Parabola Calculator

Enter the parabola equation or the known data to get all its equations (standard, general, and vertex form), its elements (focus, vertex, directrix, axis of symmetry, latus rectum, intercepts), and an interactive graph of the curve.

What data do you know?

Vertex (V)

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Focus (F)

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Vertex (V)

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Directrix

Vertex (V)

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Point (P)

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Focus (F)

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Directrix

Point 1

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Point 2

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Point 3

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Quick Examples

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Solved Exercises

Find all the equations and elements of the parabola given its equation $ y=x^2-4x+3. $

Parabola equations

Standard form

$$ y = x^{2}-4 x+3 $$

General form

$$ -x^{2}+4 x+y-3 = 0 $$

Vertex form

$$ \left(x-2\right)^2 = \left(y+1\right) $$

Parabola elements

Orientation: Vertical opening up.

Vertex: $ V \left(2, -1\right) $

Focus: $ F \left(2, -\dfrac{3}{4}\right) \approx \left(2, -0.75\right) $

Directrix equation: $ y = -\dfrac{5}{4} = -1.25 $

Axis of symmetry equation: $ x = 2 $

Latus rectum length: $ L_R = 1 $

Parameter: $ p = \dfrac{1}{4} = 0.25 $

Focal length: $ |p| = \dfrac{1}{4} = 0.25 $

Eccentricity: $ e = 1 $

x-intercepts (roots)

$$ x_1 = 1 \\[1em] x_2 = 3 $$

y-intercepts

$$ y_1 = 3 $$

Domain: $ D = \mathbb{R} $

Range: $ R = \left[-1, +\infty\right) $

Graph of a vertical parabola on the Cartesian plane, with a vertex away from the origin, equation, focus, and directrix. — Graph of the parabola

Given the parabola with equation $ x^{2}+6 x+8 y+25 = 0 $, determine its elements.

Equations

Standard form equation

$$ y = -\dfrac{x^{2}}{8}-\dfrac{3 x}{4}-\dfrac{25}{8} $$

General form equation

$$ x^{2}+6 x+8 y+25 = 0 $$

Vertex form equation

$$ \left(x+3\right)^2 = -8\left(y+2\right) $$

Elements

Orientation: Vertical opening down.

Vertex: $ V \left(-3, -2\right) $

Focus: $ F \left(-3, -4\right) $

Directrix equation: $ y = 0 $

Axis of symmetry: $ x = -3 $

Latus rectum length: $ L_R = 8 $

Parameter: $ p = -2 $

Focal length: $ |p| = 2 $

Eccentricity: $ e = 1 $

x-axis intercepts

Do not exist in real numbers.

y-axis intercepts

$$ y_1 = -\dfrac{25}{8} = -3.125 $$

Domain: $ D = \mathbb{R} $

Range: $ R = \left(-\infty, -2\right] $

Graph on the Cartesian plane of a vertical parabola that opens down, with a vertex away from the origin, equation, focus, and directrix. — Graph of the curve

Perform a complete analysis of the horizontal parabola with equation $ (y+2)^2 = -12(x-1) $.

Parabola formulas

Standard form

$$ x = -\dfrac{y^{2}}{12}-\dfrac{y}{3}+\dfrac{2}{3} $$

General form

$$ y^{2}+4 y+12 x-8 = 0 $$

Vertex form

$$ \left(y+2\right)^2 = -12\left(x-1\right) $$

Parabola elements

Orientation: Horizontal opening left.

Vertex: $ V \left(1, -2\right) $

Focus: $ F \left(-2, -2\right) $

Directrix: $ x = 4 $

Axis of symmetry: $ y = -2 $

Latus rectum: $ L_R = 12 $

Parameter: $ p = -3 $

Focal length: $ |p| = 3 $

Eccentricity: $ e = 1 $

x intercepts

$$ x_1 = \dfrac{2}{3} \approx 0.67 $$

y intercepts

$$ y_1 = \dfrac{4 \sqrt{3}-4}{2} \approx 1.46 \\[1em] y_2 = \dfrac{-4 \sqrt{3}-4}{2} \approx -5.46 $$

Graph on the Cartesian plane of a horizontal parabola that opens to the left, with a vertex away from the origin, equation with fractional coefficients, focus, and directrix. — Parabola on the Cartesian plane

Find the equation of the parabola given the vertex (2, 3) and the focus (2, 5).

Parabola equations

Standard form equation

$$ y = \dfrac{x^{2}}{8}-\dfrac{x}{2}+\dfrac{7}{2} $$

General form equation

$$ x^{2}-4 x-8 y+28 = 0 $$

Vertex form equation

$$ \left(x-2\right)^2 = 8\left(y-3\right) $$

Elements

Orientation: Vertical opening up.

Vertex: $ V \left(2, 3\right) $

Focus: $ F \left(2, 5\right) $

Directrix: $ y = 1 $

Axis of symmetry: $ x = 2 $

Latus rectum: $ L_R = 8 $

Parameter: $ p = 2 $

Focal length: $ |p| = 2 $

Eccentricity: $ e = 1 $

x-axis intercepts

Do not exist in real numbers.

y-axis intercepts

$$ y_1 = \dfrac{7}{2} = 3.5 $$

Domain: $ D = \mathbb{R} $

Range: $ R = \left[3, +\infty\right) $

Graph of a vertical parabola that opens up in a Cartesian system, calculated from its vertex and focus. — Graph of the curve on the plane

Find the equation and elements of a parabola given the focus (3, 2) and directrix x = -1.

Equations

Standard form

$$ x = \dfrac{y^{2}}{8}-\dfrac{y}{2}+\dfrac{3}{2} $$

General form

$$ y^{2}-4 y-8 x+12 = 0 $$

Vertex form

$$ \left(y-2\right)^2 = 8\left(x-1\right) $$

Parabola elements

Orientation and opening: Horizontal opening right.

Vertex: $ V \left(1, 2\right) $

Focus: $ F \left(3, 2\right) $

Directrix equation: $ x = -1 $

Equation of the axis of symmetry: $ y = 2 $

Latus rectum: $ L_R = 8 $

Parameter: $ p = 2 $

Focal length: $ |p| = 2 $

Eccentricity: $ e = 1 $

x-intercepts

$$ x_1 = \dfrac{3}{2} = 1.5 $$

y-intercepts

Do not exist in real numbers.

Graph on the Cartesian plane of a horizontal parabola that opens to the right, with a vertex away from the origin, standard equation, focus coordinates, and directrix. Calculated from its focus and directrix.

Find the equation of the parabola given 3 points: (1, 0), (2, 3) and (3, 8). Vertical orientation

Equations

Standard equation

$$ y = x^{2}-1 $$

General equation

$$ -x^{2}+y+1 = 0 $$

Vertex form equation

$$ x^2 = \left(y+1\right) $$

Elements

Orientation: Vertical opening up.

Vertex: $ V \left(0, -1\right) $

Focus: $ F \left(0, -\dfrac{3}{4}\right) \approx \left(0, -0.75\right) $

Directrix: $ y = -\dfrac{5}{4} = -1.25 $

Axis of symmetry: $ x = 0 $

Latus rectum: $ L_R = 1 $

Parameter: $ p = \dfrac{1}{4} = 0.25 $

Focal length: $ |p| = \dfrac{1}{4} = 0.25 $

Eccentricity: $ e = 1 $

x-axis intercepts

$$ x_1 = -1 \\[1em] x_2 = 1 $$

y-axis intercepts

$$ y_1 = -1 $$

Domain: $ D = \mathbb{R} $

Range: $ R = \left[-1, +\infty\right) $

Graph on the Cartesian plane of a vertical parabola that opens up, obtained from three points, with a vertex away from the origin, equation, focus, and directrix. — Graph of the parabola on the plane

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Daniel Machado

Professor of Mathematics, graduated from the Faculty of Exact, Chemical and Natural Sciences of the National University of Misiones (UNAM). Developer and creator of RigelUp, dedicated to building tools for mathematical learning.