| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch transformations from algebraic function |
| Difficulty | Moderate -0.8 This is a straightforward transformation question requiring students to sketch ln(x) and apply standard transformations (reflection in x-axis, horizontal translation). The transformations are routine C3 content with no problem-solving required—students simply apply learned rules for |f(x)| and -f(x-4), identify x-intercepts by setting y=0, and state vertical asymptotes. Easier than average due to its mechanical nature and clear structure. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.06d Natural logarithm: ln(x) function and properties |
| Answer | Marks | Guidance |
|---|---|---|
| \(\ln\) graph crossing \(x\)-axis at \((1,0)\) and asymptote at \(x=0\) | B1 | Correct shape and position passing through \((1,0)\). Graph must start to rhs of \(y\)-axis in quadrant 4 with large gradient, decreasing through \((1,0)\). No obvious maximum. Condone point marked \((0,1)\) on correct axis. Do not withhold this mark if (\(x=0\)) asymptote is incorrect or not given. |
| Answer | Marks | Guidance |
|---|---|---|
| Shape including cusp | B1ft | Correct shape including the cusp wholly in quadrant 1. Shape to rhs of cusp should have decreasing gradient with no obvious maximum. Shape to lhs should not bend backwards past \((1,0)\). Tolerate 'linear' looking section but not one with incorrect curvature. Follow through on incorrect sketch in part (i) as long as it was above and below \(x\)-axis. |
| Touches or crosses the \(x\)-axis at \((1,0)\) | B1ft | Allow for curve passing through point marked '1' on \(x\)-axis. Condone point marked on correct axis as \((0,1)\). Follow through on incorrect intersection in part (i). |
| Asymptote given as \(x=0\) | B1 | Award for asymptote given/marked as \(x=0\). Do not allow if given/marked as 'the \(y\)-axis'. There must be a graph present and an asymptote on the graph at \(x=0\). Accept if \(x=0\) is drawn separately to the \(y\)-axis. |
| Answer | Marks | Guidance |
|---|---|---|
| Shape | B1 | Gradient always negative and becoming less steep. Must be approximately infinite at lh end and no obvious minimum. lh end must not bend 'forwards' to make a C shape. Position not important for this mark. |
| Crosses at \((5, 0)\) | B1ft | Graph crosses (or touches) \(x\)-axis at \((5,0)\). Allow for curve passing through point marked '5' on \(x\)-axis. Condone point marked on correct axis as \((0,5)\). Follow through on incorrect intersection in part (i); allow for \((i)+4, 0)\). |
| Asymptote given as \(x=4\) | B1 | Given/marked as \(x=4\). Must be a graph present and asymptote on graph in correct place to rhs of \(y\)-axis. |
## Question 2(i):
| $\ln$ graph crossing $x$-axis at $(1,0)$ and asymptote at $x=0$ | B1 | Correct shape and position passing through $(1,0)$. Graph must start to rhs of $y$-axis in quadrant 4 with large gradient, decreasing through $(1,0)$. No obvious maximum. Condone point marked $(0,1)$ on correct axis. **Do not withhold this mark if ($x=0$) asymptote is incorrect or not given.** |
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## Question 2(ii):
| Shape including cusp | B1ft | Correct shape **including the cusp** wholly in quadrant 1. Shape to rhs of cusp should have decreasing gradient with no obvious maximum. Shape to lhs should not bend backwards past $(1,0)$. Tolerate 'linear' looking section but not one with incorrect curvature. Follow through on incorrect sketch in part (i) as long as it was above and below $x$-axis. |
| Touches or crosses the $x$-axis at $(1,0)$ | B1ft | Allow for curve passing through point marked '1' on $x$-axis. Condone point marked on correct axis as $(0,1)$. Follow through on incorrect intersection in part (i). |
| Asymptote given as $x=0$ | B1 | Award for asymptote given/marked as $x=0$. Do not allow if given/marked as 'the $y$-axis'. There must be a graph present and an asymptote on the graph at $x=0$. Accept if $x=0$ is drawn separately to the $y$-axis. |
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## Question 2(iii):
| Shape | B1 | Gradient always negative and becoming less steep. Must be approximately infinite at lh end and no obvious minimum. lh end must not bend 'forwards' to make a C shape. Position not important for this mark. |
| Crosses at $(5, 0)$ | B1ft | Graph crosses (or touches) $x$-axis at $(5,0)$. Allow for curve passing through point marked '5' on $x$-axis. Condone point marked on correct axis as $(0,5)$. Follow through on incorrect intersection in part (i); allow for $(i)+4, 0)$. |
| Asymptote given as $x=4$ | B1 | Given/marked as $x=4$. Must be a graph present and asymptote on graph in correct place to rhs of $y$-axis. |
*If graphs not labelled as (i), (ii), (iii) mark them in the order given.*
**(7 marks total for Question 2)**2. Given that
$$\mathrm { f } ( x ) = \ln x , \quad x > 0$$
sketch on separate axes the graphs of\\
(i) $\quad y = \mathrm { f } ( x )$,\\
(ii) $y = | \mathrm { f } ( x ) |$,\\
(iii) $y = - \mathrm { f } ( x - 4 )$.
Show, on each diagram, the point where the graph meets or crosses the $x$-axis. In each case, state the equation of the asymptote.\\
\hfill \mbox{\textit{Edexcel C3 2013 Q2 [7]}}