| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Multiple transformation descriptions |
| Difficulty | Moderate -0.3 This question tests standard transformations of curves (horizontal stretch and modulus function) with straightforward application. While requiring understanding of two different transformations, both are routine C3 content: f(2x) gives horizontal stretch factor 1/2, and |f(x)| reflects negative portions. The key features are clearly given (asymptote, x-intercept at (1,0), increasing function), making this slightly easier than average but still requiring proper understanding of transformations. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.02x Combinations of transformations: multiple transformations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct shape: increasing function with decreasing gradient, asymptotic to \(y\)-axis, wholly in quadrants 1 and 4 | B1 | Condone linear looking functions in first quadrant. No obvious maximum point. |
| Crosses \(x\)-axis at \(\left(\frac{1}{2}, 0\right)\) | B1 | Accept \(\frac{1}{2}\), \(0.5\) or \(\left(0, \frac{1}{2}\right)\) marked on correct axis. Graph must be present. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct shape wholly contained in quadrant 1. Shape to right of cusp must have no obvious maximum | B1 | Accept linear or positive with decreasing gradient. Gradient to left of cusp/minimum must always be negative. Must not form a 'C' shape or have incorrect curvature. |
| Curve touches or crosses \(x\)-axis at \((1, 0)\) | B1 | Allow curve passing through point marked '1' on \(x\)-axis. Condone \((0,1)\) marked on correct axis. |
| Cusp at \((1, 0)\), not a minimum | B1 | Must be a cusp, not a minimum |
## Question 2:
**Part (a):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct shape: increasing function with decreasing gradient, asymptotic to $y$-axis, wholly in quadrants 1 and 4 | B1 | Condone linear looking functions in first quadrant. No obvious maximum point. |
| Crosses $x$-axis at $\left(\frac{1}{2}, 0\right)$ | B1 | Accept $\frac{1}{2}$, $0.5$ or $\left(0, \frac{1}{2}\right)$ marked on correct axis. Graph must be present. |
**Part (b):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct shape wholly contained in quadrant 1. Shape to right of cusp must have no obvious maximum | B1 | Accept linear or positive with decreasing gradient. Gradient to left of cusp/minimum must always be negative. Must not form a 'C' shape or have incorrect curvature. |
| Curve touches or crosses $x$-axis at $(1, 0)$ | B1 | Allow curve passing through point marked '1' on $x$-axis. Condone $(0,1)$ marked on correct axis. |
| Cusp at $(1, 0)$, not a minimum | B1 | Must be a cusp, not a minimum |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a80a71cb-42e0-4587-8f8e-bacd69b8d07a-03_499_1099_210_443}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x ) , x > 0$, where f is an increasing function of $x$. The curve crosses the $x$-axis at the point $( 1,0 )$ and the line $x = 0$ is an asymptote to the curve.
On separate diagrams, sketch the curve with equation
\begin{enumerate}[label=(\alph*)]
\item $y = \mathrm { f } ( 2 x ) , x > 0$
\item $y = | \mathrm { f } ( x ) | , x > 0$
Indicate clearly on each sketch the coordinates of the point at which the curve crosses or meets the $x$-axis.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2013 Q2 [5]}}