Find the quadratic function that is best fit for f(x) defined by the table below Xf(x)0024014159863595864071010,009

Expert Answer to "Find the quadratic function that is best fit..."

Josalynn

Answered question

2020-10-18

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

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

Answer & Explanation

toroztatG

Skilled2020-10-19Added 98 answers

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) $a = 100.29$
$b = 2.04$
$c = 1.25$
$y = 100.29 x^{2} - 2.04 x + 1.25$

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