How this instrument works
Quadratic regression fits a curve of the form y = ax² + bx + c to a set of paired x, y data points, choosing the coefficients a, b, and c that minimize the sum of squared vertical distances between the curve and every actual data point — the same least-squares principle behind ordinary linear regression, extended to allow a curved rather than straight relationship. It's the right tool when a scatter plot of your data clearly bends rather than following a straight line, but the bend is simple enough to be captured by a single curve with one peak or one trough.
The a coefficient controls how sharply the parabola curves and which way it opens — positive a opens upward, a U-shape, useful for costs or errors that both rise and fall around a minimum; negative a opens downward, an upside-down U, useful for outcomes that rise then fall around a maximum, like a process yield with an optimal operating point. R², the coefficient of determination, reports how much of the variation in y this quadratic curve actually explains, from 0, no better than just guessing the mean of y, to 1, the curve passes through every point exactly.
With exactly three data points, a quadratic fit passes through all of them perfectly, R² = 1, the same way two points always define a perfect line. With more than three points, the fit becomes a genuine least-squares compromise, and R² tells you how good that compromise actually is — a high R² close to 1 means the underlying relationship really is well described by a parabola, while a low R² suggests either a different curve shape or a genuinely noisy relationship.
- Enter your x values into X values, separated by commas, spaces, or new lines.
- Enter the corresponding y values into Y values, in the same order and count (at least 3 pairs).
- Read a, b, and c — the fitted coefficients of y = ax² + bx + c.
- Read R² — how well the fitted parabola explains the variation in your y data, from 0 to 1.
Worked example — an exact quadratic recovered from 5 points
The true relationship is y = 2x² + 3x + 1. At x = 0, 1, 2, 3, 4, this gives y = 1, 6, 15, 28, 45. Enter '0, 1, 2, 3, 4' into X values and '1, 6, 15, 28, 45' into Y values.
With 5 exact points generated from a true degree-2 relationship, the least-squares fit recovers the original coefficients exactly: a = 2, b = 3, c = 1, and R² = 1.0 — a perfect fit, since every point lies exactly on the fitted parabola with no leftover scatter to explain.
Questions
How do I know if quadratic regression is the right fit?
Start by plotting your data — if it clearly curves in one consistent direction, bending up then leveling, or rising to a peak then falling, a quadratic fit is a reasonable candidate. Compare R² between a linear fit and a quadratic fit on the same data: if the quadratic fit's R² is meaningfully higher, the extra curvature term is capturing real structure. If your data has more than one bend, a quadratic won't be enough, and a higher-degree polynomial or a different model may fit better.
What does a negative a coefficient mean?
A negative a means the fitted parabola opens downward, like an upside-down U — the curve rises to a maximum and then falls. This is typical of relationships with a single optimal point, such as a process yield that improves with more of some input up to a point and then declines with too much of it. A positive a means the parabola opens upward instead, with a minimum rather than a maximum.
Why does R² equal exactly 1.0 with only 3 or so data points?
A quadratic curve has three free parameters, a, b, and c, so it can always be made to pass through exactly 3 points with no error at all — much like two points always define a perfect straight line. This isn't evidence of a strong underlying relationship; it's just that 3 points don't provide enough information to distinguish genuinely quadratic from coincidentally fitting a parabola. A meaningful R² requires more data points than the number of coefficients being fit.
How is this different from the general polynomial-regression calculator?
They use the identical least-squares fitting method under the hood, and give the same a, b, c and R² for the same data at degree 2 — this page is a fixed-degree-2 version dedicated specifically to quadratic curves, while polynomial-regression lets you choose any degree from 1 (linear) through 5, useful if you're not sure a quadratic is the right shape and want to compare fits at different degrees on the same data.