Fit a second degree parabolic curve to the following data:

x	1	2	3	4	5	6	7	8	9
y	2	6	7	8	10	11	11	10	9

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written 7.8 years ago by

We write X such that $ \sum X=0 $ . Here N=9 (odd)

x	$X_i$	$Y_i$	$X_i^2$	$X_i^3$	$X_i^4$	$X_iY_i$	$X_i^2Y_i$
1	-4	2	16	-64	256	-8	32
2	-3	6	9	-27	81	-8	54
3	-2	7	4	-8	16	-14	28
4	-1	8	1	-1	1	-8	8
5	0	10	0	0	0	0	0
6	1	11	1	1	1	11	11
7	2	11	4	8	16	22	44
8	3	10	9	27	81	30	90
9	4	9	16	64	256	36	144
N=9	$\sum X_i=0$	$ \sum Y_i = 74$	$ \sum X_i^2=60$	$\sum X_i^3=0$	$\sum X_i^4=708$	$\sum X_iY_i=51$	$\sum X_i^2Y_i=411$

{Here, we made the equation } We use the method called “the method of least squares”.

The Equation of parabola is $y=a+bx+cx^2$

Hence, the normal equations are

$ \sum Y_i \;=\; Na + b \sum X_i + c \sum X_i^2 \\ \; \\ \; \\ \sum X_i Y_i \;=\; a \sum X_i + b \sum X_i^2 + c \sum X_i^3 \\ \; \\ \; \\ \sum X_i^2Y_i \;=\; a \sum X_i^2 + b \sum X_i^3 + c \sum X_i^4 $

[The principle of least squares states that the parabola should be such that the distances of the given points from the parabola measured along the y axis must be minimum].

$ \therefore 74=9a+b(0)+60c \; \; \therefore 9a+60c=74 \; \; \ldots (i) \\ \; \\ 51=a(0)+60b+0c \; \; \therefore 60b=51 \; \; \; \therefore b \;=\; \dfrac{51}{60} \;=\; 0.85 \\ \; \\ 411=60a+0b+708c \; \; \therefore 60a+708c=411 \; \; \; \; \ldots (ii) $

Solving (i) and (ii) simultaneously, we get
a=10.004 , c=-0.267

The Equation of parabola is therefore,

$ y=10.004+0.85X-0.267X^2 \\ \; \\ =10.004+0.85(x-5)-0.267(x-5)^2 \\ \; \\ =10.004+0.85x-4.25-0.267(x^2-10x+25) \\ \; \\ =10.004+0.85x-4.25-0.267x^2+2.67x-6.675 \\ \; \\ \therefore y \;=\; =-0.921+3.52x-0.267x^2 $

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