Parabola Vertex & Focus Calculator
Vertex, focus and directrix from a quadratic.
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ⓘ Live results
Results
- Vertex
- (2, -1)
- Focus
- (2, -0.75)
- Directrix
- y = -1.25
- Axis of symmetry
- x = 2
Quick answer
The vertex is at (-b/2a, f(-b/2a)); the focus and directrix are offset by 1/(4a) from the vertex along the axis of symmetry.
vertex x = -b/(2a), focus offset = 1/(4a)
How to use the Parabola Vertex & Focus Calculator
- 1Enter coefficients a, b, c of y = ax^2+bx+c.
- 2The vertex x is -b/2a; substitute to get vertex y.
- 3The focus and directrix sit 1/(4a) above and below the vertex.
Worked example
Using the values the calculator loads with:
Inputs
- a1
- b-4
- c3
Results
- Vertex(2, -1)
- Focus(2, -0.75)
- Directrixy = -1.25
- Axis of symmetryx = 2
Frequently asked questions
What if a is negative?
The parabola opens downward and the focus sits below the vertex.
Does this work for x=ay^2+by+c?
No, swap x and y roles for a sideways-opening parabola.
About this calculator
- Built and reviewed by James Whitfield · How we build and test calculators
- Report an error
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.