Quadratic Regression Calculator
Fit a parabola y = ax² + bx + c to your data — for relationships that rise then fall, or curve away from a straight line.
Quadratic regression is still least squares; the model just includes an x² term.
How the quadratic regression calculator works
Quadratic regression is still least squares; the model just includes an x² term. The three normal equations are solved directly by Gaussian elimination on a 3×3 system.
The vertex, at x = −b/2a, is usually the point of interest: the dose that maximises response, the speed that minimises fuel use, the temperature at peak yield.
Formula: y = ax² + bx + c; vertex at x = −b / 2a
Worked examples
| Inputs | a (x² coefficient) | Note |
|---|---|---|
| A clean parabola | -0.880952 | peaks near x = 4 |
| An exact fit | 2 | y = 2x² + 0x + 1 |
| A straight line | 0 | a ≈ 0 — a quadratic is unnecessary |
FAQFrequently asked questions
When should I fit a quadratic?
When the relationship curves — rises then falls, or accelerates. A residual plot from a straight-line fit showing a clear arc is the usual sign.
What is the vertex for?
It locates the turning point: the optimum dose, speed or temperature. It is often the whole reason for fitting a curve.
Does a quadratic always fit better?
Its R² can never be lower than a straight line's, because the line is a special case. That is why a higher R² alone does not justify the extra term.
Can I extrapolate from it?
Very cautiously. A parabola shoots off to infinity in both directions, so predictions outside your data range go wrong fast.
What if a is nearly zero?
The curvature is negligible and a straight line is the better model — simpler, and it will predict better out of sample.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.