Quadratic Regression Calculator
Model curved relationships with a quadratic function y = a + bx + cx². Enter paired data to compute coefficients, R², and fitted values.
Quadratic model
y = 3.0000 + (2.0000)x + (1.0000)x²
R² = 1.0000
| x | Actual y | Predicted y | Residual |
|---|---|---|---|
| 0 | 3.0000 | 3.0000 | -0.0000 |
| 1 | 6.0000 | 6.0000 | 0.0000 |
| 2 | 11.0000 | 11.0000 | 0.0000 |
| 3 | 18.0000 | 18.0000 | 0.0000 |
| 4 | 27.0000 | 27.0000 | 0.0000 |
How to Use This Calculator
- Enter observations as x-y pairs.
- Review the fitted quadratic equation and R² to assess fit.
- Inspect residuals to detect patterns or potential outliers.
- Use the prediction box to estimate y for new x values.
Formula
Model: y = a + bx + cx²
Normal equations solve (XᵀX)β = Xᵀy, where rows of X are [1, x, x²]
R² = 1 − Σ(y − ŷ)² / Σ(y − ȳ)²
Full Description
Quadratic regression captures parabolic trends by fitting a second-degree polynomial. It is often used to model acceleration, peak phenomena, or diminishing returns where curvature is symmetric or nearly so.
Compare the quadratic fit with linear or higher-order models to balance simplicity and accuracy.
Frequently Asked Questions
Can I force the curve through the origin?
This calculator estimates a free intercept. For constrained fits, adjust data or solve customized equations.
Why might R² be 1?
When exactly three points are provided that lie on a unique quadratic, the fit is perfect and R² = 1.
Is the vertex easy to compute?
Yes. Vertex x-coordinate = −b / (2c). Use the fitted coefficients to locate peaks or troughs.
What about extrapolation?
Quadratics can grow rapidly outside observed ranges. Verify that extrapolated predictions are sensible.