| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Transformations of modulus graphs from given f(x) sketch |
| Difficulty | Moderate -0.3 This is a standard C3 transformations question requiring application of well-defined rules for modulus and function transformations. While it tests understanding of three different transformation types, each follows a routine procedure: reflecting negative y-values for |f(x)|, creating symmetry about the y-axis for f(|x|), and applying horizontal/vertical stretches for 2f(3x). The question is slightly easier than average because it's purely procedural with no problem-solving or novel insight required, though students must carefully track coordinate transformations. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.02x Combinations of transformations: multiple transformations |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Shape including cusp | B1 | Shape (inc cusp) with graph in just quadrants 1 and 2. Do not be overly concerned about relative gradients, but the left hand section of the back of the curve should not bend back beyond the cusp |
| \((-1.5, 0)\) and \((0, 5)\) | B1 | This is independent, and for the curve touching the \(x\)-axis at (-1.5, 0) and crossing the \(y\)-axis at (0.5) |
| (2) | (2) | |
| (b) Shape | B1 | For a U shaped curve symmetrical about the y-axis |
| \((0,5)\) | B1 | (0.5) lies on the curve |
| (2) | (2) | |
| (c) Shape | B1 | Correct shape- do not be overly concerned about relative gradients. Look for a similar shape to \(f(x)\) |
| \((0,10)\) | B1 | Curve crosses the y-axis at \((0, 10)\). The curve must appear in both quadrants 1 and 2 |
| \((-0.5, 0)\) | B1 | Curve crosses the \(x\)-axis at \((-0.5, 0)\). The curve must appear in quadrants 3 and 2. |
| (3) | (7 marks) |
| (a) Shape including cusp | B1 | Shape (inc cusp) with graph in just quadrants 1 and 2. Do not be overly concerned about relative gradients, but the left hand section of the back of the curve should not bend back beyond the cusp |
|---|---|---|
| $(-1.5, 0)$ and $(0, 5)$ | B1 | This is independent, and for the curve touching the $x$-axis at (-1.5, 0) and crossing the $y$-axis at (0.5) |
| **(2)** | **(2)** | |
| (b) Shape | B1 | For a U shaped curve symmetrical about the y-axis |
| $(0,5)$ | B1 | (0.5) lies on the curve |
| **(2)** | **(2)** | |
| (c) Shape | B1 | Correct shape- do not be overly concerned about relative gradients. Look for a similar shape to $f(x)$ |
| $(0,10)$ | B1 | Curve crosses the y-axis at $(0, 10)$. The curve must appear in both quadrants 1 and 2 |
| $(-0.5, 0)$ | B1 | Curve crosses the $x$-axis at $(-0.5, 0)$. The curve must appear in quadrants 3 and 2. |
| **(3)** | **(7 marks)** | |
**Notes:**
- (a) Note that this appears as M1A1 on EPEN: B1 Shape (inc cusp) with graph in just quadrants 1 and 2. Do not be overly concerned about relative gradients, but the left hand section of the back of the curve should not bend back beyond the cusp. B1 This is independent, and for the curve touching the $x$-axis at (-1.5, 0) and crossing the $y$-axis at (0,5).
- (b) Note that this appears as M1A1 on EPEN: B1 For a U shaped curve symmetrical about the y-axis. B1 (0.5) lies on the curve.
- (c) Note that this appears as M1B1B1 on EPEN: B1 Correct shape- do not be overly concerned about relative gradients. Look for a similar shape to $f(x)$. B1 Curve crosses the $y$-axis at $(0, 10)$. The curve must appear in both quadrants 1 and 2. B1 Curve crosses the $x$-axis at $(-0.5, 0)$. The curve must appear in quadrants 3 and 2.
**In all parts accept the following for any co-ordinate. Using (0,3) as an example, accept both (3,0) or 3 written on the y axis (as long as the curve passes through the point).**
**Special case with (a) and (b) completely correct but the wrong way around mark - SC(a) 0,1 SC(b) 0,1. Otherwise follow scheme.**
---4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{3fbdfb55-5dd5-44ab-b031-d39e64bdfc3b-06_560_1145_210_386}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows part of the curve with equation $y = \mathrm { f } ( x )$\\
The curve passes through the points $P ( - 1.5,0 )$ and $Q ( 0,5 )$ as shown.\\
On separate diagrams, sketch the curve with equation
\begin{enumerate}[label=(\alph*)]
\item $y = | f ( x ) |$
\item $y = \mathrm { f } ( | x | )$
\item $y = 2 f ( 3 x )$
Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2012 Q4 [7]}}