14.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ce06b37a-aa57-4256-bec8-7277c8a9fc65-40_611_1214_219_548}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where
$$f ( x ) = ( x - 2 ) ^ { 2 } ( 2 x + 1 ) , \quad x \in \mathbb { R }$$
The curve crosses the \(x\)-axis at \(\left( - \frac { 1 } { 2 } , 0 \right)\), touches it at \(( 2,0 )\) and crosses the \(y\)-axis at ( 0,4 ). There is a maximum turning point at the point marked \(P\).
- Use \(\mathrm { f } ^ { \prime } ( x )\) to find the exact coordinates of the turning point \(P\).
A second curve \(C _ { 2 }\) has equation \(y = \mathrm { f } ( x + 1 )\).
- Write down an equation of the curve \(C _ { 2 }\) You may leave your equation in a factorised form.
- Use your answer to part (b) to find the coordinates of the point where the curve \(C _ { 2 }\) meets the \(y\)-axis.
- Write down the coordinates of the two turning points on the curve \(C _ { 2 }\)
- Sketch the curve \(C _ { 2 }\), with equation \(y = \mathrm { f } ( x + 1 )\), giving the coordinates of the points where the curve crosses or touches the \(x\)-axis.