| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch translations and stretches/reflections on separate diagrams |
| Difficulty | Moderate -0.3 This is a standard transformations question requiring students to apply three basic transformations (reflection in x-axis, horizontal stretch, horizontal translation) to given key points. While it requires careful tracking of multiple points through each transformation, these are routine C1/C2 techniques with no problem-solving or novel insight needed—slightly easier than average due to its mechanical nature. |
| Spec | 1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| W shape | B1 | |
| Crosses \(x\)-axis at \((0,0)\) and \((6,0)\) only | B1 | Accept if curve passes through origin and \((6,0)\); condone \((0,6)\) marked on \(y\)-axis |
| All 3 turning points correct: maximum \((3,-2)\), minima \((1,-6)\) and \((5,-6)\) | B1 | Do not allow coordinates transposed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Same shape passing through (or starting at) the origin; two maxima and one minimum in Quadrant 1 | B1 | |
| Cuts \(x\)-axis at \((12,0)\) only | B1 | See part (a) for allowable alternatives |
| All 3 turning points correct: minimum \((6,2)\), maxima \((2,6)\) and \((10,6)\) | B1 | Do not allow coordinates transposed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Translation left or right; award if \(y\)-coordinates of turning points remain at 6 and 2, and at least one \(x\)-coordinate has changed | B1 | Curve does not need to meet or cross \(x\)-axis |
| Curve crosses \(x\)-axis at \((2,0)\) and \((-4,0)\) only | B1 | |
| All 3 turning points correct: maximum \((1,6)\) in Q1, maximum \((-3,6)\) in Q2, minimum \((-1,2)\) in Q2 | B1 |
## Question 12:
### Part (a): $y = -f(x)$
| Answer/Working | Marks | Guidance |
|---|---|---|
| W shape | B1 | |
| Crosses $x$-axis at $(0,0)$ and $(6,0)$ only | B1 | Accept if curve passes through origin and $(6,0)$; condone $(0,6)$ marked on $y$-axis |
| All 3 turning points correct: maximum $(3,-2)$, minima $(1,-6)$ and $(5,-6)$ | B1 | Do not allow coordinates transposed |
Special case: Sketching $y = f(-x)$ correctly with all coordinates correct scores B1B0B0. Maximum points $(-1,6)$ and $(-5,6)$, crosses $x$-axis at origin and $(-6,0)$, minimum point $(-3,2)$.
### Part (b): $y = f\!\left(\frac{1}{2}x\right)$
| Answer/Working | Marks | Guidance |
|---|---|---|
| Same shape passing through (or starting at) the origin; two maxima and one minimum in Quadrant 1 | B1 | |
| Cuts $x$-axis at $(12,0)$ only | B1 | See part (a) for allowable alternatives |
| All 3 turning points correct: minimum $(6,2)$, maxima $(2,6)$ and $(10,6)$ | B1 | Do not allow coordinates transposed |
### Part (c): $y = f(x+4)$
| Answer/Working | Marks | Guidance |
|---|---|---|
| Translation left or right; award if $y$-coordinates of turning points remain at 6 and 2, and at least one $x$-coordinate has changed | B1 | Curve does not need to meet or cross $x$-axis |
| Curve crosses $x$-axis at $(2,0)$ and $(-4,0)$ only | B1 | |
| All 3 turning points correct: maximum $(1,6)$ in Q1, maximum $(-3,6)$ in Q2, minimum $(-1,2)$ in Q2 | B1 | |
Special case: $y = f(x)+4$ with all coordinates correct scores B1B0B0. Maximum points $(1,10)$ and $(5,10)$, minimum point $(3,6)$.
---12.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{ea81408b-e292-4529-b1e2-e3246503a3ac-17_679_1241_274_500}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of part of the curve with equation $y = \mathrm { f } ( x )$.\\
The curve crosses the $x$-axis at the origin and at the point $( 6,0 )$. The curve has maximum points at $( 1,6 )$ and $( 5,6 )$ and has a minimum point at $( 3,2 )$.
On separate diagrams sketch the curve with equation
\begin{enumerate}[label=(\alph*)]
\item $y = - \mathrm { f } ( x )$
\item $y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)$
\item $y = \mathrm { f } ( x + 4 )$
On each diagram show clearly the coordinates of the maximum and minimum points, and the coordinates of the points where the curve crosses the $x$-axis.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2015 Q12 [9]}}