Graph -9x-3y^2=6

$- 9 x - 3 y^{2} = 6$

Step 1

Add $3 y^{2}$ to both sides of the equation.

$- 9 x = 6 + 3 y^{2}$

Step 2

Divide each term in $- 9 x = 6 + 3 y^{2}$ by $- 9$ and simplify.

Divide each term in $- 9 x = 6 + 3 y^{2}$ by $- 9$ .

$\frac{- 9 x}{- 9} = \frac{6}{- 9} + \frac{3 y^{2}}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 x}{- 9} = \frac{6}{- 9} + \frac{3 y^{2}}{- 9}$

Divide $x$ by $1$ .

$x = \frac{6}{- 9} + \frac{3 y^{2}}{- 9}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $6$ and $- 9$ .

Factor $3$ out of $6$ .

$x = \frac{3 (2)}{- 9} + \frac{3 y^{2}}{- 9}$

Cancel the common factors.

Factor $3$ out of $- 9$ .

$x = \frac{3 \cdot 2}{3 \cdot - 3} + \frac{3 y^{2}}{- 9}$

Cancel the common factor.

$x = \frac{3 \cdot 2}{3 \cdot - 3} + \frac{3 y^{2}}{- 9}$

Rewrite the expression.

$x = \frac{2}{- 3} + \frac{3 y^{2}}{- 9}$

Move the negative in front of the fraction.

$x = - \frac{2}{3} + \frac{3 y^{2}}{- 9}$

Cancel the common factor of $3$ and $- 9$ .

Factor $3$ out of $3 y^{2}$ .

$x = - \frac{2}{3} + \frac{3 (y^{2})}{- 9}$

Cancel the common factors.

Factor $3$ out of $- 9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 3

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Reorder $- \frac{2}{3}$ and $- \frac{y^{2}}{3}$ .

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

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

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

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

$b = 0$

$c = - \frac{2}{3}$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{0}{2 (- \frac{1}{3})}$

Simplify the right side.

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

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{1}{3})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{1}{3})}$

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

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

Multiply $- 1$ by $3$ .

$d = 0 \cdot - 3$

Multiply $0$ by $- 3$ .

$d = 0$

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

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

$e = - \frac{2}{3} - \frac{0^{2}}{4 (- \frac{1}{3})}$

Simplify the right side.

Simplify each term.

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

$e = - \frac{2}{3} - \frac{0}{4 (- \frac{1}{3})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - \frac{2}{3} - \frac{0}{- 4 (\frac{1}{3})}$

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

$e = - \frac{2}{3} - \frac{0}{\frac{- 4}{3}}$

Move the negative in front of the fraction.

$e = - \frac{2}{3} - \frac{0}{- \frac{4}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{2}{3} - (0 (- \frac{3}{4}))$

Multiply $0 (- \frac{3}{4})$ .

Multiply $- 1$ by $0$ .

$e = - \frac{2}{3} - (0 (\frac{3}{4}))$

Multiply $0$ by $\frac{3}{4}$ .

$e = - \frac{2}{3} - 0$

Multiply $- 1$ by $0$ .

$e = - \frac{2}{3} + 0$

Add $- \frac{2}{3}$ and $0$ .

$e = - \frac{2}{3}$

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

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

Set $x$ equal to the new right side.

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

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

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

$h = - \frac{2}{3}$

$k = 0$

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

Opens Left

Find the vertex $(h, k)$ .

$(- \frac{2}{3}, 0)$

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

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

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

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

Simplify.

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

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

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

Move the negative in front of the fraction.

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

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

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

Multiply the numerator by the reciprocal of the denominator.

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

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

$- \frac{3}{4}$

Find the focus.

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

$(h + p, k)$

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

$(- \frac{17}{12}, 0)$

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

$y = 0$

Find the directrix.

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.

$x = h - p$

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

$x = \frac{1}{12}$

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

Direction: Opens Left

Vertex: $(- \frac{2}{3}, 0)$

Focus: $(- \frac{17}{12}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{1}{12}$

Direction: Opens Left

Vertex: $(- \frac{2}{3}, 0)$

Focus: $(- \frac{17}{12}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{1}{12}$

Step 4

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.

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

Replace the variable $x$ with $- 2.\overline{6}$ in the expression.

$f (- 2.\overline{6}) = \sqrt{- 2 - 3 \cdot - 2.\overline{6}}$

Simplify the result.

Multiply $- 3$ by $- 2.\overline{6}$ .

$f (- 2.\overline{6}) = \sqrt{- 2 + 8}$

Add $- 2$ and $8$ .

$f (- 2.\overline{6}) = \sqrt{6}$

The final answer is $\sqrt{6}$ .

$y = \sqrt{6}$

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

Replace the variable $x$ with $- 2.\overline{6}$ in the expression.

$f (- 2.\overline{6}) = - \sqrt{- 2 - 3 \cdot - 2.\overline{6}}$

Simplify the result.

Multiply $- 3$ by $- 2.\overline{6}$ .

$f (- 2.\overline{6}) = - \sqrt{- 2 + 8}$

Add $- 2$ and $8$ .

$f (- 2.\overline{6}) = - \sqrt{6}$

The final answer is $- \sqrt{6}$ .

$y = - \sqrt{6}$

Substitute the $x$ value $- 1.\overline{6}$ into $f (x) = \sqrt{- 2 - 3 x}$ . In this case, the point is $(- 1.\overline{6}, \sqrt{3})$ .

Replace the variable $x$ with $- 1.\overline{6}$ in the expression.

$f (- 1.\overline{6}) = \sqrt{- 2 - 3 \cdot - 1.\overline{6}}$

Simplify the result.

Multiply $- 3$ by $- 1.\overline{6}$ .

$f (- 1.\overline{6}) = \sqrt{- 2 + 5}$

Add $- 2$ and $5$ .

$f (- 1.\overline{6}) = \sqrt{3}$

The final answer is $\sqrt{3}$ .

$y = \sqrt{3}$

Substitute the $x$ value $- 1.\overline{6}$ into $f (x) = - \sqrt{- 2 - 3 x}$ . In this case, the point is $(- 1.\overline{6}, - \sqrt{3})$ .

Replace the variable $x$ with $- 1.\overline{6}$ in the expression.

$f (- 1.\overline{6}) = - \sqrt{- 2 - 3 \cdot - 1.\overline{6}}$

Simplify the result.

Multiply $- 3$ by $- 1.\overline{6}$ .

$f (- 1.\overline{6}) = - \sqrt{- 2 + 5}$

Add $- 2$ and $5$ .

$f (- 1.\overline{6}) = - \sqrt{3}$

The final answer is $- \sqrt{3}$ .

$y = - \sqrt{3}$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2.667 & 2.45 \\ - 2.667 & - 2.45 \\ - 1.667 & 1.73 \\ - 1.667 & - 1.73 \\ - \frac{2}{3} & 0 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(- \frac{2}{3}, 0)$

Focus: $(- \frac{17}{12}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{1}{12}$

$\begin{array}{c} x & y \\ - 2.667 & 2.45 \\ - 2.667 & - 2.45 \\ - 1.667 & 1.73 \\ - 1.667 & - 1.73 \\ - \frac{2}{3} & 0 \end{array}$

Step 6

Graph -9x-3y^2=6

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