Parabola Calculator

Enter y=ax²+bx+c to find the vertex, focus, directrix, axis, latus rectum, and intercepts, with a parabola plot.

Input

Enter the coefficients a, b and c of the quadratic y=ax²+bx+c to find the vertex, focus, directrix, axis of symmetry, latus rectum and intercepts.

Coefficient a

Coefficient b

Coefficient c

The equation has the form y=ax²+bx+c. If a is 0 the curve is a line, not a parabola.

Vertex (h, k)

(1, -4)

Opens upward

Focus

(1, -3.75)

Directrix

y=-4.25

Axis of symmetry

x=1

x intercepts

-1, 3

y intercept

-3

Latus rectum length

The vertex is h=−b/2a, k=c−b²/4a. With focal length p=1/4a, the focus is (h, k+p), the directrix is y=k−p, and the latus rectum is |1/a|.

The vertex comes from completing the square y=a(x−h)²+k, giving h=−b/2a and k=c−b²/4a.
The axis of symmetry is the vertical line x=h, and the parabola is mirror symmetric about it.
Using the focal length p=1/4a, the focus is (h, k+p) and the directrix is y=k−p.
The latus rectum length is |1/a|, which describes how wide the parabola opens.
The number of x intercepts depends on the discriminant D=b²−4ac: two when D>0, one when D=0, none when D is negative.
The y intercept is the value c, obtained by setting x=0.
When the coefficient a is 0 the curve is a straight line, not a parabola, so it cannot be computed.

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