For relationships that bend once
A quadratic regression fits ŷ = a + bx + cx²: a parabola. The x² term buys exactly one bend. Braking distances, projectile heights, dose-response curves that rise then fall, the cost curve that dips at efficient scale: single-bend relationships are everywhere, and a straight line through them is wrong in a systematic way, overpredicting at both ends and underpredicting in the middle (or the reverse).
Stopping distances (metres) measured at speeds 20 to 70 km/h: 8.5, 17, 31, 46.5, 68, 90.5. A straight line explains a respectable R² = 0.975, which sounds fine until you fit the parabola: ŷ = 0.157 + 0.0457v + 0.01786v², R² = 0.9996. Physics says distance grows with the square of speed, and the fit agrees. At 55 km/h the curve predicts 56.7 m; the straight line, pulled by the ends, would have said about 52.
Reading the coefficients
The c coefficient steers: positive opens the parabola upward (growth accelerating), negative opens it downward (growth peaking then falling). The b and a coefficients lose their simple slope-and-start readings once the curve bends; the honest summary of a quadratic is its picture plus a few predictions at chosen x values. The calculator plots the curve through your points for exactly this reason.
Two traps. First, extrapolation: a parabola accelerates away from the data in both directions, faster than any line, so predictions outside the observed range degrade quickly. Second, reflex: adding the squared term to data that is really straight buys a wiggle that fits noise. Compare adjusted R², and trust the bend when a mechanism suggests one.
Quadratic versus higher curves
Cubic and higher polynomials bend more times and fit training data ever more closely, while meaning ever less. Each added power is a modelling decision that needs a reason. One bend with a physical story, like the stopping distance, is a model. Four bends chasing scatter is a drawing.
Frequently asked questions
How do I know if I need the quadratic term?
Plot the data. A single visible bend, residuals from the straight line that form an arc, or a known mechanism (distance and speed squared) all argue for it. Compare adjusted R-squared values too.
What does a negative x-squared coefficient mean?
The curve opens downward: the outcome rises, peaks, and falls as x grows. The peak sits at x = -b / (2c), readable straight from the equation.
Why does the linear R-squared sometimes look high on curved data?
A rising curve still has a strong upward trend, and the line captures the trend while missing the shape. R-squared rewards the trend. The residual pattern exposes the miss.