Parabola Calculator
Focus, directrix, vertex and latus rectum of a parabola.
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, -0.5)
- Directrix
- y = -1.5
- Focal length
- 0.5
- Latus rectum length
- 2
- Opens
- Upward
- Axis of symmetry
- x = 2
Quick answer
For y = ax² + bx + c the vertex is at x = −b/2a and the focus sits 1 ÷ (4a) above it, with the directrix the same distance below.
focal length p = 1 ÷ (4a)
At a glance
| What it does | Focus, directrix, vertex and latus rectum of a parabola. |
|---|---|
| Category | Math |
| Inputs needed | a, b, c |
| Main output | Vertex |
| Formula | focal length p = 1 ÷ (4a) |
| Cost | Free — no sign-up, no download |
How to use the Parabola Calculator
- 1Enter the coefficients of y = ax² + bx + c.
- 2Read the vertex, then the focus and directrix.
- 3Use the latus rectum width to sketch the curve accurately through three points.
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
- a0.5
- b-2
- c1
Results
- Vertex(2, -1)
- Focus(2, -0.5)
- Directrixy = -1.5
- Focal length0.5
- Latus rectum length2
- OpensUpward
- Axis of symmetryx = 2
Frequently asked questions
Why does a satellite dish use a parabola?
Every ray parallel to the axis reflects through the focus, so the receiver sits exactly at the focal point.
What is the latus rectum?
The chord through the focus parallel to the directrix; its length is 1 ÷ |a| and it fixes how wide the curve opens.
What do I need to enter into the Parabola 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 Calculator work out the answer?
For y = ax² + bx + c the vertex is at x = −b/2a and the focus sits 1 ÷ (4a) above it, with the directrix the same distance below. It applies the formula focal length p = 1 ÷ (4a) and shows the working so you can check each step by hand.
Is the Parabola 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.