Quadratic Best Fit Through Given Points

Find the quadratic equation through the origin that is a best fit for these three points: $(1, 1)$, $(2, 5)$, and $(-1, -2)$.

In this problem, we are asked to find the quadratic equation that best fits a set of specific points, including the origin. This involves understanding how to construct a quadratic equation given certain conditions, which in this scenario, include the requirement for the curve to pass through the origin and best fit the other given points. One approach is to assume a general form for the quadratic equation and then determine the unknown coefficients by solving a system of equations derived from the conditions specified by the points.

The key concept here is utilizing systems of linear equations to determine the coefficients of the quadratic curve. By substituting the given points into the assumed equation, you generate multiple linear equations which can then be solved using methods such as substitution, elimination, or matrix operations. This exercise emphasizes understanding how fitting a curve to data points can be formalized as solving a linear system, a fundamental concept in linear algebra and statistics.

Moreover, working through problems like this highlights the importance of least squares fitting in real-world applications where perfect fits through all data points are usually not achievable. This concept extends to fitting high-degree polynomial functions and understanding trade-offs in complexity, highlighting issues like overfitting in statistical models.