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 $