| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Composite transformation sketch |
| Difficulty | Moderate -0.8 This is a straightforward application of standard transformation rules: a horizontal stretch followed by a vertical translation. Part (a) requires sketching transformed key points, and part (b) requires writing g(x) = f(2x) + 1, both routine procedures covered extensively in P1 with no problem-solving or novel insight required. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.02x Combinations of transformations: multiple transformations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Three points at the bottom of their transformed graph plotted at \(y = 2\) | B1 | All 5 points of the graph must be connected. |
| Bottom three points of \(\mathsf{M}\) at \(x = 0,\ x = 1\) & \(x = 2\) | B1 | Must be this shape. |
| All correct | B1 | Condone extra cycles outside \(0 \leqslant x \leqslant 2\). SC: If B0 B0 scored, B1 available for \(\Lambda\) in one of correct positions or all 5 points correctly plotted and not connected or correctly sized shape in the wrong position. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([g(x) =]\ f(2x) + 1\) | B1 B1 | Award marks for their final answer: \(f(2x)\) B1, \(+ 1\) B1. Condone \(y =\) or \(f(x) =\). |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Three points at the bottom of their transformed graph plotted at $y = 2$ | B1 | All 5 points of the graph must be connected. |
| Bottom three points of $\mathsf{M}$ at $x = 0,\ x = 1$ & $x = 2$ | B1 | Must be this shape. |
| All correct | B1 | Condone extra cycles outside $0 \leqslant x \leqslant 2$. **SC:** If B0 B0 scored, B1 available for $\Lambda$ in one of correct positions **or** all 5 points correctly plotted and not connected **or** correctly sized shape in the wrong position. |
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $[g(x) =]\ f(2x) + 1$ | B1 B1 | Award marks for their final answer: $f(2x)$ B1, $+ 1$ B1. Condone $y =$ or $f(x) =$. |
---5 The graph with equation $y = \mathrm { f } ( x )$ is transformed to the graph with equation $y = \mathrm { g } ( x )$ by a stretch in the $x$-direction with factor 0.5 , followed by a translation of $\binom { 0 } { 1 }$.
\begin{enumerate}[label=(\alph*)]
\item The diagram below shows the graph of $y = \mathrm { f } ( x )$.
On the diagram sketch the graph of $y = \mathrm { g } ( x )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-07_613_1527_623_342}
\item Find an expression for $\mathrm { g } ( x )$ in terms of $\mathrm { f } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q5 [5]}}