Fitting a Parabola (Quadratic Regression)

Fitting a Parabola (or Quadratic Curve Fitting) uses the Least Squares Method to fit a second-degree polynomial equation $y = a + bx + cx^2$ to a set of data points where the trend exhibits curvature.

Normal Equations for Parabola ( $y = a + bx + cx^2$ ):

\sum y = n \cdot a + b \sum x + c \sum x^2

\sum xy = a \sum x + b \sum x^2 + c \sum x^3

\sum x^2 y = a \sum x^2 + b \sum x^3 + c \sum x^4

Where:

$n$ is the number of data points.
$a, b, c$ are the intercept, linear, and quadratic coefficients.

Example of Fitting a Parabola

Given the following dataset of 5 observation points:

X	0	1	2	3	4
Y	1.0	1.8	3.3	6.0	9.5

Step 1: Calculate Power Sums

\sum x = 10, \quad \sum y = 21.6

\sum x^2 = 30, \quad \sum x^3 = 100, \quad \sum x^4 = 354

\sum xy = 64.4, \quad \sum x^2 y = 221.0

Step 2 & 3: Solve Matrix System

a = 0.9857, \quad b = 0.5286, \quad c = 0.4071

Conclusion

\boxed{y = 0.9857 + 0.5286 x + 0.4071 x^2}

Therefore, using Quadratic Curve Fitting, the parabolic equation of best fit for the given dataset is y = 0.9857 + 0.5286x + 0.4071x².

Interactive Fitting a Parabola Calculator

Fitting a Parabola (Quadratic Fit)

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Unit 5•23 of 28

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Fitting a Parabola (Quadratic Regression)

Normal Equations for Parabola ( $y = a + bx + cx^2$ ):

\sum y = n \cdot a + b \sum x + c \sum x^2

\sum xy = a \sum x + b \sum x^2 + c \sum x^3

\sum x^2 y = a \sum x^2 + b \sum x^3 + c \sum x^4

Where:

$n$ is the number of data points.
$a, b, c$ are the intercept, linear, and quadratic coefficients.

Example of Fitting a Parabola

Given the following dataset of 5 observation points:

X	0	1	2	3	4
Y	1.0	1.8	3.3	6.0	9.5

Step 1: Calculate Power Sums

\sum x = 10, \quad \sum y = 21.6

\sum x^2 = 30, \quad \sum x^3 = 100, \quad \sum x^4 = 354

\sum xy = 64.4, \quad \sum x^2 y = 221.0

Step 2 & 3: Solve Matrix System

a = 0.9857, \quad b = 0.5286, \quad c = 0.4071

Conclusion

\boxed{y = 0.9857 + 0.5286 x + 0.4071 x^2}

Therefore, using Quadratic Curve Fitting, the parabolic equation of best fit for the given dataset is y = 0.9857 + 0.5286x + 0.4071x².

Interactive Fitting a Parabola Calculator

Fitting a Parabola (Quadratic Fit)

PreviousFitting Straight Lines

Unit 5•23 of 28

NextLeast Squares Method

Fitting a Parabola (Quadratic Regression)

Example of Fitting a Parabola

Step 1: Calculate Power Sums

Step 2 & 3: Solve Matrix System

Conclusion

Interactive Fitting a Parabola Calculator

Fitting a Parabola (Quadratic Fit)

NETZ Engine

Fitting a Parabola (Quadratic Regression)

Example of Fitting a Parabola

Step 1: Calculate Power Sums

Step 2 & 3: Solve Matrix System

Conclusion

Interactive Fitting a Parabola Calculator

Fitting a Parabola (Quadratic Fit)