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If you film a basketball shot in slow motion and track its position every tenth of a second, the points don't fall along a straight line — they form an arching curve.
Real-world data is rarely perfect, so how do we find the exact mathematical curve that best matches a scatter plot of curved data?

What is Quadratic Regression?

A quadratic function is an equation in the standard form y=ax2+bx+cy = ax^2 + bx + c, whose graph creates a U-shaped curve called a parabola.
When data curves upward or downward with a single peak or valley, quadratic regression is the statistical process of finding the values of aa, bb, and cc that create the best-fitting parabola through those data points.
In 1801, mathematician Carl Friedrich Gauss used this principle of minimizing vertical errors — known as the least squares method — to successfully predict the orbit of the dwarf planet Ceres after astronomers lost sight of it.
How do modern graphing calculators and tools perform this calculation in seconds?

Step-by-Step Technology Workflow

Whether you use a handheld graphing calculator or online graphing software, finding a quadratic regression equation follows three simple steps.
In a handheld calculator, enter your independent variable in list L1L_1 and dependent variable in L2L_2, then calculate QuadReg.
In modern graphing tools, create a table with columns x1x_1 and y1y_1, then type the regression statement y1∼ax12+bx1+cy_1 \sim a x_1^2 + b x_1 + c using the tilde symbol (∼)(\sim) to represent 'is modeled by'.