Properties of the parabola $$$y = - 2 x^{2} + 6 x - 3$$$

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

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 $$$y = \frac{1}{4 \left(\frac{11}{8} - \frac{3}{2}\right)} \left(x - \frac{3}{2}\right)^{2} + \frac{3}{2}$$$.

Thus, $$$h = \frac{3}{2}$$$, $$$k = \frac{3}{2}$$$, $$$f = \frac{11}{8}$$$.

The standard form is $$$y = - 2 x^{2} + 6 x - 3$$$.

The general form is $$$- 2 x^{2} + 6 x - y - 3 = 0$$$.

The vertex form is $$$y = - 2 \left(x - \frac{3}{2}\right)^{2} + \frac{3}{2}$$$.

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: $$$\frac{3}{2} - \frac{11}{8} = d - \frac{3}{2}$$$.

Thus, the directrix is $$$y = \frac{13}{8}$$$.

The axis of symmetry is the line perpendicular to the directrix that passes through the vertex and the focus: $$$x = \frac{3}{2}$$$.

The focal length is the distance between the focus and the vertex: $$$\frac{1}{8}$$$.

The focal parameter is the distance between the focus and the directrix: $$$\frac{1}{4}$$$.

The latus rectum is parallel to the directrix and passes through the focus: $$$y = \frac{11}{8}$$$.

The endpoints of the latus rectum can be found by solving the system $$$\begin{cases} - 2 x^{2} + 6 x - y - 3 = 0 \\ y = \frac{11}{8} \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the latus rectum are $$$\left(\frac{5}{4}, \frac{11}{8}\right)$$$, $$$\left(\frac{7}{4}, \frac{11}{8}\right)$$$.

The length of the latus rectum (focal width) is four times the distance between the vertex and the focus: $$$\frac{1}{2}$$$.

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-intercepts: $$$\left(\frac{3}{2} - \frac{\sqrt{3}}{2}, 0\right)$$$, $$$\left(\frac{\sqrt{3}}{2} + \frac{3}{2}, 0\right)$$$

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

y-intercept: $$$\left(0, -3\right)$$$.

Standard form/equation: $$$y = - 2 x^{2} + 6 x - 3$$$A.

General form/equation: $$$- 2 x^{2} + 6 x - y - 3 = 0$$$A.

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

Focus-directrix form/equation: $$$\left(x - \frac{3}{2}\right)^{2} + \left(y - \frac{11}{8}\right)^{2} = \left(y - \frac{13}{8}\right)^{2}$$$A.

Intercept form/equation: $$$y = - 2 \left(x - \frac{3}{2} + \frac{\sqrt{3}}{2}\right) \left(x - \frac{3}{2} - \frac{\sqrt{3}}{2}\right)$$$A.

Graph: see the graphing calculator.

Vertex: $$$\left(\frac{3}{2}, \frac{3}{2}\right) = \left(1.5, 1.5\right)$$$A.

Focus: $$$\left(\frac{3}{2}, \frac{11}{8}\right) = \left(1.5, 1.375\right)$$$A.

Directrix: $$$y = \frac{13}{8} = 1.625$$$A.

Axis of symmetry: $$$x = \frac{3}{2} = 1.5$$$A.

Latus rectum: $$$y = \frac{11}{8} = 1.375$$$A.

Endpoints of the latus rectum: $$$\left(\frac{5}{4}, \frac{11}{8}\right) = \left(1.25, 1.375\right)$$$, $$$\left(\frac{7}{4}, \frac{11}{8}\right) = \left(1.75, 1.375\right)$$$A.

Length of the latus rectum (focal width): $$$\frac{1}{2} = 0.5$$$A.

Focal parameter: $$$\frac{1}{4} = 0.25$$$A.

Focal length: $$$\frac{1}{8} = 0.125$$$A.

Eccentricity: $$$1$$$A.

x-intercepts: $$$\left(\frac{3}{2} - \frac{\sqrt{3}}{2}, 0\right)\approx \left(0.633974596215561, 0\right)$$$, $$$\left(\frac{\sqrt{3}}{2} + \frac{3}{2}, 0\right)\approx \left(2.366025403784439, 0\right)$$$A.

y-intercept: $$$\left(0, -3\right)$$$A.

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

Range: $$$\left(-\infty, \frac{3}{2}\right] = \left(-\infty, 1.5\right]$$$A.