OCR C1 — Question 8 9 marks

Exam BoardOCR
ModuleC1 (Core Mathematics 1)
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicFunction Transformations
TypeIdentify transformation from equations
DifficultyStandard +0.3 Part (i) is straightforward identification of a vertical stretch by factor 3. Part (ii) is routine sketching with asymptote identification. Part (iii) requires setting up a tangency condition (discriminant = 0) and solving a quadratic, which is standard C1 technique but involves multiple steps and algebraic manipulation.
Spec1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02w Graph transformations: simple transformations of f(x)1.07m Tangents and normals: gradient and equations
8. (i) Describe fully a single transformation that maps the graph of \(y = \frac { 1 } { x }\) onto the graph of \(y = \frac { 3 } { x }\).
(ii) Sketch the graph of \(y = \frac { 3 } { x }\) and write down the equations of any asymptotes.
(iii) Find the values of the constant \(c\) for which the straight line \(y = c - 3 x\) is a tangent to the curve \(y = \frac { 3 } { x }\).
AnswerMarks Guidance
(i) Stretch by factor of 3 in y-direction about x-axis or stretch by factor of 3 in x-direction about y-axisB2
(ii) Graph showing curve with asymptotes at \(x = 0\) and \(y = 0\)B2 Asymptotes: \(x = 0\) and \(y = 0\)
(iii) \(\frac{3}{x} = c - 3x\)M1
\(3 = cx - 3x^2\)M1
\(3x^2 - cx + 3 = 0\)
Tangent \(\therefore\) equal roots, \(b^2 - 4ac = 0\)M1 A1
\((-c)^2 - (4 \times 3 \times 3) = 0\)M1
\(c^2 = 36\)
\(c = \pm 6\)A1 (9) 
**(i)** Stretch by factor of 3 in y-direction about x-axis or stretch by factor of 3 in x-direction about y-axis | B2 |

**(ii)** Graph showing curve with asymptotes at $x = 0$ and $y = 0$ | B2 | Asymptotes: $x = 0$ and $y = 0$

**(iii)** $\frac{3}{x} = c - 3x$ | M1 |
$3 = cx - 3x^2$ | M1 |
$3x^2 - cx + 3 = 0$ | |
Tangent $\therefore$ equal roots, $b^2 - 4ac = 0$ | M1 A1 |
$(-c)^2 - (4 \times 3 \times 3) = 0$ | M1 |
$c^2 = 36$ | |
$c = \pm 6$ | A1 | (9)

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8. (i) Describe fully a single transformation that maps the graph of $y = \frac { 1 } { x }$ onto the graph of $y = \frac { 3 } { x }$.\\
(ii) Sketch the graph of $y = \frac { 3 } { x }$ and write down the equations of any asymptotes.\\
(iii) Find the values of the constant $c$ for which the straight line $y = c - 3 x$ is a tangent to the curve $y = \frac { 3 } { x }$.\\

\hfill \mbox{\textit{OCR C1  Q8 [9]}}
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