Quadratic Regression Calculator (3 Points)
Fit the exact parabola y = ax² + bx + c through three points, then predict y at any x — the unique second-degree curve on your data.
How it works
Solve y = a·x² + b·x + c so all three points satisfy it · a, b, c come from the divided differences of the data
Three points with distinct x-values pin down exactly one parabola, just as two points pin down one line. This calculator solves the resulting 3×3 system directly using divided differences, so there is no fitting error — the curve passes through every point precisely. With more than three points the fit would instead be a least-squares approximation; here the "regression" is the exact interpolating quadratic.
Worked example
Through (0, 1), (1, 3) and (2, 9): the constant is c = 1, then a + b = 2 and 4a + 2b = 8 give a = 2, b = 0. The parabola is y = 2x² + 1, so at x = 3 it predicts 2 × 9 + 1 = 19.
Sources: Polynomial regression — Wikipedia
Frequently asked questions
Is this a least-squares regression?
No — with exactly three points the quadratic passes through all of them, so it is an exact interpolation with zero residual. True least-squares regression fits many points and minimises the total squared error, generally not passing through any single one.
Why must the x-values be different?
Two points sharing an x would demand two different y-values from one curve at the same input, which is impossible for a function. Distinct x-values keep the system solvable and the parabola unique.
What if the three points lie on a straight line?
Then the x² coefficient a comes out as 0 and the fit collapses to that line — the calculator handles it gracefully and reports a ≈ 0.
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This calculator is for educational purposes. Double-check important results before acting on them.