Properties of the parabola $$$y^{2} = - 3 x$$$

Find the vertex, focus, directrix, axis of symmetry, latus rectum, length of the latus rectum (focal width), focal parameter, focal length, eccentricity, x-intercepts, y-intercepts, domain, and range of the parabola $$$y^{2} = - 3 x$$$.

The equation of a parabola is $$$x = \frac{1}{4 \left(f - h\right)} \left(y - k\right)^{2} + h$$$, where $$$\left(h, k\right)$$$ is the vertex and $$$\left(f, k\right)$$$ is the focus.

Our parabola in this form is $$$x = \frac{1}{4 \left(- \frac{3}{4} - 0\right)} \left(y - 0\right)^{2} + 0$$$.

Thus, $$$h = 0$$$, $$$k = 0$$$, $$$f = - \frac{3}{4}$$$.

The standard form is $$$x = - \frac{y^{2}}{3}$$$.

The general form is $$$3 x + y^{2} = 0$$$.

The vertex form is $$$x = - \frac{y^{2}}{3}$$$.

The directrix is $$$x = d$$$.

To find $$$d$$$, use the fact that the distance from the focus to the vertex is the same as the distance from the vertex to the directrix: $$$0 - \left(- \frac{3}{4}\right) = d - 0$$$.

Thus, the directrix is $$$x = \frac{3}{4}$$$.

The axis of symmetry is the line perpendicular to the directrix that passes through the vertex and the focus: $$$y = 0$$$.

The focal length is the distance between the focus and the vertex: $$$\frac{3}{4}$$$.

The focal parameter is the distance between the focus and the directrix: $$$\frac{3}{2}$$$.

The latus rectum is parallel to the directrix and passes through the focus: $$$x = - \frac{3}{4}$$$.

The endpoints of the latus rectum can be found by solving the system $$$\begin{cases} 3 x + y^{2} = 0 \\ x = - \frac{3}{4} \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the latus rectum are $$$\left(- \frac{3}{4}, - \frac{3}{2}\right)$$$, $$$\left(- \frac{3}{4}, \frac{3}{2}\right)$$$.

The length of the latus rectum (focal width) is four times the distance between the vertex and the focus: $$$3$$$.

The eccentricity of a parabola is always $$$1$$$.

The x-intercepts can be found by setting $$$y = 0$$$ in the equation and solving for $$$x$$$ (for steps, see intercepts calculator).

x-intercept: $$$\left(0, 0\right)$$$.

The y-intercepts can be found by setting $$$x = 0$$$ in the equation and solving for $$$y$$$: (for steps, see intercepts calculator).

y-intercept: $$$\left(0, 0\right)$$$.

Standard form/equation: $$$x = - \frac{y^{2}}{3}$$$A.

General form/equation: $$$3 x + y^{2} = 0$$$A.

Vertex form/equation: $$$x = - \frac{y^{2}}{3}$$$A.

Focus-directrix form/equation: $$$y^{2} + \left(x + \frac{3}{4}\right)^{2} = \left(x - \frac{3}{4}\right)^{2}$$$A.

Intercept form/equation: $$$x = - \frac{y^{2}}{3}$$$A.

Graph: see the graphing calculator.

Vertex: $$$\left(0, 0\right)$$$A.

Focus: $$$\left(- \frac{3}{4}, 0\right) = \left(-0.75, 0\right)$$$A.

Directrix: $$$x = \frac{3}{4} = 0.75$$$A.

Axis of symmetry: $$$y = 0$$$A.

Latus rectum: $$$x = - \frac{3}{4} = -0.75$$$A.

Endpoints of the latus rectum: $$$\left(- \frac{3}{4}, - \frac{3}{2}\right) = \left(-0.75, -1.5\right)$$$, $$$\left(- \frac{3}{4}, \frac{3}{2}\right) = \left(-0.75, 1.5\right)$$$A.

Length of the latus rectum (focal width): $$$3$$$A.

Focal parameter: $$$\frac{3}{2} = 1.5$$$A.

Focal length: $$$\frac{3}{4} = 0.75$$$A.

Eccentricity: $$$1$$$A.

x-intercept: $$$\left(0, 0\right)$$$A.

y-intercept: $$$\left(0, 0\right)$$$A.

Domain: $$$\left(-\infty, 0\right]$$$A.

Range: $$$\left(-\infty, \infty\right)$$$A.