Edexcel P1 2023 June — Question 9 9 marks

Exam BoardEdexcel
ModuleP1 (Pure Mathematics 1)
Year2023
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicTrig Graphs & Exact Values
TypeSketch or identify transformed graph
DifficultyModerate -0.3 This is a standard P1 question on identifying and transforming trigonometric graphs. Part (i) requires recognizing a sine/cosine function from its graph and applying a horizontal translation; part (ii) involves identifying another trig function and applying a vertical translation. While it requires understanding of transformations and careful tracking of key points, these are routine A-level skills with no novel problem-solving required. Slightly easier than average due to being mostly pattern recognition and standard transformations.
Spec1.02w Graph transformations: simple transformations of f(x)1.05f Trigonometric function graphs: symmetries and periodicities
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-24_536_933_255_568} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of the graph of the trigonometric function with equation \(y = \mathrm { f } ( x )\)
  1. Write down an expression for \(\mathrm { f } ( x )\) On a separate diagram,
  2. sketch, for \(- 2 \pi < x < 2 \pi\), the graph of the curve with equation \(y = \mathrm { f } \left( x + \frac { \pi } { 4 } \right)\) Show clearly the coordinates of all the points where the curve intersects the coordinate axes.
    (ii) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-24_378_1251_1617_408} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows part of the graph of the trigonometric function with equation \(y = \mathrm { g } ( x )\)
  3. Write down an expression for \(\mathrm { g } ( x )\) On a separate diagram,
  4. sketch, for \(- 2 \pi < x < 2 \pi\), the graph of the curve with equation \(y = \mathrm { g } ( x ) - 2\) Show clearly the coordinates of the \(y\) intercept.
Question 9:
Part (i)(a):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\((y =)\, 3\cos(x)\)M1 Identifies curve as cosine function of form \(\alpha\cos(\beta x)\). Also allow sine function of form \(\alpha\sin\!\left(\beta x + \frac{\pi}{2}\right)\) oe
Fully correct expressionA1 No requirement for \(y = ...\) or \(f(x) = ...\). Allow e.g. \(3\cos\theta\)
Part (i)(b):
AnswerMarks Guidance
Answer/WorkingMark Guidance
Same shape translated left or rightB1 Applies horizontal translation \(y = f(x)\) either direction. Maximum at \((0,3)\) must have moved left or right of \(y\)-axis. Should be one minimum to left and one to right of \(y\)-axis
All \(x\)-intercepts labelled correctlyB1 Must be the only \(x\)-intercepts and curve must pass through these points. Allow coordinate pairs e.g. \(\left(-\frac{7\pi}{4}, 0\right)\). Condone 3sf decimals: \(-5.50,\ -2.36,\ 0.785,\ 3.93\). Condone degrees: \(-315°,\ -135°,\ 45°,\ 225°\)
Correct \(y\)-intercept \(\dfrac{3\sqrt{2}}{2}\)B1 Allow as shown or as e.g. \(\left(0, \frac{3\sqrt{2}}{2}\right)\). Must not be a clear maximum
Part (ii)(a):
AnswerMarks Guidance
Answer/WorkingMark Guidance
\((y =)\, \sin(2x)\)M1 Identifies curve as sine function of form \(\alpha\sin(\beta x)\). Also allow cosine of form \(\alpha\cos\!\left(\beta x \pm \frac{\pi}{2}\right)\) oe
Fully correct expressionA1 No requirement for \(y = ...\) or \(g(x) = ...\). Allow e.g. \(\sin 2\theta\)
Part (ii)(b):
AnswerMarks Guidance
Answer/WorkingMark Guidance
Same shape translated down, lying entirely below \(x\)-axisB1 Applies vertical translation down to \(y = g(x)\). Must have same number of cycles as \(g(x)\), look the same, and lie entirely below \(x\)-axis. Do not allow clear "zig-zag" with cusps
Correct \(y\)-intercept \(-2\) labelledB1 Allow as shown or e.g. \((0,-2)\); allow \((-2, 0)\) if in correct position; allow shown by intercept passing through line \(y = -2\) 
# Question 9:

## Part (i)(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(y =)\, 3\cos(x)$ | M1 | Identifies curve as cosine function of form $\alpha\cos(\beta x)$. Also allow sine function of form $\alpha\sin\!\left(\beta x + \frac{\pi}{2}\right)$ oe |
| Fully correct expression | A1 | No requirement for $y = ...$ or $f(x) = ...$. Allow e.g. $3\cos\theta$ |

## Part (i)(b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Same shape translated left or right | B1 | Applies horizontal translation $y = f(x)$ either direction. Maximum at $(0,3)$ must have moved left or right of $y$-axis. Should be one minimum to left and one to right of $y$-axis |
| All $x$-intercepts labelled correctly | B1 | Must be the only $x$-intercepts and curve must pass through these points. Allow coordinate pairs e.g. $\left(-\frac{7\pi}{4}, 0\right)$. Condone 3sf decimals: $-5.50,\ -2.36,\ 0.785,\ 3.93$. Condone degrees: $-315°,\ -135°,\ 45°,\ 225°$ |
| Correct $y$-intercept $\dfrac{3\sqrt{2}}{2}$ | B1 | Allow as shown or as e.g. $\left(0, \frac{3\sqrt{2}}{2}\right)$. Must not be a clear maximum |

## Part (ii)(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(y =)\, \sin(2x)$ | M1 | Identifies curve as sine function of form $\alpha\sin(\beta x)$. Also allow cosine of form $\alpha\cos\!\left(\beta x \pm \frac{\pi}{2}\right)$ oe |
| Fully correct expression | A1 | No requirement for $y = ...$ or $g(x) = ...$. Allow e.g. $\sin 2\theta$ |

## Part (ii)(b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| Same shape translated down, lying entirely below $x$-axis | B1 | Applies vertical translation down to $y = g(x)$. Must have same number of cycles as $g(x)$, look the same, and lie entirely below $x$-axis. Do not allow clear "zig-zag" with cusps |
| Correct $y$-intercept $-2$ labelled | B1 | Allow as shown or e.g. $(0,-2)$; allow $(-2, 0)$ if in correct position; allow shown by intercept passing through line $y = -2$ |

---
\begin{enumerate}
  \item (i)
\end{enumerate}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-24_536_933_255_568}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}

Figure 3 shows part of the graph of the trigonometric function with equation $y = \mathrm { f } ( x )$\\
(a) Write down an expression for $\mathrm { f } ( x )$

On a separate diagram,\\
(b) sketch, for $- 2 \pi < x < 2 \pi$, the graph of the curve with equation $y = \mathrm { f } \left( x + \frac { \pi } { 4 } \right)$

Show clearly the coordinates of all the points where the curve intersects the coordinate axes.\\
(ii)

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{a5a5dd8b-1438-4698-929a-c5e3d5ed0694-24_378_1251_1617_408}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}

Figure 4 shows part of the graph of the trigonometric function with equation $y = \mathrm { g } ( x )$\\
(a) Write down an expression for $\mathrm { g } ( x )$

On a separate diagram,\\
(b) sketch, for $- 2 \pi < x < 2 \pi$, the graph of the curve with equation $y = \mathrm { g } ( x ) - 2$

Show clearly the coordinates of the $y$ intercept.

\hfill \mbox{\textit{Edexcel P1 2023 Q9 [9]}}
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