| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Multiple separate transformations (sketch-based, modulus involved) |
| Difficulty | Moderate -0.3 This is a standard C3 transformations question requiring recall of transformation rules (inverse, modulus, stretch/translation) and careful plotting of key points. Part (a) is trivial reading from the graph. Parts (b)-(d) are routine applications of well-practiced transformation techniques with no novel problem-solving required, though students must track multiple coordinate changes accurately. |
| 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 |
|---|---|---|
| \(ff(-3) = f(0) = 2\) | M1, A1 | A full method of finding \(ff(-3)\). \(f(0)\) is acceptable but \(f(-3)=0\) is not. Accept a solution obtained from two substitutions into the equation \(y = \frac{2}{3}x + 2\) as the line passes through both points. Do not allow for \(y = \ln(x+4)\), which only passes through one of the points. |
| Answer | Marks | Guidance |
|---|---|---|
| Shape | B1 | For the correct shape. Award this mark for an increasing function in quadrants 3, 4 and 1 only. Do not award if the curve bends back on itself or has a clear minimum. |
| (0,-3) and (2,0) | B1 | This is independent to the first mark and for the graph passing through \((0,-3)\) and \((2, 0)\). Accept \(-3\) and \(2\) marked on the correct axes. Accept \(P'=(0,-3)\), \(Q'=(2,0)\) stated elsewhere as long as P' and Q' are marked in the correct place on the graph. There must be a graph for this to be awarded. |
| Answer | Marks | Guidance |
|---|---|---|
| Shape | B1 | Award for a correct shape 'roughly' symmetrical about the \(y\)- axis. It must have a cusp and a gradient that 'decreases' either side of the cusp. Do not award if the graph has a clear maximum \((0,0)\) lies on their graph. Accept the graph passing through the origin without seeing \((0,0)\) marked. |
| (0,0) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Shape | B1 | Shape. The position is not important. The gradient should be always positive but decreasing. There should not be a clear maximum point. |
| (-6,0) or (0,4) | B1 | The graph passes through \((0,4)\) or \((-6,0)\). See part (b) for allowed variations. |
| (-6,0) and (0,4) | B1 | The graph passes through \((0,4)\) and \((-6,0)\). See part (b) for allowed variations. |
| (9 marks) |
**(a)**
$ff(-3) = f(0) = 2$ | M1, A1 | A full method of finding $ff(-3)$. $f(0)$ is acceptable but $f(-3)=0$ is not. Accept a solution obtained from two substitutions into the equation $y = \frac{2}{3}x + 2$ as the line passes through both points. Do not allow for $y = \ln(x+4)$, which only passes through one of the points. | (2 marks)
**(b)**
| | Shape | B1 | For the correct shape. Award this mark for an increasing function in quadrants 3, 4 and 1 only. Do not award if the curve bends back on itself or has a clear minimum. |
| | (0,-3) and (2,0) | B1 | This is independent to the first mark and for the graph passing through $(0,-3)$ and $(2, 0)$. Accept $-3$ and $2$ marked on the correct axes. Accept $P'=(0,-3)$, $Q'=(2,0)$ stated elsewhere as long as **P' and Q' are marked in the correct place on the graph**. **There must be a graph for this to be awarded.** | (2 marks)
**(c)**
| | Shape | B1 | Award for a correct shape 'roughly' symmetrical about the $y$- axis. It must have a cusp and a gradient that 'decreases' either side of the cusp. Do not award if the graph has a clear maximum $(0,0)$ lies on their graph. Accept the graph passing through the origin without seeing $(0,0)$ marked. |
| | (0,0) | B1 | | (2 marks)
**(d)**
| | Shape | B1 | Shape. The position is not important. The gradient should be always positive but decreasing. There should not be a clear maximum point. |
| | (-6,0) or (0,4) | B1 | The graph passes through $(0,4)$ **or** $(-6,0)$. See part (b) for allowed variations. |
| | (-6,0) and (0,4) | B1 | The graph passes through $(0,4)$ **and** $(-6,0)$. See part (b) for allowed variations. | (3 marks)
| (9 marks) |
---3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c78b0245-5c5a-407f-ad8a-602949a76e05-04_620_1095_223_420}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows part of the curve with equation $y = \mathrm { f } ( x ) , x \in \mathbb { R }$.\\
The curve passes through the points $Q ( 0,2 )$ and $P ( - 3,0 )$ as shown.
\begin{enumerate}[label=(\alph*)]
\item Find the value of ff(-3).
On separate diagrams, sketch the curve with equation
\item $y = \mathrm { f } ^ { - 1 } ( x )$,
\item $y = \mathrm { f } ( | x | ) - 2$,
\item $y = 2 \mathrm { f } \left( \frac { 1 } { 2 } x \right)$.
Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2013 Q3 [9]}}