| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | March |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Sequence of transformations order |
| Difficulty | Moderate -0.8 Part (a) is routine completing the square with a coefficient factored out. Part (b) requires identifying transformations from completed square form, which is a standard textbook exercise testing recall of transformation rules. The circle/line problem appears unrelated to the transformation question. Overall, this is below average difficulty—straightforward application of well-practiced techniques with no problem-solving insight required. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02w Graph transformations: simple transformations of f(x)1.03d Circles: equation (x-a)^2+(y-b)^2=r^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2[\{(x-2)^2\}\ \{+3\}]\) | B1 B1 | B1 for \(a=2\), B1 for \(b=3\). \(2(x-2)^2+6\) gains B1B0 |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| {Translation} \(\begin{pmatrix}\{2\}\\\{3\}\end{pmatrix}\) OR {Stretch} {\(y\) direction} {factor 2} | B2,1,0 | B2 for fully correct, B1 with two elements correct. \(\{\}\) indicates different elements |
| {Stretch} {\(y\) direction} {factor 2} OR {Translation} \(\begin{pmatrix}\{2\}\\\{6\}\end{pmatrix}\) | B2,1,0 | B2 for fully correct, B1 with two elements correct. \(\{\}\) indicates different elements |
| 4 |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2[\{(x-2)^2\}\ \{+3\}]$ | B1 B1 | B1 for $a=2$, B1 for $b=3$. $2(x-2)^2+6$ gains B1B0 |
| | **2** | |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| {Translation} $\begin{pmatrix}\{2\}\\\{3\}\end{pmatrix}$ OR {Stretch} {$y$ direction} {factor 2} | B2,1,0 | B2 for fully correct, B1 with two elements correct. $\{\}$ indicates different elements |
| {Stretch} {$y$ direction} {factor 2} OR {Translation} $\begin{pmatrix}\{2\}\\\{6\}\end{pmatrix}$ | B2,1,0 | B2 for fully correct, B1 with two elements correct. $\{\}$ indicates different elements |
| | **4** | |5
\begin{enumerate}[label=(\alph*)]
\item Express $2 x ^ { 2 } - 8 x + 14$ in the form $2 \left[ ( x - a ) ^ { 2 } + b \right]$.\\
The functions $f$ and $g$ are defined by
$$\begin{aligned}
& \mathrm { f } ( x ) = x ^ { 2 } \quad \text { for } x \in \mathbb { R } \\
& \mathrm {~g} ( x ) = 2 x ^ { 2 } - 8 x + 14 \quad \text { for } x \in \mathbb { R }
\end{aligned}$$
\item Describe fully a sequence of transformations that maps the graph of $y = \mathrm { f } ( x )$ onto the graph of $y = \mathrm { g } ( x )$, making clear the order in which the transformations are applied.\\
\includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-08_679_1043_260_552}
The circle with equation $( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 85$ and the straight line with equation $y = 3 x - 20$ are shown in the diagram. The line intersects the circle at $A$ and $B$, and the centre of the circle is at $C$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q5 [6]}}