Quadratic Regression Calculator

The Quadratic Regression Calculator fits a second-degree polynomial $$y = ax^2 + bx + c$$ to five data points using the least squares method. Enter paired (x, y) values and a prediction point, and the calculator determines the best-fit parabola, coefficient of determination (R²), vertex coordinates, and predicted value.

Quadratic regression is appropriate when the relationship between variables follows a curved, parabolic pattern rather than a straight line. Common applications include projectile motion, profit optimization, dose-response curves in pharmacology, and modeling phenomena with a single maximum or minimum.

The least squares method minimizes the sum of squared residuals: $$\sum_{i=1}^{n}(y_i - ax_i^2 - bx_i - c)^2$$. Taking partial derivatives with respect to $$a$$, $$b$$, and $$c$$ and setting them to zero yields a system of three normal equations that can be expressed in matrix form and solved using Cramer's rule or matrix inversion.

The normal equations form a 3×3 linear system: $$\begin{pmatrix} \sum x_i^4 & \sum x_i^3 & \sum x_i^2 \\ \sum x_i^3 & \sum x_i^2 & \sum x_i \\ \sum x_i^2 & \sum x_i & n \end{pmatrix} \begin{pmatrix} a \\ b \\ c \end{pmatrix} = \begin{pmatrix} \sum x_i^2 y_i \\ \sum x_i y_i \\ \sum y_i \end{pmatrix}$$. The determinant of the coefficient matrix must be nonzero for a unique solution to exist.

The vertex of the fitted parabola is at $$x_v = -\frac{b}{2a}$$, $$y_v = a x_v^2 + b x_v + c$$. When $$a > 0$$, the parabola opens upward and the vertex is a minimum; when $$a < 0$$, it opens downward and the vertex is a maximum. The vertex has practical significance in optimization problems such as finding the price that maximizes revenue or the angle that maximizes projectile range.

The coefficient of determination $$R^2 = 1 - \frac{SS_{res}}{SS_{tot}}$$ measures goodness of fit. With three parameters fitted to five points, R² may be high simply due to flexibility. Adjusted R² accounts for the number of parameters: $$R^2_{adj} = 1 - \frac{(1 - R^2)(n - 1)}{n - p - 1}$$ where $$p$$ is the number of predictors. Always compare quadratic fit against linear and higher-order alternatives to select the most parsimonious model that adequately describes the data.