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# In the science lab, Wanda Ngo measures the following values of $x$ and $y$ :\$\begin{array}{ll}\frac{x}{11} & \frac{y}{341} \\ 18 & 411 \\ 23 & 471\end{array}$ Wanda wants to know whether the function could be linear. Tell her the answer, and how you decided. If it is linear, find the particular equation. If it is not linear, assume it is quadratic and find the particular equation.

Solution

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If the function is linear, then the slope will always be the same.

If we calculate the slope between points 1 and 2:

$m_{1,2} = \frac{411-341}{18-11} = 10$

Between points 2 and 3:

$m_{2,3} = \frac{471-411}{23-18} = 12$

As the slopes change, then the function \textcolor[RGB]{120,0,0} cannot be linear. We assume, therefore, it is quadratic in the form

$y = ax^2+bx+c$

Substituting the given points into the equation gives us the following system of equations:

\begin{align*} 121a+11b+c&=341\\ 324a+18b+c&=411\\ 529a+23b+c&=471 \end{align*}

which we solve through diagonalization:

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