| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Multiple separate transformations (sketch-based, modulus involved) |
| Difficulty | Standard +0.3 This is a standard C3 transformations question requiring sketching of composite transformations and describing transformation sequences. While it involves multiple parts including modulus functions and composite transformations like f(|2x|) and f(2x+2), these are routine textbook exercises at this level. The most challenging aspect is correctly handling the sequence in part (c), but this follows standard patterns taught in C3. Overall, slightly easier than average due to the predictable nature of the transformations. |
| Spec | 1.02s Modulus graphs: sketch graph of |ax+b|1.02w Graph transformations: simple transformations of f(x)1.02x Combinations of transformations: multiple transformations |
| Answer | Marks | Guidance |
|---|---|---|
| ## Part (a): Sketch \(y = - | f(x) | \) |
| Answer | Mark | Guidance |
| Correct shape: original curve reflected in x-axis for parts below x-axis, parts above x-axis reflected down | B1 | Shape correct - all of curve on or below x-axis |
| x-intercepts at \(x = -3\) and \(x = 2\) correct | B1 | Intercepts marked |
| Point \((4, -3)\) remains at \((4, -3)\) (or equivalent minimum shown correctly) | B1 | |
| ## Part (b): Sketch \(y = f( | 2x | )\) |
| Answer | Mark | Guidance |
| Correct right-hand half of curve compressed horizontally by factor \(\frac{1}{2}\), with x-intercept at \(x=1\), passing through \((2, -3)\) | B1 | |
| Left-hand side is reflection of right-hand side in y-axis, with x-intercept at \(x = -1\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Translation by \(\begin{pmatrix}-1\\0\end{pmatrix}\) (i.e. shift left 1) | M1 A1 | Must be described as translation |
| Stretch parallel to x-axis, scale factor \(\frac{1}{2}\) | M1 A1 | Must be described as stretch, sf \(\frac{1}{2}\) in x-direction |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| After translation: \((3, -3)\) | M1 | Applying first transformation to point |
| After stretch: \(\left(\frac{3}{2}, -3\right)\) | A1 | Correct final coordinates |
# Question 4:
## Part (a): Sketch $y = -|f(x)|$
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct shape: original curve reflected in x-axis for parts below x-axis, parts above x-axis reflected down | B1 | Shape correct - all of curve on or below x-axis |
| x-intercepts at $x = -3$ and $x = 2$ correct | B1 | Intercepts marked |
| Point $(4, -3)$ remains at $(4, -3)$ (or equivalent minimum shown correctly) | B1 | |
## Part (b): Sketch $y = f(|2x|)$
| Answer | Mark | Guidance |
|--------|------|----------|
| Correct right-hand half of curve compressed horizontally by factor $\frac{1}{2}$, with x-intercept at $x=1$, passing through $(2, -3)$ | B1 | |
| Left-hand side is reflection of right-hand side in y-axis, with x-intercept at $x = -1$ | B1 | |
## Part (c)(i): Two transformations mapping $y=f(x)$ onto $y=f(2x+2)$
| Answer | Mark | Guidance |
|--------|------|----------|
| Translation by $\begin{pmatrix}-1\\0\end{pmatrix}$ (i.e. shift left 1) | M1 A1 | Must be described as translation |
| Stretch parallel to x-axis, scale factor $\frac{1}{2}$ | M1 A1 | Must be described as stretch, sf $\frac{1}{2}$ in x-direction |
Note: Translation first, then stretch (order matters)
## Part (c)(ii): Image of $P(4,-3)$
| Answer | Mark | Guidance |
|--------|------|----------|
| After translation: $(3, -3)$ | M1 | Applying first transformation to point |
| After stretch: $\left(\frac{3}{2}, -3\right)$ | A1 | Correct final coordinates |
---4 The sketch shows part of the curve with equation $y = \mathrm { f } ( x )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-08_536_1054_367_539}
\begin{enumerate}[label=(\alph*)]
\item On Figure 2 below, sketch the curve with equation $y = - | \mathrm { f } ( x ) |$.
\item On Figure 3 on the page opposite, sketch the curve with equation $y = \mathrm { f } ( | 2 x | )$.
\item \begin{enumerate}[label=(\roman*)]
\item Describe a sequence of two geometrical transformations that maps the graph of $y = \mathrm { f } ( x )$ onto the graph of $y = \mathrm { f } ( 2 x + 2 )$.
\item Find the coordinates of the image of the point $P ( 4 , - 3 )$ under the sequence of transformations given in part (c)(i).
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-09_778_1032_424_529}
\end{center}
\end{figure}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2014 Q4 [11]}}