Quadratic Regression Calculator

Quadratic regression finds the parabola that most closely follows a collection of paired observations. Its model is y = ax² + bx + c, where a controls the direction and strength of curvature, b controls the linear trend, and c is the predicted value when x equals zero. Unlike interpolation, regression does not force the curve through every point. It minimizes the sum of squared vertical residuals, so it is useful when measurements include ordinary noise or natural variation. The calculator builds the normal equations from the sums of x, x², x³, x⁴, y, xy, and x²y, then solves the resulting three-by-three linear system. This is the least-squares solution for a second-degree polynomial. At least three points are required, and the x values must contain enough variation to identify all three coefficients. More observations usually produce a more stable model, especially when the data are noisy. R-squared describes how much of the observed variation in y is explained by the fitted curve. A value near 1 indicates that the quadratic model tracks the data closely, while a value near 0 indicates little improvement over predicting the mean of y. A high R-squared is not proof that the relationship is causal or that a parabola is the correct scientific model. Always inspect residuals, consider the measurement process, and compare plausible alternatives. Quadratic regression is appropriate for patterns with one bend, such as a projectile path, accelerating growth, a U-shaped cost curve, or response data that rise and then fall. It is less suitable for periodic behavior, sharp discontinuities, or relationships with several turning points. Predictions inside the observed x range are interpolation and are generally safer than extrapolating beyond it, where the squared term can quickly dominate. Enter each point as an x-y pair and separate pairs with semicolons. The optional prediction field evaluates the fitted equation at a chosen x. Coefficients and predictions are rounded for readability, while the calculation uses full floating-point precision. Keep units consistent across observations and avoid rounding source measurements prematurely for the most meaningful curve fit.