UNIT 5 • CURVE FITTING & STATISTICS

Fitting a Parabola
Quadratic Polynomial Regression

When data exhibits non-linear curvature, Quadratic Polynomial Regression fits a 2nd-degree parabola $y = a + bx + cx^2$ by constructing a 3×3 system of normal equations to minimize squared residual errors!

PARABOLIC 3×3 NORMAL SYSTEMQUADRATIC LEAST SQUARES

y = a + b x + c x^2

\begin{aligned} \sum y &= n a + b \sum x + c \sum x^2 \\ \sum x y &= 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 \end{aligned}

1. Intercept (a)Constant offset elevation at

x = 0

2. Linear Slope (b)Initial linear tangent slope coefficient.

3. Curvature (c)Quadratic term controlling parabola concavity (

c > 0

opens upward).

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Fit a second-degree parabola $y = a + bx + cx^2$ to the 5 points:

x	0	1	2	3	4
y	1.0	1.8	3.3	6.0	9.5

STEP 01 • TABULATE STATISTICAL POWER SUMS

x	y	x²	x³	x⁴	x·y	x²·y
0	1.0	0	0	0	0.0	0.0
1	1.8	1	1	1	1.8	1.8
2	3.3	4	8	16	6.6	13.2
3	6.0	9	27	81	18.0	54.0
4	9.5	16	64	256	38.0	152.0
Sum = 10	Sum = 21.6	Sum = 30	Sum = 100	Sum = 354	Sum = 64.4	Sum = 221.0

STEP 02 • SOLVE 3×3 NORMAL MATRIX SYSTEM

Construct system with $n = 5$ :

5 a + 10 b + 30 c = 21.6

10 a + 30 b + 100 c = 64.4

30 a + 100 b + 354 c = 221.0

a = 1.0229, \quad b = 0.2343, \quad c = 0.4714

FINAL RESULT

Best-Fit Parabola: y = 1.0229 + 0.2343 x + 0.4714 x²

\boxed{y = 1.0229 + 0.2343 x + 0.4714 x^2}

Using quadratic least squares regression, the parabolic model of best fit is y = 1.0229 + 0.2343x + 0.4714x²!

LABORATORY

Interactive Live Calculator & Visualizer

QUADRATIC REGRESSION LABORATORY

Fitting a Parabola Engine

PreviousFitting Straight Lines

Unit 5•23 of 28

NextLeast Squares Method

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 5 • CURVE FITTING & STATISTICS

Fitting a Parabola
Quadratic Polynomial Regression

PARABOLIC 3×3 NORMAL SYSTEMQUADRATIC LEAST SQUARES

y = a + b x + c x^2

\begin{aligned} \sum y &= n a + b \sum x + c \sum x^2 \\ \sum x y &= 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 \end{aligned}

1. Intercept (a)Constant offset elevation at

x = 0

2. Linear Slope (b)Initial linear tangent slope coefficient.

3. Curvature (c)Quadratic term controlling parabola concavity (

c > 0

opens upward).

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Fit a second-degree parabola $y = a + bx + cx^2$ to the 5 points:

x	0	1	2	3	4
y	1.0	1.8	3.3	6.0	9.5

STEP 01 • TABULATE STATISTICAL POWER SUMS

x	y	x²	x³	x⁴	x·y	x²·y
0	1.0	0	0	0	0.0	0.0
1	1.8	1	1	1	1.8	1.8
2	3.3	4	8	16	6.6	13.2
3	6.0	9	27	81	18.0	54.0
4	9.5	16	64	256	38.0	152.0
Sum = 10	Sum = 21.6	Sum = 30	Sum = 100	Sum = 354	Sum = 64.4	Sum = 221.0

STEP 02 • SOLVE 3×3 NORMAL MATRIX SYSTEM

Construct system with $n = 5$ :

5 a + 10 b + 30 c = 21.6

10 a + 30 b + 100 c = 64.4

30 a + 100 b + 354 c = 221.0

a = 1.0229, \quad b = 0.2343, \quad c = 0.4714

FINAL RESULT

Best-Fit Parabola: y = 1.0229 + 0.2343 x + 0.4714 x²

\boxed{y = 1.0229 + 0.2343 x + 0.4714 x^2}

Using quadratic least squares regression, the parabolic model of best fit is y = 1.0229 + 0.2343x + 0.4714x²!

LABORATORY

Interactive Live Calculator & Visualizer

QUADRATIC REGRESSION LABORATORY

Fitting a Parabola Engine

PreviousFitting Straight Lines

Unit 5•23 of 28

NextLeast Squares Method

Fitting a Parabola Quadratic Polynomial Regression

Step-by-Step Worked Example

Best-Fit Parabola: y = 1.0229 + 0.2343 x + 0.4714 x²

Interactive Live Calculator & Visualizer

Fitting a Parabola Engine

NETZ Engine

Fitting a Parabola Quadratic Polynomial Regression

Step-by-Step Worked Example

Best-Fit Parabola: y = 1.0229 + 0.2343 x + 0.4714 x²

Interactive Live Calculator & Visualizer

Fitting a Parabola Engine

Fitting a Parabola
Quadratic Polynomial Regression

Fitting a Parabola
Quadratic Polynomial Regression