Challenging +1.8 This AEA question requires understanding multiple function transformations (horizontal stretch, composition with absolute value), interpreting a transformed graph to deduce parameters, and applying inverse transformations. While systematic, it demands strong conceptual grasp of transformations and careful multi-step reasoning beyond standard A-level, though not requiring novel mathematical insight.
Find the coordinates of the points \(P , Q\) and \(R\).
Sketch, on separate diagrams, the graphs of
\(y = \mathrm { f } ( 2 x )\),
\(y = \mathrm { f } ( | x | + 1 )\),
indicating on each sketch the coordinates of any maximum points and the intersections with the \(x\)-axis.
(6)
\begin{figure}[h]
\end{figure}
Figure 2 shows a sketch of part of the curve \(C\), with equation \(y = \mathrm { f } ( x - v ) + w\), where \(v\) and \(w\) are constants. The \(x\)-axis is a tangent to \(C\) at the minimum point \(T\), and \(C\) intersects the \(y\)-axis at \(S\). The line joining \(S\) to the maximum point \(U\) is parallel to the \(x\)-axis.
Find the value of \(v\) and the value of \(w\) and hence find the roots of the equation
$$f ( x - v ) + w = 0$$
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the points $P , Q$ and $R$.
\item Sketch, on separate diagrams, the graphs of
\begin{enumerate}[label=(\roman*)]
\item $y = \mathrm { f } ( 2 x )$,
\item $y = \mathrm { f } ( | x | + 1 )$,\\
indicating on each sketch the coordinates of any maximum points and the intersections with the $x$-axis.\\
(6)
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{f9d3e02c-cef2-435b-9cda-76c43fcac575-5_1015_1464_232_337}
\end{center}
\end{figure}
Figure 2 shows a sketch of part of the curve $C$, with equation $y = \mathrm { f } ( x - v ) + w$, where $v$ and $w$ are constants. The $x$-axis is a tangent to $C$ at the minimum point $T$, and $C$ intersects the $y$-axis at $S$. The line joining $S$ to the maximum point $U$ is parallel to the $x$-axis.
\end{enumerate}\item Find the value of $v$ and the value of $w$ and hence find the roots of the equation
$$f ( x - v ) + w = 0$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2005 Q6 [19]}}