Parabola Calculator

This calculator will find either the equation of the parabola from the given parameters or the vertex, focus, directrix, axis of symmetry, latus rectum, length of the latus rectum, focal parameter, focal length (distance), eccentricity, x-intercepts, y-intercepts, domain, and range of the entered parabola. Also, it will graph the parabola. Steps are available.

Related calculators: Circle Calculator, Ellipse Calculator, Hyperbola Calculator, Conic Section Calculator

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Find the vertex, focus, directrix, axis of symmetry, latus rectum, length of the latus rectum, focal parameter, focal length, eccentricity, x-intercepts, y-intercepts, domain, and range of the parabola $y = \left(x - 2\right)^{2} + 5$.

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

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 length of the latus rectum 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: $y = x^{2} - 4 x + 9$A.

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

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

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

Length of the latus rectum: $1$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.