| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Express function using transformations |
| Difficulty | Moderate -0.8 This is a routine completing-the-square question with standard transformations. Parts (a) and (b) are textbook exercises requiring only algebraic manipulation. Part (c) requires connecting the two forms but follows directly from the previous work. Part (d) tests basic understanding of function transformations (stretch and translation), which is standard P1 content with no problem-solving insight needed. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks |
|---|---|
| \((x-2)^2 + 5\) | B1 |
| Answer | Marks |
|---|---|
| \(2\left(\left\{(x+1)^2\right\} + \{5\}\right)\) | B2, 1, 0 |
| Answer | Marks | Guidance |
|---|---|---|
| \([g(x) =]\ 2f(x+3)\) or \(k=2,\ h=3\) | B1 | In correct form. B0 if contradiction |
| Answer | Marks | Guidance |
|---|---|---|
| Translation \(\begin{pmatrix}-3\\0\end{pmatrix}\) | B2, 1, 0 FT | FT on *their* \(x+3\) or \(h=3\) |
| Stretch \(\{y\) direction, factor \(2\}\) | B2, 1, 0 FT | FT on *their* \(2\) or \(k=2\) |
## Question 9:
### Part 9(a):
| $(x-2)^2 + 5$ | B1 | |
|---|---|---|
### Part 9(b):
| $2\left(\left\{(x+1)^2\right\} + \{5\}\right)$ | B2, 1, 0 | |
|---|---|---|
### Part 9(c):
| $[g(x) =]\ 2f(x+3)$ or $k=2,\ h=3$ | B1 | In correct form. B0 if contradiction |
|---|---|---|
### Part 9(d):
| Translation $\begin{pmatrix}-3\\0\end{pmatrix}$ | B2, 1, 0 FT | FT on *their* $x+3$ or $h=3$ |
|---|---|---|
| Stretch $\{y$ direction, factor $2\}$ | B2, 1, 0 FT | FT on *their* $2$ or $k=2$ |
---9 Functions f and g are both defined for $x \in \mathbb { R }$ and are given by
$$\begin{aligned}
& \mathrm { f } ( x ) = x ^ { 2 } - 4 x + 9 \\
& \mathrm {~g} ( x ) = 2 x ^ { 2 } + 4 x + 12
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in the form $( x - a ) ^ { 2 } + b$.
\item Express $\mathrm { g } ( x )$ in the form $2 \left[ ( x + c ) ^ { 2 } + d \right]$.
\item Express $\mathrm { g } ( x )$ in the form $k \mathrm { f } ( x + h )$, where $k$ and $h$ are integers.
\item Describe fully the two transformations that have been combined to transform the graph of $y = \mathrm { f } ( x )$ to the graph of $y = \mathrm { g } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q9 [8]}}