Quadratic Regression Calculator

The vertex is the answer, and it is usually missing

Nobody fits a parabola to a dose-response curve because they want three coefficients. They want to know which dose is best.

That is −b/2c, exactly. It is where the derivative b + 2cx equals zero, so it is the single point at which the fitted relationship stops improving and starts getting worse. One division, and almost no calculator prints it.

On the fertiliser preset it lands at 173.93 kg/ha. The experiment applied 0, 40, 80, 120, 160, 200 and 240. The optimum is none of them — it sits between the two doses on either side of the peak, which is precisely the information a table of results cannot give you.

The comparison that matters is curve against curve. The highest number in that y column is 82, observed at 160. The curve’s value at 160 is 81.64 and its maximum is 81.90 at 173.93, so the 82 is a point sitting 0.36 above the fitted line — noise, not a better dose. Reading the largest observation as the optimum is reading the noise as the signal.

The units of c tell you how sharp the optimum is. A c close to zero makes a shallow curve where being some way off the vertex costs almost nothing; a large one makes a sharp peak where it costs a great deal. The tool prints 2c, the second derivative, because that is the quantity with the direct interpretation.

And the vertex is an estimate, with its own uncertainty. It is a ratio of two fitted coefficients, so its error is larger than either of theirs and grows quickly as c approaches zero. Treat 173.93 as a region rather than a number, and read the confidence interval logic as applying to it too.

A parabola always turns, whether your data did or not

This is the failure mode that makes quadratic fits dangerous in a way linear fits are not, and it is structural rather than a matter of data quality.

Every parabola reverses exactly once. There is no quadratic that rises forever, and none that flattens out. Fit one to data that is merely decelerating and you have asserted a peak, somewhere, whether or not anything in your measurements suggested one.

The ad-spend preset is built to show it. Revenue rises across every spend level from 5 to 40 and never falls. The fitted vertex is at 42.18 — just past the end of the data — so the model predicts that spending 45 earns less than spending 40, and 80 earns 161, below every value ever observed.

Nothing in that data supports the reversal. It comes from the shape. A curve that flattens is indistinguishable from a curve that is about to turn over, across a range that stops before the turn, and the parabola resolves the ambiguity by assumption.

So the tool states where the vertex sits. Inside the measured range, the reversal is something your data saw and the vertex is a finding. Outside it, the reversal is the model’s arithmetic, and any prediction past it should be discarded rather than discounted.

When the curve only flattens, fit something that only flattens. A logarithmic fit rises forever at a decreasing rate and never turns, which is usually the honest shape for diminishing returns. Choosing between them is a claim about the process, not about which scores higher.

A higher R² here is arithmetic, not evidence

The quadratic will beat the line on your data. It will beat it on any data. That is a theorem, not a result.

Adding a term can never raise the residual sum of squares, because the straight line is a quadratic with c = 0 and least squares would have chosen it if nothing better existed. R² is therefore non-decreasing in model complexity by construction.

Which makes a small improvement worthless as evidence. Going from 0.910 to 0.914 is what fitting an extra parameter does to noise. It is not a sign that the relationship curves.

A large gap is a different matter. The speed-and-economy preset scores 0.068 as a straight line and 0.998 as a parabola. A line explains seven percent of a relationship that is almost perfectly described by a curve — because the two halves of the data run in opposite directions and a line splits the difference.

That is also why a near-zero correlation is not evidence of no relationship. A symmetric rise and fall has a Pearson correlation close to zero by construction, and a coefficient of determination to match. The relationship can be exact and the linear summary will still report nothing.

Adjusted R² is the figure that can fall. It charges for the parameter, so it declines when the extra term buys less than a useless variable would on average. Between two nested models it is the comparison worth making — and the residual plot is worth more than either, because it showswhere the line is wrong rather than only how much.

What a, b and c actually mean

The three coefficients are reported everywhere and interpreted almost nowhere, which matters because two of them mean something quite specific and one of them frequently means nothing at all.

c is half the second derivative. It is the curvature, and it is constant everywhere on the curve — the one thing a parabola asserts about the whole domain. On the projectile preset 2c comes out at −9.662, which is the fitted acceleration in metres per second squared, against a true gravitational value of −9.807. A fit to nine noisy heights recovers g to within 1.5%.

b is the slope at x = 0, and only there. Not the average slope, not the trend. On the fertiliser data b = 0.4607 is the marginal yield of the first kilogram of fertiliser, which is meaningful because zero is in the range. On data running from 2020 to 2026 it would be the slope in the year zero, which is not a quantity anybody wants.

a is the fitted value at x = 0, with the same caveat. When zero is far outside the data it is an extrapolation, sometimes an absurd one. Centring x on its mean before fitting makes a and b the value and slope at the centre of the data, which is almost always the more useful parameterisation.

Centring also fixes a numerical problem. With large x the normal equations involve Σx⁴, and for x near 2,026 that is around 1.7×10¹³ per point. The system becomes ill-conditioned and the coefficients lose precision long before anything looks wrong. Subtracting the mean first costs nothing and removes the problem.

The vertex survives all of this. It moves with the coordinates, so if you centre x you add the mean back. But unlike a and b, it means the same thing regardless of where you put the origin, which is another reason to report it rather than the raw coefficients.

The roots, and when they are meaningless

The tool reports where the curve crosses zero, because for a trajectory that is the landing point and for a profit curve those are the break-even prices.

On the projectile preset the roots are −0.066 and 4.540. The second is the flight time in seconds; the first is the moment the projectile would have left the ground had it started there, and is a fitting artefact of a launch at 1.5 metres rather than a physical event.

That asymmetry is the general rule. One root is usually inside or just beyond the data and means something; the other is often far outside it and means nothing. The fertiliser curve has roots at −74.7 and 422.6 kg/ha, and a negative fertiliser dose is not a quantity.

No real roots is a real answer. When b² − 4ac is negative the curve never reaches zero, and the tool says so rather than returning complex numbers that would be arithmetically correct and practically irrelevant.

The stable formula matters more than it looks. The textbook quadratic formula loses precision badly when 4ac is small relative to b²: one root comes out as the difference of two nearly equal numbers and can lose most of its significant digits. The tool forms the larger root first and reaches the other through the product of roots, which is exact where the naive form is not.

A root is still an extrapolation unless it is inside your data. It is the point where a fitted curve predicts zero, on a model that was only ever tested over the range you measured. The quadratic formula solves the equation exactly; whether the equation describes reality at that x is a separate question.

When a parabola is the wrong shape

A quadratic is the default for “it curves”, and it is wrong in a few recognisable ways that a fit statistic will not tell you about.

It is symmetric. The curve falls away from its vertex at exactly the same rate on both sides. Most real optima are not like that — under-fertilising and over-fertilising have different costs, and a parabola forces them to match. If your residuals are positive on one side of the peak and negative on the other, that is the symmetry failing.

It has no asymptote. Processes that approach a ceiling need a shape that approaches a ceiling: logistic, Michaelis-Menten, or an exponential approach to a plateau. A parabola imitates one for a while and then turns over.

Higher-degree polynomials are rarely the fix. A cubic adds an inflection, a quartic adds two, and each one buys a higher R² while making the extrapolation wilder. Beyond a cubic you are almost always fitting noise, and the tell is coefficients that change sharply when one point is removed.

Compare families rather than degrees. Linear, quadratic, exponential, logarithmic and power are five genuinely different shapes, and the R² calculator fits all five to the same data at once. Picking among shapes is a more useful question than picking among polynomial degrees.

Let the subject matter decide where it can. A trajectory is a parabola because of physics. A dose-response curve is usually not, and fitting one is a convenience that happens to put the optimum in roughly the right place. Knowing which situation you are in is worth more than any comparison of fit statistics.

Whatever you fit, look at the residuals. Structure left over after a quadratic — a run of same-signed residuals, a widening band — is the model telling you the shape is wrong, and it shows up there long before it shows up in R². That is what the residual calculator is for.

Sources and methodology

References for polynomial least squares and for reading its output.

• NIST/SEMATECH e-Handbook — polynomial models, and the conditioning problem with uncentred x
• Anscombe, “Graphs in Statistical Analysis”, The American Statistician 27(1), 1973 — the second dataset is an exact parabola with a respectable linear R²
• NIST — standard acceleration of gravity, 9.80665 m/s², against the 9.662 the projectile preset recovers
• R² Calculator — five model families on one dataset, with adjusted R²
• Residual Calculator — the plot that shows a wrong shape before a fit statistic does
• How we calculate — methodology

Method. The coefficients come from the exact 3×3 normal equations solved by Cramer’s rule rather than from an iterative fit, because a quadratic is linear in its parameters and a closed form is both faster and exact. The vertex is −b/2c and the roots use the numerically stable pairing, forming the larger-magnitude root first and reaching the other through the product of roots, which avoids the catastrophic cancellation the textbook formula suffers when 4ac is small beside b². The suite checks over 500 random parabolas that the reported vertex is always where the derivative vanishes and that every reported root satisfies the polynomial to a relative residual below 10⁻¹², and pins the presets’ figures: an optimum of 173.93 kg/ha, an ad-spend vertex at 42.18 that forces a predicted fall the data never showed, and a fitted acceleration of −9.662 m/s² from nine projectile heights. That engine is verified on every change against 188 assertions. The count and the per-case breakdown are published on the formula verification page.

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