Input Three Points
Example Data Table
Click Use to populate the form.
| Set | (x₁, y₁) | (x₂, y₂) | (x₃, y₃) | Equation | |
|---|---|---|---|---|---|
| A | (0, 1) | (1, 4) | (2, 9) | y = 1x² + 2x + 1 | |
| B | (-1, 6) | (0, 1) | (2, 9) | y = 2x² - 2x + 1 | |
| C | (-2, 5) | (1, 2) | (3, 10) | y = 1x² - 2x + 1 |
Formula Used
We model the curve as y = a x² + b x + c and determine a, b, c by solving a Vandermonde system, or equivalently, expanding the Lagrange interpolating polynomial through three given points.
- Distinct x-values guarantee a unique solution.
- Vertex:
h = -b/(2a),k = f(h). - Discriminant:
Δ = b² - 4ac; roots via quadratic formula.
How to Use
- Enter three points with distinct x-values.
- Choose decimal precision for display rounding.
- Click Calculate to compute coefficients and forms.
- Review vertex, axis, discriminant, and roots.
- Export a summary using Download CSV or Download PDF.
FAQs
Equal x-values make the Vandermonde matrix singular, preventing a unique quadratic from being determined.
Then the best-fit through three points is linear:
y = bx + c. The tool reports linear details accordingly.Yes. If the discriminant is negative, the curve has complex-conjugate zeros; no real x-intercepts appear.
Computation uses exact floating arithmetic internally; display rounding follows your decimal setting for readability.
No, any ordering yields the same quadratic provided x-values remain distinct.
Yes. The verification block evaluates
f(x) at your inputs and displays the targets alongside.| Title | Quadratic Function From 3 Points Calculator |
|---|---|
| Inputs | x1=, y1=, x2=, y2=, x3=, y3=, decimals=6 |