

Replace the variable $x$ with $- 11$ in the expression.

$f (- 11) = \frac{{(- 11)}^{2}}{9} + 2 (- 11) - 1$

Step 3.5

Simplify the result.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

Raise $- 11$ to the power of $2$ .

$f (- 11) = \frac{121}{9} + 2 (- 11) - 1$

Step 3.5.1.2

Multiply $2$ by $- 11$ .

$f (- 11) = \frac{121}{9} - 22 - 1$

Step 3.5.2

Find the common denominator.

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Step 3.5.2.1

Write $- 22$ as a fraction with denominator $1$ .

$f (- 11) = \frac{121}{9} + \frac{- 22}{1} - 1$

Step 3.5.2.2

Multiply $\frac{- 22}{1}$ by $\frac{9}{9}$ .

$f (- 11) = \frac{121}{9} + \frac{- 22}{1} \cdot \frac{9}{9} - 1$

Step 3.5.2.3

Multiply $\frac{- 22}{1}$ by $\frac{9}{9}$ .

$f (- 11) = \frac{121}{9} + \frac{- 22 \cdot 9}{9} - 1$

Step 3.5.2.4

Write $- 1$ as a fraction with denominator $1$ .

$f (- 11) = \frac{121}{9} + \frac{- 22 \cdot 9}{9} + \frac{- 1}{1}$

Step 3.5.2.5

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

$f (- 11) = \frac{121}{9} + \frac{- 22 \cdot 9}{9} + \frac{- 1}{1} \cdot \frac{9}{9}$

Step 3.5.2.6

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

$f (- 11) = \frac{121}{9} + \frac{- 22 \cdot 9}{9} + \frac{- 1 \cdot 9}{9}$

Step 3.5.3

Combine the numerators over the common denominator.

$f (- 11) = \frac{121 - 22 \cdot 9 - 1 \cdot 9}{9}$

Step 3.5.4

Simplify each term.

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Step 3.5.4.1

Multiply $- 22$ by $9$ .

$f (- 11) = \frac{121 - 198 - 1 \cdot 9}{9}$

Step 3.5.4.2

Multiply $- 1$ by $9$ .

$f (- 11) = \frac{121 - 198 - 9}{9}$

Step 3.5.5

Simplify the expression.

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Step 3.5.5.1

Subtract $198$ from $121$ .

$f (- 11) = \frac{- 77 - 9}{9}$

Step 3.5.5.2

Subtract $9$ from $- 77$ .

$f (- 11) = \frac{- 86}{9}$

Step 3.5.5.3

Move the negative in front of the fraction.

$f (- 11) = - \frac{86}{9}$

Step 3.5.6

The final answer is $- \frac{86}{9}$ .

$- \frac{86}{9}$

Step 3.6

The $y$ value at $x = - 11$ is $- \frac{86}{9}$ .

$y = - \frac{86}{9}$

Step 3.7

Replace the variable $x$ with $- 6$ in the expression.

$f (- 6) = \frac{{(- 6)}^{2}}{9} + 2 (- 6) - 1$

Step 3.8

Simplify the result.

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Step 3.8.1

Simplify each term.

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Step 3.8.1.1

Raise $- 6$ to the power of $2$ .

$f (- 6) = \frac{36}{9} + 2 (- 6) - 1$

Step 3.8.1.2

Divide $36$ by $9$ .

$f (- 6) = 4 + 2 (- 6) - 1$

Step 3.8.1.3

Multiply $2$ by $- 6$ .

$f (- 6) = 4 - 12 - 1$

Step 3.8.2

Simplify by subtracting numbers.

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Step 3.8.2.1

Subtract $12$ from $4$ .

$f (- 6) = - 8 - 1$

Step 3.8.2.2

Subtract $1$ from $- 8$ .

$f (- 6) = - 9$

Step 3.8.3

The final answer is $- 9$ .

$- 9$

Step 3.9

The $y$ value at $x = - 6$ is $- 9$ .

$y = - 9$

Step 3.10

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

$f (- 8) = \frac{{(- 8)}^{2}}{9} + 2 (- 8) - 1$

Step 3.11

Simplify the result.

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Step 3.11.1

Simplify each term.

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Step 3.11.1.1

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

$f (- 8) = \frac{64}{9} + 2 (- 8) - 1$

Step 3.11.1.2

Multiply $2$ by $- 8$ .

$f (- 8) = \frac{64}{9} - 16 - 1$

Step 3.11.2

Find the common denominator.

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Step 3.11.2.1

Write $- 16$ as a fraction with denominator $1$ .

$f (- 8) = \frac{64}{9} + \frac{- 16}{1} - 1$

Step 3.11.2.2

Multiply $\frac{- 16}{1}$ by $\frac{9}{9}$ .

$f (- 8) = \frac{64}{9} + \frac{- 16}{1} \cdot \frac{9}{9} - 1$

Step 3.11.2.3

Multiply $\frac{- 16}{1}$ by $\frac{9}{9}$ .

$f (- 8) = \frac{64}{9} + \frac{- 16 \cdot 9}{9} - 1$

Step 3.11.2.4

Write $- 1$ as a fraction with denominator $1$ .

$f (- 8) = \frac{64}{9} + \frac{- 16 \cdot 9}{9} + \frac{- 1}{1}$

Step 3.11.2.5

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

$f (- 8) = \frac{64}{9} + \frac{- 16 \cdot 9}{9} + \frac{- 1}{1} \cdot \frac{9}{9}$

Step 3.11.2.6

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

$f (- 8) = \frac{64}{9} + \frac{- 16 \cdot 9}{9} + \frac{- 1 \cdot 9}{9}$

Step 3.11.3

Combine the numerators over the common denominator.

$f (- 8) = \frac{64 - 16 \cdot 9 - 1 \cdot 9}{9}$

Step 3.11.4

Simplify each term.

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Step 3.11.4.1

Multiply $- 16$ by $9$ .

$f (- 8) = \frac{64 - 144 - 1 \cdot 9}{9}$

Step 3.11.4.2

Multiply $- 1$ by $9$ .

$f (- 8) = \frac{64 - 144 - 9}{9}$

Step 3.11.5

Simplify the expression.

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Step 3.11.5.1

Subtract $144$ from $64$ .

$f (- 8) = \frac{- 80 - 9}{9}$

Step 3.11.5.2

Subtract $9$ from $- 80$ .

$f (- 8) = \frac{- 89}{9}$

Step 3.11.5.3

Move the negative in front of the fraction.

$f (- 8) = - \frac{89}{9}$

Step 3.11.6

The final answer is $- \frac{89}{9}$ .

$- \frac{89}{9}$

Step 3.12

The $y$ value at $x = - 8$ is $- \frac{89}{9}$ .

$y = - \frac{89}{9}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 11 & - \frac{86}{9} \\ - 10 & - \frac{89}{9} \\ - 9 & - 10 \\ - 8 & - \frac{89}{9} \\ - 6 & - 9 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: $(- 9, - 10)$

Focus: $(- 9, - \frac{31}{4})$

Axis of Symmetry: $x = - 9$

Directrix: $y = - \frac{49}{4}$

$\begin{array}{c} x & y \\ - 11 & - \frac{86}{9} \\ - 10 & - \frac{89}{9} \\ - 9 & - 10 \\ - 8 & - \frac{89}{9} \\ - 6 & - 9 \end{array}$

Step 5