Parabola Vertex & Focus Calculator
Vertex, focus and directrix from a quadratic.
Enter your values
Enter the a you are working with.
Enter the b you are working with.
Enter the c you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- 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)
At a glance
| What it does | Vertex, focus and directrix from a quadratic. |
|---|---|
| Category | Math |
| Inputs needed | a, b, c |
| Main output | Vertex |
| Formula | vertex x = -b/(2a), focus offset = 1/(4a) |
| Cost | Free — no sign-up, no download |
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.
Inputs explained
- a
- Enter the a you are working with.
- b
- Enter the b you are working with.
- c
- Enter the c you are working with.
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.
What do I need to enter into the Parabola Vertex & Focus Calculator?
Just 3 values: a, b, c. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Parabola Vertex & Focus Calculator work out the 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. It applies the formula vertex x = -b/(2a), focus offset = 1/(4a) and shows the working so you can check each step by hand.
Is the Parabola Vertex & Focus Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.