Quadratic Regression Calculator

Compute the quadratic polynomial y = ax² + bx + c that best fits your data using the method of least squares. Visualize the fitted curve and data points, obtain R², and analyze residuals.

Data Points (X, Y)
#XY
Examples:
? Perfect parabola: y = x²
? Projectile motion
? Revenue optimization
? Scattered with noise
? Reset default
Private & secure: All computations happen locally in your browser. No data is uploaded to any server.

What is Quadratic Regression?

Quadratic regression is a form of polynomial regression where the relationship between the independent variable x and the dependent variable y is modeled as a second-degree polynomial: y = ax² + bx + c + ε. The least squares method minimizes the sum of squared residuals (vertical distances between observed y-values and predicted y-values). This technique is widely used when the data exhibits a parabolic trend — increasing then decreasing, or a single extremum (peak or valley).

Mathematical Derivation

Given n points (xᵢ, yᵢ), we minimize S = Σ (yᵢ - (axᵢ² + bxᵢ + c))². Taking partial derivatives leads to a system of three linear equations (normal equations): 

Σxᵢ⁴ a + Σxᵢ³ b + Σxᵢ² c = Σ xᵢ² yᵢ
Σxᵢ³ a + Σxᵢ² b + Σxᵢ c = Σ xᵢ yᵢ
Σxᵢ² a + Σxᵢ b + n·c = Σ yᵢ

Solving this system (via Gaussian elimination) yields the coefficients a, b, c. The coefficient of determination R² = 1 − (SSres/SStot) measures how well the model explains the variance.

Applications & Real‑World Use Cases

  • Physics: Projectile motion (height vs. horizontal distance), free fall under gravity (t² relationship).
  • Economics & Finance: Revenue optimization where profit follows concave parabola, cost-volume-profit analysis.
  • Engineering: Stress-strain behavior, calibration curves, sensor response.
  • Biology: Drug dosage response (dose-effect curves with maximum efficacy).
  • Machine Learning: Baseline polynomial regression for feature engineering.
Case Study: Projectile Motion

A physics student measures the height of a ball launched vertically: data points (time in seconds, height in meters): (0, 1.2), (0.5, 4.1), (1.0, 5.8), (1.5, 5.9), (2.0, 4.2), (2.5, 1.1). Quadratic regression yields a downward opening parabola modeling the trajectory. The vertex indicates maximum height and time of apex, while the coefficient 'a' relates to gravitational acceleration. Our calculator instantly provides the best-fit parabola, helping students verify theoretical predictions.

Interpretation of Coefficients & R²

Parameter Meaning
a (quadratic) Curvature; if a > 0 → convex (upward), if a < 0 → concave (downward). Large |a| means steeper curve.
b (linear) Slope at x=0; influences shift and tilt.
c (intercept) Predicted y when x = 0.
R² Proportion of variance explained. Values close to 1 indicate excellent fit; 0.7–0.9 suggests good fit; below 0.5 indicates poor explanatory power.

Limitations & Assumptions

Quadratic regression assumes a parabolic relationship; it is sensitive to outliers. Overfitting may occur with limited data, and extrapolation beyond the data range can be unreliable. Always check residual plots for patterns. For highly non‑parabolic data, higher-degree polynomials or other models might be more appropriate.

Frequently Asked Questions

At least 3 non-collinear points (distinct x values). More points improve statistical reliability. Our tool supports up to 50 points.

In standard OLS, R² ranges from 0 to 1. Negative R² would indicate a model worse than the mean baseline, but our implementation ensures R² ≥ 0.

Yes, after computing coefficients, you can manually plug any x into y = ax² + bx + c. Future versions may include a prediction feature.

Adjusted R² penalizes the addition of predictors; for quadratic model with 3 coefficients, it’s useful when comparing models.

This quadratic regression tool implements robust numeric methods (Gaussian elimination with partial pivoting) and follows statistical best practices. All formulas are derived from standard textbooks (Montgomery & Peck, "Introduction to Linear Regression Analysis"). Reviewed and validated against R's `lm()` function for accuracy. Last updated: June 2026.

References: Weisstein, Eric W. "Least Squares Fitting—Polynomial." MathWorld; NIST/SEMATECH e-Handbook of Statistical Methods.