Graph y^2=-3x

$y^{2} = - 3 x$

Step 1

Rewrite the equation as $- 3 x = y^{2}$ .

$- 3 x = y^{2}$

Step 2

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

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

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

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 3

Find the properties of the given parabola.

Rewrite the equation in vertex form.

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

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

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

$b = 0$

$c = 0$

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

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

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

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

Simplify the right side.

Simplify each term.

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

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

Simplify the denominator.

Multiply $- 1$ by $4$ .

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

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

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

Move the negative in front of the fraction.

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

Multiply the numerator by the reciprocal of the denominator.

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

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

Multiply $- 1$ by $0$ .

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

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

$e = 0 - 0$

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 $- \frac{1}{3} y^{2}$ .

$- \frac{1}{3} y^{2}$

Set $x$ equal to the new right side.

$x = - \frac{1}{3} y^{2}$

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

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

$h = 0$

$k = 0$

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.

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

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.

$(- \frac{3}{4}, 0)$

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.

$x = \frac{3}{4}$

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

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{3}{4}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{3}{4}$

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{3}{4}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{3}{4}$

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 $- 2$ into $f (x) = \sqrt{- 3 x}$ . In this case, the point is $(- 2, 2.44948974)$ .

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

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

Simplify the result.

Multiply $- 3$ by $- 2$ .

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

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

$y = \sqrt{6}$

Convert $\sqrt{6}$ to decimal.

$= 2.44948974$

Substitute the $x$ value $- 2$ into $f (x) = - \sqrt{- 3 x}$ . In this case, the point is $(- 2, - 2.44948974)$ .

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

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

Simplify the result.

Multiply $- 3$ by $- 2$ .

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

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

$y = - \sqrt{6}$

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

$= - 2.44948974$

Substitute the $x$ value $- 1$ into $f (x) = \sqrt{- 3 x}$ . In this case, the point is $(- 1, 1.7320508)$ .

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

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

Simplify the result.

Multiply $- 3$ by $- 1$ .

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

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

$y = \sqrt{3}$

Convert $\sqrt{3}$ to decimal.

$= 1.7320508$

Substitute the $x$ value $- 1$ into $f (x) = - \sqrt{- 3 x}$ . In this case, the point is $(- 1, - 1.7320508)$ .

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

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

Simplify the result.

Multiply $- 3$ by $- 1$ .

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

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

$y = - \sqrt{3}$

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

$= - 1.7320508$

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & 2.45 \\ - 2 & - 2.45 \\ - 1 & 1.73 \\ - 1 & - 1.73 \\ 0 & 0 \end{array}$

Step 5

Graph the parabola using its properties and the selected points.

Direction: Opens Left

Vertex: $(0, 0)$

Focus: $(- \frac{3}{4}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = \frac{3}{4}$

$\begin{array}{c} x & y \\ - 2 & 2.45 \\ - 2 & - 2.45 \\ - 1 & 1.73 \\ - 1 & - 1.73 \\ 0 & 0 \end{array}$

Step 6

Graph y^2=-3x

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