Math

Parabola Calculator (Vertex, Focus, Directrix, Vertex Form)

Q: Where is a satellite dish's receiver?

At the focus: a parabolic reflector sends all incoming parallel rays to it.

Enter the coefficients a, b and c.

By TheCalcTools Updated October 17, 2026 Formula checked and tested

Your numbers

Your results

Vertex

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Vertex form

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Focus

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Directrix

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x-intercepts

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Opens

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Parabola Calculator (Vertex, Focus, Directrix, Vertex Form) by TheCalcTools

How the Parabola Calculator (Vertex, Focus, Directrix, Vertex Form) works

Completing the square turns y = ax² + bx + c into vertex form y = a(x − h)² + k with h = −b/(2a) and k = c − b²/(4a). The axis of symmetry is x = h.

The focus is 1/(4a) above the vertex (below if a < 0) and the directrix the same distance on the other side. Every point on the parabola is equally far from the focus and the directrix.

The formula

h = −b/(2a), k = c − b²/(4a); focus (h, k + 1/(4a)), directrix y = k − 1/(4a)

Worked example

Using the default values shown in the calculator:

a	1
b	-4
c	3

y = x² − 4x + 3 has vertex (2, -1) and vertex form y = 1(x − 2)² − 1.
Focus (2, -0.75), directrix y = -1.25; it crosses the x-axis at 1 and 3.

Result:

Vertex	(2, -1)
Vertex form	y = 1(x − 2)² − 1
Focus	(2, -0.75)
Directrix	y = -1.25
x-intercepts	1 and 3
Opens	Upward (minimum at the vertex)

Frequently asked questions

Where is a satellite dish's receiver?