How this calculator works
Fit a parabola to paired observations. The least-squares solution minimises squared vertical residuals and includes an intercept. At least three distinct x values identify the coefficients; more than three pairs are needed to estimate residual variance. A centred and scaled QR fit reduces round-off from large predictor offsets.
Formula
ŷ = b0 + b1x + b2x²; choose coefficients to minimise Σ(y - ŷ)².
Worked example
Points (-1,1), (0,0), (1,1), (2,4) give b0 = 0, b1 = 0, b2 = 1. At x = 3 the prediction is 9.
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Frequently asked questions
What controls the curvature?
The coefficient b2 is positive for an upward-opening parabola and negative for a downward-opening one.
Do three points prove a quadratic relationship?
Three distinct x values determine a quadratic exactly. More observations and a scientific explanation are needed to assess the model.
What if b2 is nearly zero?
The fitted quadratic is close to a straight line over the data range.