Q.Find an equation of the parabolic form $y = ax^2 + bx + c$ passing through $(0, 0)$, $(1, 1)$ and $(2, 20)$ using Lagranges Interpolation.

Apply Lagrange's formula to $(0,0),(1,1),(2,20)$; the first term is zero, and the rest simplify to $y = 9x^2 - 8x$.

Lagrange's interpolation formula for the points $x_0=0,\ x_1=1,\ x_2=2$ with $y_0=0,\ y_1=1,\ y_2=20$:

$y = y_0\frac{(x-x_1)(x-x_2)}{(x_0-x_1)(x_0-x_2)} + y_1\frac{(x-x_0)(x-x_2)}{(x_1-x_0)(x_1-x_2)} + y_2\frac{(x-x_0)(x-x_1)}{(x_2-x_0)(x_2-x_1)}.$

Term 1 ($y_0 = 0$): vanishes.

Term 2 ($y_1 = 1$):

$1 \cdot \frac{(x-0)(x-2)}{(1-0)(1-2)} = \frac{x(x-2)}{(1)(-1)} = -(x^2 - 2x) = -x^2 + 2x.$

Term 3 ($y_2 = 20$): …