| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Multiple transformation descriptions |
| Difficulty | Moderate -0.3 This is a straightforward C1 curve sketching question requiring basic skills: expanding to find intercepts, applying a standard horizontal translation (replace x with x-2), and identifying a vertical stretch with scale factor 1/2. All parts are routine textbook exercises with no problem-solving or novel insight required, making it slightly easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Negative cubic with max and min] | B1 | Negative cubic with a max and a min. Must clearly be a cubic – must not stop at or before \(x\) axis, do not allow straight line sections drawn with ruler/tending to extra turning points etc. Must not be a finite plot. |
| [Cubic meets \(y\)-axis at \((0,0)\) only] | B1 | Cubic that meets \(y\)-axis at \((0,0)\) only |
| [Double root at \((0,0)\), single root at \((3,0)\)] | B1 | Double root at \((0,0)\) and single root at \((3,0)\) and no other roots |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = (x-2)^2(5-x)\) or \(y = 3(x-2)^2 - (x-2)^3\) | M1 | Translates curve by \(+2\) or \(-2\) parallel to the \(x\)-axis; must be consistent; e.g. for M1 \((x-2)^2(3-(x-2))\) but not \((x-2)^2(3-x-2)\) |
| A1 | Fully correct, must have "\(y=\)". ISW expansions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Stretch | B1 | Must use the word "stretch". Do not accept "in/on/across/up the \(y\) axis". |
| Scale factor one-half parallel to the \(y\)-axis | B1 | Must have "factor" or "scale factor". For "parallel to the \(y\) axis" allow "vertically", "in the \(y\) direction". Allow second B1 after "squash" etc. but not after "translate" etc. |
## Question 7:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Negative cubic with max and min] | B1 | Negative cubic with a max and a min. Must clearly be a cubic – must not stop at or before $x$ axis, do not allow straight line sections drawn with ruler/tending to extra turning points etc. Must not be a finite plot. |
| [Cubic meets $y$-axis at $(0,0)$ only] | B1 | Cubic that meets $y$-axis at $(0,0)$ only |
| [Double root at $(0,0)$, single root at $(3,0)$] | B1 | Double root at $(0,0)$ and single root at $(3,0)$ and no other roots |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = (x-2)^2(5-x)$ or $y = 3(x-2)^2 - (x-2)^3$ | M1 | Translates curve by $+2$ or $-2$ parallel to the $x$-axis; must be consistent; e.g. for M1 $(x-2)^2(3-(x-2))$ but not $(x-2)^2(3-x-2)$ |
| | A1 | Fully correct, must have "$y=$". ISW expansions |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Stretch | B1 | Must use the word "stretch". Do not accept "in/on/across/up the $y$ axis". |
| Scale factor one-half parallel to the $y$-axis | B1 | Must have "factor" or "scale factor". For "parallel to the $y$ axis" allow "vertically", "in the $y$ direction". Allow second B1 after "squash" etc. but not after "translate" etc. |
---7 (i) Sketch the curve $y = x ^ { 2 } ( 3 - x )$ stating the coordinates of points of intersection with the axes.\\
(ii) The curve $y = x ^ { 2 } ( 3 - x )$ is translated by 2 units in the positive direction parallel to the $x$-axis. State the equation of the curve after it has been translated.\\
(iii) Describe fully a transformation that transforms the curve $y = x ^ { 2 } ( 3 - x )$ to $y = \frac { 1 } { 2 } x ^ { 2 } ( 3 - x )$.
\hfill \mbox{\textit{OCR C1 2016 Q7 [7]}}