9.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{bc7fb499-9462-40ae-88f4-87fc60f6a005-18_542_551_212_790}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}

The curve $C$ has equation $y = \mathrm { f } ( x )$, where

$$f ( x ) = x ^ { 3 } \sqrt { 4 x + 7 } \quad x \geq - \frac { 7 } { 4 }$$
\begin{enumerate}[label=(\alph*)]
\item Show that

$$\mathrm { f } ^ { \prime } ( x ) = \frac { k x ^ { 2 } ( 2 x + 3 ) } { \sqrt { 4 x + 7 } }$$

where $k$ is a constant to be found.

The point $P$, shown in Figure 3, is the minimum turning point on $C$.
\item Find the coordinates of $P$.
\item Hence find the range of the function g defined by

$$\operatorname { g } ( x ) = - 4 \mathrm { f } ( x ) \quad x \geq - \frac { 7 } { 4 }$$

The point $Q$ with coordinates $\left( \frac { 1 } { 2 } , \frac { 3 } { 8 } \right)$ lies on $C$.
\item Find the coordinates of the point to which $Q$ is mapped when $C$ is transformed to the curve with equation

$$y = 40 f \left( x - \frac { 3 } { 2 } \right) - 8$$
\end{enumerate}

\hfill \mbox{\textit{SPS SPS SM Pure 2025 Q9 [9]}}