Graph x=-3y^2

$x = - 3 y^{2}$

Step 1

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

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

$a = - 3$

$b = 0$

$c = 0$

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

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

Simplify the right side.

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

Factor $2$ out of $0$ .

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

Cancel the common factors.

Factor $2$ out of $2 \cdot - 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $0$ and $- 3$ .

Factor $3$ out of $0$ .

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

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Multiply $- 1$ by $0$ .

$d = 0$

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

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $- 3$ .

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

Divide $0$ by $- 12$ .

$e = 0 - 0$

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Add $0$ and $0$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 y^{2}$ .

$- 3 y^{2}$

Set $x$ equal to the new right side.

$x = - 3 y^{2}$

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

$a = - 3$

$h = 0$

$k = 0$

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

Opens Left

Find the vertex $(h, k)$ .

$(0, 0)$

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.

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

Simplify.

Multiply $4$ by $- 3$ .

$\frac{1}{- 12}$

Move the negative in front of the fraction.

$- \frac{1}{12}$

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.

$(- \frac{1}{12}, 0)$

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: $(0, 0)$

Focus: $(- \frac{1}{12}, 0)$

Axis of Symmetry: $y = 0$

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

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{1}{12}, 0)$

Axis of Symmetry: $y = 0$

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

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.

Substitute the $x$ value $- 2$ into $f (x) = \sqrt{- \frac{x}{3}}$ . In this case, the point is $(- 2, 0.81649658)$ .

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

$f (- 2) = \sqrt{- \frac{- 2}{3}}$

Simplify the result.

Move the negative in front of the fraction.

$f (- 2) = \sqrt{1 (\frac{2}{3})}$

Combine exponents.

Factor out negative.

$f (- 2) = \sqrt{\frac{2}{3}}$

Multiply $- 1$ by $- 1$ .

$f (- 2) = \sqrt{1 (\frac{2}{3})}$

Multiply $\frac{2}{3}$ by $1$ .

$f (- 2) = \sqrt{\frac{2}{3}}$

Rewrite $\sqrt{\frac{2}{3}}$ as $\frac{\sqrt{2}}{\sqrt{3}}$ .

$f (- 2) = \frac{\sqrt{2}}{\sqrt{3}}$

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (- 2) = \frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (- 2) = \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$f (- 2) = \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$f (- 2) = \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$f (- 2) = \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$f (- 2) = \frac{\sqrt{2} \sqrt{3}}{3}$

Evaluate the exponent.

$f (- 2) = \frac{\sqrt{2} \sqrt{3}}{3}$

Simplify the numerator.

Combine using the product rule for radicals.

$f (- 2) = \frac{\sqrt{2 \cdot 3}}{3}$

Multiply $2$ by $3$ .

$f (- 2) = \frac{\sqrt{6}}{3}$

The final answer is $\frac{\sqrt{6}}{3}$ .

$y = \frac{\sqrt{6}}{3}$

Convert $\frac{\sqrt{6}}{3}$ to decimal.

$= 0.81649658$

Substitute the $x$ value $- 2$ into $f (x) = - \sqrt{- \frac{x}{3}}$ . In this case, the point is $(- 2, - 0.81649658)$ .

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

$f (- 2) = - \sqrt{- \frac{- 2}{3}}$

Simplify the result.

Move the negative in front of the fraction.

$f (- 2) = - \sqrt{1 (\frac{2}{3})}$

Combine exponents.

Factor out negative.

$f (- 2) = - \sqrt{\frac{2}{3}}$

Multiply $- 1$ by $- 1$ .

$f (- 2) = - \sqrt{1 (\frac{2}{3})}$

Multiply $\frac{2}{3}$ by $1$ .

$f (- 2) = - \sqrt{\frac{2}{3}}$

Rewrite $\sqrt{\frac{2}{3}}$ as $\frac{\sqrt{2}}{\sqrt{3}}$ .

$f (- 2) = - \frac{\sqrt{2}}{\sqrt{3}}$

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (- 2) = - (\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$f (- 2) = - \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$f (- 2) = - \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$f (- 2) = - \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$f (- 2) = - \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$f (- 2) = - \frac{\sqrt{2} \sqrt{3}}{3}$

Evaluate the exponent.

$f (- 2) = - \frac{\sqrt{2} \sqrt{3}}{3}$

Simplify the numerator.

Combine using the product rule for radicals.

$f (- 2) = - \frac{\sqrt{2 \cdot 3}}{3}$

Multiply $2$ by $3$ .

$f (- 2) = - \frac{\sqrt{6}}{3}$

The final answer is $- \frac{\sqrt{6}}{3}$ .

$y = - \frac{\sqrt{6}}{3}$

Convert $- \frac{\sqrt{6}}{3}$ to decimal.

$= - 0.81649658$

Substitute the $x$ value $- 1$ into $f (x) = \sqrt{- \frac{x}{3}}$ . In this case, the point is $(- 1, 0.57735026)$ .

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

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

Simplify the result.

Move the negative in front of the fraction.

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

Combine exponents.

Factor out negative.

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

Multiply $- 1$ by $- 1$ .

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

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

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

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

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

Any root of $1$ is $1$ .

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

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

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

Combine and simplify the denominator.

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

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

Raise $\sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

$y = \frac{\sqrt{3}}{3}$

Convert $\frac{\sqrt{3}}{3}$ to decimal.

$= 0.57735026$

Substitute the $x$ value $- 1$ into $f (x) = - \sqrt{- \frac{x}{3}}$ . In this case, the point is $(- 1, - 0.57735026)$ .

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

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

Simplify the result.

Move the negative in front of the fraction.

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

Combine exponents.

Factor out negative.

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

Multiply $- 1$ by $- 1$ .

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

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

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

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

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

Any root of $1$ is $1$ .

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

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

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

Combine and simplify the denominator.

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

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

Raise $\sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

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

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

Convert $- \frac{\sqrt{3}}{3}$ to decimal.

$= - 0.57735026$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 0.82 \\ - 2 & - 0.82 \\ - 1 & 0.58 \\ - 1 & - 0.58 \\ 0 & 0 \end{array}$

Step 3

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{1}{12}, 0)$

Axis of Symmetry: $y = 0$

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

$\begin{array}{c} x & y \\ - 2 & 0.82 \\ - 2 & - 0.82 \\ - 1 & 0.58 \\ - 1 & - 0.58 \\ 0 & 0 \end{array}$

Step 4

Graph x=-3y^2

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