Parabola Calculator
Find the vertex, focus, directrix and axis of symmetry of a parabola y=ax²+bx+c.
About Parabola Calculator
A parabola y = ax² + bx + c has its vertex at x = −b/(2a). The focus lies 1/(4|a|) units from the vertex along the axis of symmetry, and the directrix is the same distance on the opposite side.
Formulas
Vertex
x_v = −b/(2a), y_v = f(x_v)
- a
- Leading coefficient
- b
- Linear coefficient
Focal Distance
f = 1/(4|a|)
- a
- Leading coefficient
Worked example
With the default values below, Parabola Calculator returns Vertex X of 2.0000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- a (x² coefficient)
- 1
- b (x coefficient)
- -4
- c (constant)
- 3
Results
- Vertex X
- 2.0000
- Vertex Y
- -1.0000
- Focus Y
- -0.7500
- Directrix (y = )
- -1.2500
- Focal Distance
- 0.2500
Step by step
- 1Vertex x-coordinate= 2.0000
- 2Vertex y-coordinate= -1.0000
- 3Focal distance= 0.2500
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What does the sign of 'a' determine?
If a > 0 the parabola opens upward; if a < 0 it opens downward.
How do I find the axis of symmetry?
It is the vertical line x = −b/(2a), passing through the vertex.