Fitting Data
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Given \(y\,\,\alpha \,\,{x^n}\) and the data in the table, establish the relationship between the two variables.
$$
\begin{aligned}y &\propto x^{n}\\
y&=k x^{n}\\
12&=k \times 2^{n} {\color{red}\ldots (1)}\\
48&=k \times 4^{n} {\color{red}\ldots (2)}\\
{\color{red}(2)\div{\color{red} (1)} } \quad 4&=4^{n} \div 2^{n}\\
4&=2^{2 n} \div 2^{n}\\
2^{2}&=2^{n}\\
\therefore\ n&=2\\
\text{In } {\color{red}(1)}\ 12&=k \times 2^{2}\\
\therefore\ k&=3\\
\therefore\ y&=3 x^{2}
\end{aligned}
$$
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