Fitting Data
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Establish the relationship between the two variables for the given table of values.
\begin{align}
&y\ \text{values are increasing}\\
&\therefore\ y=kx^{n}\\
&y=8,\ x=2 \rightarrow 8=k \times 2^{n}....(1)\\
&y=32,\ x=4 \quad 32=k \times 4^{n}...(2)\\
&\begin{aligned}
(2) \div(1) \quad 4&=4^{n} \div 2^{n}\\
4&=2^{2 n} \div 2^{n}\\
2^{2}&=2^{n}\\
\therefore\ n&=2\\
\text{In (1)}\quad 8&=k \times 2^{2}\\
4k&=8 \rightarrow k=2\\
\therefore\ y&=2x^{2}
\end{aligned}
\end{align}
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