| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Express function using transformations |
| Difficulty | Moderate -0.3 This question requires completing the square (a standard P1 technique) and identifying transformations from the algebraic form. While it involves multiple steps and connecting algebra to geometry, the techniques are routine and the question guides students through the process systematically. Slightly easier than average due to its structured nature and use of well-practiced methods. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f(x) = (x-1)^2 + 4\) | B1 | |
| \(g(x) = (x+2)^2 + 9\) | B1 | |
| \(g(x) = f(x+3) + 5\) | B1 B1 | B1 for each correct element. Accept \(p = 3, q = 5\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Translation or Shift | B1 | |
| \(\begin{pmatrix}-3\\5\end{pmatrix}\) or acceptable explanation | B1 FT | If given as 2 single translations both must be described correctly e.g. \(\begin{pmatrix}-3\\0\end{pmatrix}\) & \(\begin{pmatrix}0\\5\end{pmatrix}\). FT from their \(f(x+p)+q\) or their \(f(x) \rightarrow g(x)\). Do not accept \(\begin{pmatrix}1\\4\end{pmatrix}\) or \(\begin{pmatrix}-2\\9\end{pmatrix}\) |
## Question 6:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f(x) = (x-1)^2 + 4$ | B1 | |
| $g(x) = (x+2)^2 + 9$ | B1 | |
| $g(x) = f(x+3) + 5$ | B1 B1 | B1 for each correct element. Accept $p = 3, q = 5$ |
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Translation or Shift | B1 | |
| $\begin{pmatrix}-3\\5\end{pmatrix}$ or acceptable explanation | B1 FT | If given as 2 single translations both must be described correctly e.g. $\begin{pmatrix}-3\\0\end{pmatrix}$ & $\begin{pmatrix}0\\5\end{pmatrix}$. FT from their $f(x+p)+q$ or their $f(x) \rightarrow g(x)$. Do not accept $\begin{pmatrix}1\\4\end{pmatrix}$ or $\begin{pmatrix}-2\\9\end{pmatrix}$ |
---6 Functions f and g are both defined for $x \in \mathbb { R }$ and are given by
$$\begin{aligned}
& \mathrm { f } ( x ) = x ^ { 2 } - 2 x + 5 \\
& \mathrm {~g} ( x ) = x ^ { 2 } + 4 x + 13
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item By first expressing each of $\mathrm { f } ( x )$ and $\mathrm { g } ( x )$ in completed square form, express $\mathrm { g } ( x )$ in the form $\mathrm { f } ( x + p ) + q$, where $p$ and $q$ are constants.
\item Describe fully the transformation which transforms the graph of $y = \mathrm { f } ( x )$ to the graph of $y = \mathrm { g } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2021 Q6 [6]}}