| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Algebraic to algebraic transformation description |
| Difficulty | Moderate -0.8 This is a straightforward question testing standard knowledge of function transformations. Part (a) requires identifying two basic transformations from composite notation (horizontal shift and vertical stretch), while part (b) applies two transformations to a given function using routine substitution rules. Both parts are direct recall with minimal problem-solving, making this easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.02x Combinations of transformations: multiple transformations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Translation \(\begin{pmatrix}1\\0\end{pmatrix}\) | B1 | Allow shift by 1 in \(x\)-direction or parallel to/along the \(x\)-axis or horizontally. 'Translation by 1 to the right' only scores B0 |
| Stretch | B1 | Stretch. SC B2 for amplitude doubled |
| Factor 2 in \(y\)-direction | B1 | With/by factor 2 in \(y\)-direction or parallel to/along the \(y\)-axis or vertically or with \(x\) axis invariant. 'With/by factor 2 upwards' only scores B0. Accept SF as abbreviation for scale factor |
| 3 | Note: Transformations can be in either order |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([-\sin 6x][+15x]\) or \([\sin(-6x)][+15x]\) OE | B1 B1 | Accept unsimplified version. ISW. B1 for each correct component |
| If B0, SC B1 for either \(\sin(-2x)+5x\) or \(-\sin(2x)+5x\) or \(\sin 6x - 15x\) or \(\sin\left(-\frac{2}{3}x\right)+\frac{5}{3}x\) | ||
| 2 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Translation $\begin{pmatrix}1\\0\end{pmatrix}$ | B1 | Allow shift by 1 in $x$-direction or parallel to/along the $x$-axis or horizontally. 'Translation by 1 to the right' only scores B0 |
| Stretch | B1 | Stretch. **SC B2** for amplitude doubled |
| Factor 2 in $y$-direction | B1 | With/by factor 2 in $y$-direction or parallel to/along the $y$-axis or vertically or with $x$ axis invariant. 'With/by factor 2 upwards' only scores B0. Accept SF as abbreviation for scale factor |
| | **3** | **Note:** Transformations can be in either order |
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[-\sin 6x][+15x]$ or $[\sin(-6x)][+15x]$ OE | B1 B1 | Accept unsimplified version. ISW. B1 for each correct component |
| | | If B0, **SC B1** for either $\sin(-2x)+5x$ or $-\sin(2x)+5x$ or $\sin 6x - 15x$ or $\sin\left(-\frac{2}{3}x\right)+\frac{5}{3}x$ |
| | **2** | |2
\begin{enumerate}[label=(\alph*)]
\item The graph of $y = \mathrm { f } ( x )$ is transformed to the graph of $y = 2 \mathrm { f } ( x - 1 )$.\\
Describe fully the two single transformations which have been combined to give the resulting transformation.
\item The curve $y = \sin 2 x - 5 x$ is reflected in the $y$-axis and then stretched by scale factor $\frac { 1 } { 3 }$ in the $x$-direction.
Write down the equation of the transformed curve.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q2 [5]}}