Ans: (a) $(18,-13)$ (b) $g(x)=\dfrac{-1}{3}(x-18)^2$ (c) Reflection with respect to the $y$-axis
-
$\begin{array}{rcl}
y & = & f(x) \\
& = & \dfrac{-1}{3} x^2 +12x -121 \\
& = & \dfrac{-1}{3}( x^2 -36x) -121 \\
& = & \dfrac{-1}{3}(x^2 -36x + 18^2 – 18^2) – 121 \\
& = & \dfrac{-1}{3} (x-18)^2 + 108 – 121\\
& = & \dfrac{-1}{3}(x-18)^2 -13
\end{array}$Therefore, the coordinates of the vertex of the graph of $y=f(x)$ are $(18,-13)$.
- From the result of (a), the coordinates of the vertex of the graph of $y=f(x)$ are $(18,-13)$. Hence, to obtain the graph of $y=g(x)$ by translating the graph of $y=f(x)$ vertically, we have to translate the graph of $y=f(x)$ upwards $13$ units. Therefore, we have$\begin{array}{rcl}
y & = & g(x) \\
& = & f(x) + 13 \\
& = & \left(\dfrac{-1}{3}(x-18)^2 -13\right) + 13\\
& = & \dfrac{-1}{3}(x-18)^2
\end{array}$ - Note that$\begin{array}{rcl}
f(-x) & = & \dfrac{-1}{3} (-x)^2 + 12(-x) -121 \\
& = & \dfrac{-1}{3}x^2 -12x -121
\end{array}$Hence, the mentioned transformation is reflecting $y=f(x)$ about $y$-axis.
