| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Stationary points after transformation |
| Difficulty | Standard +0.3 Part (a) requires reading period and phase shift from a sine graph—straightforward pattern recognition. Part (b) applies standard transformation rules (horizontal stretch ×4, translation -8, vertical stretch ×3, reflection) to a stationary point, which is a routine textbook exercise requiring systematic application of known rules rather than problem-solving. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05f Trigonometric function graphs: symmetries and periodicities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(a = \frac{1}{2}\) | B1 | |
| \(b = \frac{\pi}{3}\) | B1 | \(b = \frac{\pi}{3} + 4n\pi, n \geq 0\) |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x\)-coordinate \(= \{4p\}\{-8\}\) | B1 B1 | OE, e.g. \(4(p-2)\) |
| \(y\)-coordinate \(= -3q\) | B1 | |
| 3 |
## Question 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = \frac{1}{2}$ | B1 | |
| $b = \frac{\pi}{3}$ | B1 | $b = \frac{\pi}{3} + 4n\pi, n \geq 0$ |
| | **2** | |
## Question 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x$-coordinate $= \{4p\}\{-8\}$ | B1 B1 | OE, e.g. $4(p-2)$ |
| $y$-coordinate $= -3q$ | B1 | |
| | **3** | |8\\
\includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-10_515_936_274_577}
The diagram shows part of the graph of $y = \sin ( a ( x + b ) )$, where $a$ and $b$ are positive constants.
\begin{enumerate}[label=(\alph*)]
\item State the value of $a$ and one possible value of $b$.\\
Another curve, with equation $y = \mathrm { f } ( x )$, has a single stationary point at the point $( p , q )$, where $p$ and $q$ are constants. This curve is transformed to a curve with equation
$$y = - 3 f \left( \frac { 1 } { 4 } ( x + 8 ) \right) .$$
\item For the transformed curve, find the coordinates of the stationary point, giving your answer in terms of $p$ and $q$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2023 Q8 [5]}}