Find the quadratic function that is best fit for f(x) defined by the table below

Question

$\begin{array}{cc} X & f (x) \\ 0 & 0 \\ 2 & 401 \\ 4 & 1598 \\ 6 & 3595 \\ 8 & 6407 \\ 10 & 10, 009 \end{array}$

Answer 1

Step 1 Let the quadratic equation be $f (x) = a x^{2} + b x + c$ Since there are 3 unlnowns a, b, c Let us use the least square method for the curve fit

Step 2

$\begin{array}{cc} X & y & x y & x^{2} & x^{2} y & x^{3} & x^{4} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 401 & 802 & 4 & 1604 & 8 & 16 \\ 4 & 1598 & 6392 & 16 & 25568 & 64 & 256 \\ 6 & 3595 & 21570 & 36 & 129420 & 216 & 1296 \\ 8 & 6407 & 51256 & 64 & 410048 & 512 & 4096 \\ 10 & 10009 & 100090 & 100 & 1000900 & 1000 & 10000 \\ \sum x = 30 & \sum y = 22010 & \sum x y = 180110 & \sum x^{2} = 220 & \sum x^{2} y = 1567540 & \sum x^{3} = 1800 & \sum x^{4} = 1 \end{array}$

Step 3 number of data $= n = 6$ $\sum y = a \sum x^{2} + b \sum x + n c$
$22010 = 220 a + 30 b + 6 c \dots . (1)$
$\sum x y = a \sum x^{3} + b \sum x^{2} + c \sum x$
$180110 = 1800 a + 220 b + 30 c \dots . (2)$
$\sum x^{2} y = a \sum x^{4} + b \sum x^{3} + c \sum x^{2}$
$1567540 = 15664 a + 1800 b + 220 c \dots . (3)$ Step 4 Solve equation (1), (2) and (3) $a = 100.29$
$b = 2.04$
$c = 1.25$
$y = 100.29 x^{2} - 2.04 x + 1.25$

Find the quadratic function that is best fit for f(x) defined by the table below

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