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

Point 1
Point 2
Point 3
Point 4
Point 5
Point 6
Point 7
Point 8
Point 9
Point 10

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

xy
Data Points
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

R² ≥ 0.90
Excellent fit
Very strong relationship
0.70 ≤ R² < 0.90
Good fit
Strong relationship
0.50 ≤ R² < 0.70
Moderate fit
Moderate relationship
R² < 0.50
Poor fit
Weak relationship

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.

System of Equations

na + b∑x + c∑x² = ∑y
a∑x + b∑x² + c∑x³ = ∑xy
a∑x² + b∑x³ + c∑x⁴ = ∑x²y