Polynomial Regression Calculator

The quadratic polynomial regression model is:

$$y = ax^2 + bx + c$$

This can be rewritten as a multiple regression problem by defining two predictor variables: \(Z_1 = x\) and \(Z_2 = x^2\). The normal equations form a 3×3 linear system:

$$\begin{bmatrix} n & \sum x_i & \sum x_i^2 \\ \sum x_i & \sum x_i^2 & \sum x_i^3 \\ \sum x_i^2 & \sum x_i^3 & \sum x_i^4 \end{bmatrix} \begin{bmatrix} c \\ b \\ a \end{bmatrix} = \begin{bmatrix} \sum y_i \\ \sum x_i y_i \\ \sum x_i^2 y_i \end{bmatrix}$$

The calculator computes all required sums (∑x, ∑x², ∑x³, ∑x⁴, ∑y, ∑xy, ∑x²y) and then applies Cramer's rule to solve for a, b, and c. Each coefficient is the ratio of a modified determinant to the main determinant of the coefficient matrix.

The quadratic coefficient a determines the curvature: a > 0 creates a U-shaped parabola (concave up), while a < 0 creates an inverted U (concave down). The linear coefficient b affects the tilt, and c is the y-intercept. The vertex of the parabola occurs at x = -b/(2a), which represents either the minimum or maximum point of the curve.

R² is computed as 1 - SSE/SST, where SSE is the sum of squared residuals from the quadratic fit and SST is the total sum of squares. If the data truly follows a quadratic pattern, polynomial regression will yield a much higher R² than linear regression.

When analyzing polynomial regression results:

• Coefficient a (quadratic): If a ≈ 0, the data is essentially linear and a straight line suffices. If |a| is large, there is significant curvature. Positive a means the curve opens upward; negative a means it opens downward.
• Coefficient b (linear): Represents the instantaneous rate of change at x = 0. The overall rate of change at any point x is given by the derivative dy/dx = 2ax + b.
• Coefficient c (constant): The predicted Y when X = 0.
• R²: Compare the polynomial R² with the linear R². If the quadratic model substantially improves R², the curvature is significant. If the improvement is negligible, prefer the simpler linear model (parsimony principle).

A warning about overfitting: with 5 data points and 3 parameters, a quadratic model has only 2 degrees of freedom. It will almost always fit well, but this does not mean the quadratic relationship generalizes to new data. With only 3 points, the quadratic model fits perfectly (R² = 1) regardless of the data, which is meaningless. Use polynomial regression when you have theoretical reasons to expect curvature, not just to maximize R².