| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Composite transformation sketch |
| Difficulty | Standard +0.3 This is a standard C3 transformations question requiring sketches of |f(x)| and f(x+1) with asymptotes, plus routine algebraic work finding intercepts and an inverse function. The transformations are straightforward applications of standard rules, and the inverse function calculation is a typical textbook exercise. Slightly easier than average due to the guided structure and standard techniques throughout. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) Graph with asymptote \(y = 2\), vertical asymptote \(x = 1\), passes through \((0, q)\) and \((p, 0)\) | M1 A1 | |
| (ii) Graph with asymptote \(y = 4\), vertical asymptote \(x = 0\), passes through \((p - 1, 0)\) | M2 A2 | |
| (b) \(y = 0 \Rightarrow 2x - 1 = 0 \Rightarrow x = \frac{1}{2} \therefore \rho = \frac{1}{2}\) | M1 A1 | |
| \(x = 0 \Rightarrow y = 1 \therefore q = 1\) | B1 | |
| (c) \(y = \frac{2x - 1}{x - 1}\), \(y(x - 1) = 2x - 1\) | M1 | |
| \(xy - y = 2x - 1\), \(x = \frac{y - 1}{y - 2}\) | M1 | |
| \(\therefore f^{-1}(x) = \frac{x - 1}{x - 2}\) | A1 | (12 marks) |
**(a)(i)** Graph with asymptote $y = 2$, vertical asymptote $x = 1$, passes through $(0, q)$ and $(p, 0)$ | M1 A1 |
**(ii)** Graph with asymptote $y = 4$, vertical asymptote $x = 0$, passes through $(p - 1, 0)$ | M2 A2 |
**(b)** $y = 0 \Rightarrow 2x - 1 = 0 \Rightarrow x = \frac{1}{2} \therefore \rho = \frac{1}{2}$ | M1 A1 |
$x = 0 \Rightarrow y = 1 \therefore q = 1$ | B1 |
**(c)** $y = \frac{2x - 1}{x - 1}$, $y(x - 1) = 2x - 1$ | M1 |
$xy - y = 2x - 1$, $x = \frac{y - 1}{y - 2}$ | M1 |
$\therefore f^{-1}(x) = \frac{x - 1}{x - 2}$ | A1 | (12 marks)6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3db6c0d8-2c8a-47a2-8c98-13fa191320d0-3_727_1006_244_356}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the curve with equation $y = \mathrm { f } ( x )$. The curve crosses the axes at $( p , 0 )$ and $( 0 , q )$ and the lines $x = 1$ and $y = 2$ are asymptotes of the curve.
\begin{enumerate}[label=(\alph*)]
\item Showing the coordinates of any points of intersection with the axes and the equations of any asymptotes, sketch on separate diagrams the graphs of
\begin{enumerate}[label=(\roman*)]
\item $y = | \mathrm { f } ( x ) |$,
\item $y = 2 \mathrm { f } ( x + 1 )$.
Given also that
$$\mathrm { f } ( x ) \equiv \frac { 2 x - 1 } { x - 1 } , \quad x \in \mathbb { R } , \quad x \neq 1$$
\end{enumerate}\item find the values of $p$ and $q$,
\item find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q6 [12]}}