Quadratic Regression Calculator
Find the best-fit quadratic equation (y = a + bx + cx²) for your data points
Enter Data Points
Number of significant figures to display
Enter at least 3 data points (both x and y coordinates) to calculate quadratic regression.
Currently entered: 4 valid points
Scatter Plot with Quadratic Fit
Quadratic Regression Results
Error in calculation: Cannot access 'u' before initialization
Step-by-Step Example
Example Data
Data Points: (1, 8), (2, 4), (3, 6), (4, 5)
Goal: Find the quadratic equation y = a + bx + cx² that best fits this data
Solution Process
1. Set up the system of normal equations using least squares method
2. Calculate sums: Σx, Σy, Σx², Σx³, Σx⁴, Σxy, Σx²y
3. Solve the 3×3 matrix system using Gaussian elimination
4. Extract coefficients: a (constant), b (linear), c (quadratic)
5. Calculate R² to measure goodness of fit
R² Interpretation
Coefficient Meanings
Coefficient a
• y-intercept
• Value of y when x = 0
• Vertical shift of parabola
Coefficient b
• Linear component
• Affects the slope at x = 0
• Horizontal shift of parabola
Coefficient c
• Quadratic component
• Controls parabola curvature
• If c > 0: opens upward
• If c < 0: opens downward
Quick Tips
Minimum 3 points needed for quadratic regression
More points generally improve accuracy
Check R² value for model quality
Use for data with curved relationships
Understanding Quadratic Regression
What is Quadratic Regression?
Quadratic regression finds the best-fitting parabola (quadratic curve) for a set of data points. The model has the form y = a + bx + cx², where a, b, and c are coefficients determined using the least squares method.
When to Use
- •Data shows a curved, parabolic relationship
- •Linear regression doesn't fit well
- •Physics problems with acceleration
- •Economics: cost/revenue optimization
Mathematical Method
The least squares method minimizes the sum of squared residuals. We solve a system of three normal equations derived from the partial derivatives of the sum of squared errors with respect to each coefficient.