| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Find constants from sketch features |
| Difficulty | Moderate -0.3 This is a standard P3 transformation question requiring application of horizontal/vertical stretches and reflections to key features of a curve. While it requires careful tracking of multiple points through transformations, the techniques are routine and well-practiced at this level, making it slightly easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.02x Combinations of transformations: multiple transformations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| Shape (two way stretch), graph in quadrants 1 and 3 passing through origin | B1 | Same shape passing through origin with evidence of two-way stretch. Minimum on \(x\)-axis. Neither end bends back significantly nor consists of three straight lines |
| Maximum at \((0.5, 6)\) | B1 | Condone \(\wedge\) shape at maximum. Must have sketch. May be implied by 0.5 and 6 marked on correct axes |
| Minimum at \((1.5, 0)\) | B1 | Condone \(\vee\) shape. Allow 1.5 marked on \(x\)-axis; condone \((0, 1.5)\) on \(x\)-axis |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| Shape and position: \(-x^3\) shaped curve crossing \(y\)-axis but not at origin, turning points left of \(y\)-axis | B1 | Reflection in \(y\)-axis followed by vertical translation. Don't be concerned about heights of turning points or \(y\)-intercept for this mark |
| Minimum at \((-3,-1)\) and maximum at \((-1,1)\) | B1 | Must be in correct quadrants and be turning points. These may be implied |
| Crosses \(y\)-axis at \((0,-1)\) | B1 | May be awarded for curve stopping at \(y\)-axis at \((0,-1)\). Allow \(-1\) marked on \(y\)-axis; condone \((-1,0)\) on \(y\)-axis |
# Question 2(i):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Shape (two way stretch), graph in quadrants 1 and 3 passing through origin | B1 | Same shape passing through origin with evidence of two-way stretch. Minimum on $x$-axis. Neither end bends back significantly nor consists of three straight lines |
| Maximum at $(0.5, 6)$ | B1 | Condone $\wedge$ shape at maximum. Must have sketch. May be implied by 0.5 and 6 marked on correct axes |
| Minimum at $(1.5, 0)$ | B1 | Condone $\vee$ shape. Allow 1.5 marked on $x$-axis; condone $(0, 1.5)$ on $x$-axis |
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# Question 2(ii):
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| Shape and position: $-x^3$ shaped curve crossing $y$-axis but not at origin, turning points left of $y$-axis | B1 | Reflection in $y$-axis followed by vertical translation. Don't be concerned about heights of turning points or $y$-intercept for this mark |
| Minimum at $(-3,-1)$ and maximum at $(-1,1)$ | B1 | Must be in correct quadrants and be turning points. These may be implied |
| Crosses $y$-axis at $(0,-1)$ | B1 | May be awarded for curve stopping at $y$-axis at $(0,-1)$. Allow $-1$ marked on $y$-axis; condone $(-1,0)$ on $y$-axis |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{624e9e2f-b6b8-47ce-accc-31dcd5f0554e-04_903_1148_123_399}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$, where $x \in \mathbb { R }$ and $\mathrm { f } ( x )$ is a polynomial.
The curve passes through the origin and touches the $x$-axis at the point $( 3,0 )$
There is a maximum turning point at $( 1,2 )$ and a minimum turning point at $( 3,0 )$
On separate diagrams, sketch the curve with equation\\
(i) $y = 3 f ( 2 x )$\\
(ii) $y = \mathrm { f } ( - x ) - 1$
On each sketch, show clearly the coordinates of
\begin{itemize}
\item the point where the curve crosses the $y$-axis
\item any maximum or minimum turning points
\end{itemize}
\hfill \mbox{\textit{Edexcel P3 2021 Q2 [6]}}