Properties of the parabola $$$y = \frac{x^{2}}{12}$$$

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 $$$y = \frac{x^{2}}{12}$$$.

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

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

Thus, $$$h = 0$$$, $$$k = 0$$$, $$$f = 3$$$.

The standard form is $$$y = \frac{x^{2}}{12}$$$.

The general form is $$$x^{2} - 12 y = 0$$$.

The vertex form is $$$y = \frac{x^{2}}{12}$$$.

The directrix is $$$y = d$$$.

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: $$$0 - 3 = d - 0$$$.

Thus, the directrix is $$$y = -3$$$.

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

The focal length is the distance between the focus and the vertex: $$$3$$$.

The focal parameter is the distance between the focus and the directrix: $$$6$$$.

The latus rectum is parallel to the directrix and passes through the focus: $$$y = 3$$$.

The endpoints of the latus rectum can be found by solving the system $$$\begin{cases} x^{2} - 12 y = 0 \\ y = 3 \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the latus rectum are $$$\left(-6, 3\right)$$$, $$$\left(6, 3\right)$$$.

The length of the latus rectum (focal width) is four times the distance between the vertex and the focus: $$$12$$$.

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: $$$y = \frac{x^{2}}{12}$$$A.

General form/equation: $$$x^{2} - 12 y = 0$$$A.

Vertex form/equation: $$$y = \frac{x^{2}}{12}$$$A.

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

Intercept form/equation: $$$y = \frac{x^{2}}{12}$$$A.

Graph: see the graphing calculator.

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

Focus: $$$\left(0, 3\right)$$$A.

Directrix: $$$y = -3$$$A.

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

Latus rectum: $$$y = 3$$$A.

Endpoints of the latus rectum: $$$\left(-6, 3\right)$$$, $$$\left(6, 3\right)$$$A.

Length of the latus rectum (focal width): $$$12$$$A.

Focal parameter: $$$6$$$A.

Focal length: $$$3$$$A.

Eccentricity: $$$1$$$A.

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

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

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

Range: $$$\left[0, \infty\right)$$$A.