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

The calculator will find the properties of the parabola $y = - 2 x^{2} + 6 x - 3$, with steps shown.

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

### Solution

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.