Fitting second degree parabola - Curve fitting calculator

SolutionHelp

1. Fit a straight line y = a + bx using the following data

x	5	4	3	2	1
y	1	2	3	4	5

2. Fit a straight line to the following data on production.

Year	1996	1997	1998	1999	2000
Production	40	50	62	58	60

3. Fit second degree parabola equation

y=a+bx+cx2 $y = a + bx + cx^2$ using the following data.

x	12	10	8	6	4	2
y	6	5	4	3	2	1

4. Fit cubic equation

y=a+bx+cx2+dx3 $y = a + bx + cx^2 + dx^3$ using the following data.

x	12	10	8	6	4	2
y	6	5	4	3	2	1

Example

1. Calculate Fitting second degree parabola - Curve fitting using Least square method
X Y
1 -5
2 -2
3 5
4 16
5 31
6 50
7 73

Solution:
The equation is

y=a+bx+cx2 $y = a + bx + cx^2$ and the normal equations are

∑y=an+b∑x+c∑x2 $sum y = an + b sum x + c sum x^2$

∑xy=a∑x+b∑x2+c∑x3 $sum xy = a sum x + b sum x^2 + c sum x^3$

∑x2y=a∑x2+b∑x3+c∑x4 $sum x^2y = a sum x^2 + b sum x^3 + c sum x^4$

The values are calculated using the following table

x $x$	y $y$	x2 $x^2$	x3 $x^3$	x4 $x^4$	x⋅y $x*y$	x2⋅y $x^2*y$
1	-5	1	1	1	-5	-5
2	-2	4	8	16	-4	-8
3	5	9	27	81	15	45
4	16	16	64	256	64	256
5	31	25	125	625	155	775
6	50	36	216	1296	300	1800
7	73	49	343	2401	511	3577
---	---	---	---	---	---	---
∑x=28 $sum x=28$	∑y=168 $sum y=168$	∑x2=140 $sum x^2=140$	∑x3=784 $sum x^3=784$	∑x4=4676 $sum x^4=4676$	∑x⋅y=1036 $sum x*y=1036$	∑x2⋅y=6440 $sum x^2*y=6440$

Substituting these values in the normal equations

7a+28b+140c=168 $7a+28b+140c=168$

28a+140b+784c=1036 $28a+140b+784c=1036$

140a+784b+4676c=6440 $140a+784b+4676c=6440$

Solving these 3 equations,
Total Equations are

3 $3$

7a+28b+140c=168→(1) $7a+28b+140c=168 -> (1)$

28a+140b+784c=1036→(2) $28a+140b+784c=1036 -> (2)$

140a+784b+4676c=6440→(3) $140a+784b+4676c=6440 -> (3)$

Select the equations

(1) $(1)$ and

(2) $(2)$ , and eliminate the variable

a $a$ .

7a+28b+140c=168 $7a+28b+140c=168$	×4→ $xx 4->$		28a $28a$	+ $+$	112b $112b$	+ $+$	560c $560c$	= $=$	672 $672$
		−
28a+140b+784c=1036 $28a+140b+784c=1036$	×1→ $xx 1->$		28a $28a$	+ $+$	140b $140b$	+ $+$	784c $784c$	= $=$	1036 $1036$

				- $-$	28b $28b$	- $-$	224c $224c$	= $=$	-364 $-364$	→(4) $-> (4)$

Select the equations

(1) $(1)$ and

(3) $(3)$ , and eliminate the variable

a $a$ .

7a+28b+140c=168 $7a+28b+140c=168$	×20→ $xx 20->$		140a $140a$	+ $+$	560b $560b$	+ $+$	2800c $2800c$	= $=$	3360 $3360$
		−
140a+784b+4676c=6440 $140a+784b+4676c=6440$	×1→ $xx 1->$		140a $140a$	+ $+$	784b $784b$	+ $+$	4676c $4676c$	= $=$	6440 $6440$

				- $-$	224b $224b$	- $-$	1876c $1876c$	= $=$	-3080 $-3080$	→(5) $-> (5)$

Select the equations

(4) $(4)$ and

(5) $(5)$ , and eliminate the variable

b $b$ .

-28b-224c=-364 $-28b-224c=-364$	×8→ $xx 8->$		- $-$	224b $224b$	- $-$	1792c $1792c$	= $=$	-2912 $-2912$
		−
-224b-1876c=-3080 $-224b-1876c=-3080$	×1→ $xx 1->$		- $-$	224b $224b$	- $-$	1876c $1876c$	= $=$	-3080 $-3080$

						84c $84c$	= $=$	168 $168$	→(6) $-> (6)$

Now use back substitution method
From (6)

84c=168 $84c=168$

⇒c=16884=2 $=>c=(168)/(84)=2$

From (4)

-28b-224c=-364 $-28b-224c=-364$

⇒-28b-224(2)=-364 $=>-28b-224(2)=-364$

⇒-28b-448=-364 $=>-28b-448=-364$

⇒-28b=-364+448=84 $=>-28b=-364+448=84$

⇒b=84-28=-3 $=>b=(84)/(-28)=-3$

From (1)

7a+28b+140c=168 $7a+28b+140c=168$

⇒7a+28(-3)+140(2)=168 $=>7a+28(-3)+140(2)=168$

⇒7a+196=168 $=>7a+196=168$

⇒7a=168-196=-28 $=>7a=168-196=-28$

⇒a=-287=-4 $=>a=(-28)/(7)=-4$

Solution using Elimination method.

a=-4,b=-3,c=2 $a=-4,b=-3,c=2$

Now substituting this values in the equation is

y=a+bx+cx2 $y = a + bx + cx^2$ , we get

y=-4-3x+2x2 $y = -4 -3x+2x^2$