Parabola Calculator

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 $$$y = \left(x - 2\right)^{2} + 5$$$.

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 $$$y = \frac{1}{4 \left(\frac{21}{4} - 5\right)} \left(x - 2\right)^{2} + 5$$$.

Thus, $$$h = 2$$$, $$$k = 5$$$, $$$f = \frac{21}{4}$$$.

The standard form is $$$y = x^{2} - 4 x + 9$$$.

The general form is $$$x^{2} - 4 x - y + 9 = 0$$$.

The vertex form is $$$y = \left(x - 2\right)^{2} + 5$$$.

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: $$$5 - \frac{21}{4} = d - 5$$$.

Thus, the directrix is $$$y = \frac{19}{4}$$$.

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

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

The focal parameter is the distance between the focus and the directrix: $$$\frac{1}{2}$$$.

The latus rectum is parallel to the directrix and passes through the focus: $$$y = \frac{21}{4}$$$.

The endpoints of the latus rectum can be found by solving the system $$$\begin{cases} x^{2} - 4 x - y + 9 = 0 \\ y = \frac{21}{4} \end{cases}$$$ (for steps, see system of equations calculator).

The endpoints of the latus rectum are $$$\left(\frac{3}{2}, \frac{21}{4}\right)$$$, $$$\left(\frac{5}{2}, \frac{21}{4}\right)$$$.

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

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).

Since there are no real solutions, there are no x-intercepts.

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

y-intercept: $$$\left(0, 9\right)$$$.

Standard form/equation: $$$y = x^{2} - 4 x + 9$$$A.

General form/equation: $$$x^{2} - 4 x - y + 9 = 0$$$A.

Vertex form/equation: $$$y = \left(x - 2\right)^{2} + 5$$$A.

Focus-directrix form/equation: $$$\left(x - 2\right)^{2} + \left(y - \frac{21}{4}\right)^{2} = \left(y - \frac{19}{4}\right)^{2}$$$A.

Graph: see the graphing calculator.

Vertex: $$$\left(2, 5\right)$$$A.

Focus: $$$\left(2, \frac{21}{4}\right) = \left(2, 5.25\right)$$$A.

Directrix: $$$y = \frac{19}{4} = 4.75$$$A.

Axis of symmetry: $$$x = 2$$$A.

Latus rectum: $$$y = \frac{21}{4} = 5.25$$$A.

Endpoints of the latus rectum: $$$\left(\frac{3}{2}, \frac{21}{4}\right) = \left(1.5, 5.25\right)$$$, $$$\left(\frac{5}{2}, \frac{21}{4}\right) = \left(2.5, 5.25\right)$$$A.

Length of the latus rectum (focal width): $$$1$$$A.

Focal parameter: $$$\frac{1}{2} = 0.5$$$A.

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

Eccentricity: $$$1$$$A.

x-intercepts: no x-intercepts.

y-intercept: $$$\left(0, 9\right)$$$A.

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

Range: $$$\left[5, \infty\right)$$$A.