Quadratic Regression Calculator

Quadratic regression models the relationship between an independent variable x and a dependent variable y using a second-degree polynomial equation of the form: y = ax^2 + bx + c. To determine the best-fit coefficients (a, b, and c), the calculator employs the method of ordinary least squares. This mathematical technique minimizes the sum of the squared residuals—the vertical distances between the actual data points and the predicted points on the parabola. By setting up a system of normal equations derived from partial derivatives, the algorithm solves for the coefficients simultaneously to deliver the most accurate parabolic model possible.

Imagine you are tracking the height of a launched projectile over time. You record the following coordinate pairs (x, y): (1, 5), (2, 8), and (3, 9). First, input these points into the calculator's x and y variables. The calculator sets up the least-squares matrices based on sums of x, x^2, x^3, x^4, y, xy, and x^2y. Solving the system yields the quadratic equation y = -1x^2 + 6x + 0, meaning the peak of the curve occurs at x = 3 with a maximum height of 9 units. This modeled curve allows you to predict future trajectory positions accurately.

To achieve the most reliable results, ensure you input at least three distinct data points that exhibit a clear curved or parabolic trend rather than a straight line. Avoid extrapolating too far beyond your observed data range, as polynomial models can behave unpredictably outside known inputs. Finally, double-check your data for outliers, as extreme values can heavily distort the shape of the fitted parabola.