Question

Use data points (15,104) (45,209) and (75,105) to find an appropriate equation relating x and y

Answer 1

\[y = ax^2 + bx + c\]\[a(15)^2+b(15)+c = 104\]\[a(45)^2+b(45)+c = 209\]\[a(75)^2+b(75)+c = 105\]
\[\left[\begin{matrix}15^2 & 15 & 1 \\ 45^2 & 45 & 1 \\ 75^2 & 75 & 1\end{matrix}\right]*\left[\begin{matrix}a \\ b \\ c \end{matrix}\right] = \left[\begin{matrix}104 \\ 209 \\ 105 \end{matrix}\right]\]
Using inverses to solve:
\[\left[\begin{matrix}a \\ b\\ c \end{matrix}\right] = \frac{1}{1800}\left[\begin{matrix}1 & -2 & 1 \\-120 & 180 & -60 \\ 3375 & -2250 & 675\end{matrix}\right] * \left[\begin{matrix}104 \\ 209 \\ 105 \end{matrix}\right]\]
\[\left[\begin{matrix}a \\ b\\ c \end{matrix}\right] = \left[\begin{matrix}-\frac{209}{1800} \\ \frac{157}{15} \\ -\frac{215}{8}\end{matrix}\right]\]
So:
\[y = -\frac{209}{1800}x^2 \frac{157}{15}x -\frac{215}{8}\]

Answer 2

Sorry, the last line should read:
\[-\frac{209}{1800}x^2 + \frac{157}{15}x -\frac{215}{8}\]