How to Use Quadratic Regression Calculator
The Quadratic Regression Calculator fits a parabola to your data in just a few steps:
- Enter your data — Type or paste your (x, y) pairs into the data field, one pair per line, separated by a comma or space. Click Load Sample to explore the Quadratic Regression Calculator with built-in example data.
- Click Calculate — The Quadratic Regression Calculator instantly computes the best-fit equation y = ax² + bx + c using the least squares method.
- Review the coefficients — Check a, b, and c, along with R² to see how well the quadratic curve fits your data.
- Find the vertex — The Quadratic Regression Calculator automatically reports the parabola's vertex, which is the maximum or minimum point of the fitted curve.
- Predict new values — Enter any x in the prediction field; the Quadratic Regression Calculator returns the estimated y immediately.
- Check the residuals table — Each data point is shown with its observed y, predicted ŷ, and residual (y − ŷ) so you can evaluate fit quality.
Use Clear to start over with a fresh dataset.
Formula & Theory — Quadratic Regression Calculator
The Quadratic Regression Calculator fits the model:
y = ax² + bx + c
Coefficients are found by minimising the residual sum of squares:
RSS = Σᵢ (yᵢ − aXᵢ² − bXᵢ − c)²
Setting partial derivatives to zero gives a 3×3 system of normal equations:
[ n Σx Σx² ] [c] [Σy ]
[ Σx Σx² Σx³ ] [b] = [Σxy ]
[ Σx² Σx³ Σx⁴ ] [a] [Σx²y]
The Quadratic Regression Calculator solves this system using Cramer's rule for reliability and speed.
| Symbol | Meaning |
|---|---|
| a | Coefficient of x² (controls parabola width and direction) |
| b | Coefficient of x (controls horizontal shift) |
| c | Constant term (y-intercept) |
| R² | Coefficient of determination |
| Vertex | Point (−b/2a, f(−b/2a)) — max or min of the parabola |
Vertex and Axis of Symmetry
The vertex x-coordinate is:
x_vertex = -b / (2a)
When a > 0 the parabola opens upward (minimum); when a < 0 it opens downward (maximum). The Quadratic Regression Calculator reports both the vertex coordinates automatically.
Use Cases for Quadratic Regression Calculator
The Quadratic Regression Calculator is useful whenever you expect a parabolic relationship between variables:
- Physics experiments — Projectile motion, falling objects under gravity, and many optics problems follow quadratic laws. The Quadratic Regression Calculator extracts the precise coefficients from experimental measurements.
- Economics and cost analysis — U-shaped cost curves and profit maximisation problems are natural fits for the Quadratic Regression Calculator.
- Biology and medicine — Growth and dose-response curves often have an optimal point — exactly what a Quadratic Regression Calculator is designed to find.
- Engineering design — Aerodynamic drag, parabolic antenna shapes, and beam deflection all follow y = ax² + bx + c relationships that the Quadratic Regression Calculator can model.
- Education — Students learning statistics can use the Quadratic Regression Calculator to visually understand least squares fitting and R² as a measure of model quality.
Whenever linear regression under-fits your data and you suspect a curved relationship, the Quadratic Regression Calculator is the fastest way to quantify that curvature.