| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Combined transformation sketches |
| Difficulty | Standard +0.3 This is a standard C3 transformation question requiring application of well-defined rules: horizontal translation and vertical stretch for (a), and reflection of negative parts for absolute value in (b). The transformations are routine textbook exercises with straightforward coordinate changes, making it slightly easier than average. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02w Graph transformations: simple transformations of f(x)1.02x Combinations of transformations: multiple transformations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Shape unchanged (right-hand section need not cross \(x\)-axis) | B1 | Positioning not significant for this mark |
| \(x\)-coordinates of P' and Q' are \(-5\) and \(0\) respectively | B1 | Translation 2 units left; accept \(-5\) on \(x\)-axis for P' with Q' on \(y\)-axis (marked \(-12\)) |
| \(y\)-coordinates of P' and Q' are \(0\) and \(-12\) respectively | B1 | Stretch \(\times 3\) parallel to \(y\)-axis; accept \(-12\) on \(y\)-axis for Q' with P' on \(x\)-axis (marked \(-5\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Curve below \(x\)-axis reflected in \(x\)-axis; curve above unchanged | B1 | Do not accept if clearly rounded off with zero gradient at \(x\)-axis; allow small curvature issues |
| Maximum at \((2, 4)\) | B1 | Both coordinates correct; must have graph with only one maximum; accept 2 and 4 on correct axes |
| Minimum at \((-3, 0)\) | B1 | Both coordinates correct; must have graph; tolerate two cusps if previous mark lost; accept \((0,-3)\) on correct axis |
## Question 2:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Shape unchanged (right-hand section need not cross $x$-axis) | B1 | Positioning not significant for this mark |
| $x$-coordinates of P' and Q' are $-5$ and $0$ respectively | B1 | Translation 2 units left; accept $-5$ on $x$-axis for P' with Q' on $y$-axis (marked $-12$) |
| $y$-coordinates of P' and Q' are $0$ and $-12$ respectively | B1 | Stretch $\times 3$ parallel to $y$-axis; accept $-12$ on $y$-axis for Q' with P' on $x$-axis (marked $-5$) |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Curve below $x$-axis reflected in $x$-axis; curve above unchanged | B1 | Do not accept if clearly rounded off with zero gradient at $x$-axis; allow small curvature issues |
| Maximum at $(2, 4)$ | B1 | Both coordinates correct; must have graph with only one maximum; accept 2 and 4 on correct axes |
| Minimum at $(-3, 0)$ | B1 | Both coordinates correct; must have graph; tolerate two cusps if previous mark lost; accept $(0,-3)$ on correct axis |
---\begin{tikzpicture}[>=latex, thick, x=1.2cm, y=0.7cm]
% Draw horizontal x-axis
\draw[->] (-5.5, 0) -- (6.5, 0) node[below] {$x$};
% Draw vertical y-axis
\draw[->] (0, -6) -- (0, 6.5) node[left] {$y$};
% Origin label
\node[below left] at (0, 0) {$O$};
% Plot the specific cubic curve
\draw[domain=-4.8:5.2, smooth, samples=100]
plot (\x, {0.064*\x*\x*\x + 0.096*\x*\x - 1.152*\x - 2.592});
% Label Point P
\node[above=1pt] at (-3, 0) {$P(-3, 0)$};
% Label Point Q
\node[below=2pt] at (2, -4) {$Q(2, -4)$};
\end{tikzpicture}
Figure 1 shows the graph of equation $y = \mathrm { f } ( x )$.\\
The points $P ( - 3,0 )$ and $Q ( 2 , - 4 )$ are stationary points on the graph.\\
Sketch, on separate diagrams, the graphs of
\begin{enumerate}[label=(\alph*)]
\item $y = 3 \mathrm { f } ( x + 2 )$
\item $y = | \mathrm { f } ( x ) |$
On each diagram, show the coordinates of any stationary points.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2012 Q2 [6]}}