Basic factored form sketching

5 On separate diagrams,

1. sketch the curve \(y = \frac { 1 } { x }\),
2. sketch the curve \(y = x \left( x ^ { 2 } - 1 \right)\), stating the coordinates of the points where it crosses the \(x\)-axis,
3. sketch the curve \(y = - \sqrt { } x\).

6

1. Expand and simplify \(( x - 5 ) ( x + 2 ) ( x + 5 )\).
2. Sketch the curve \(y = ( x - 5 ) ( x + 2 ) ( x + 5 )\), giving the coordinates of the points where the curve crosses the axes.

1 Sketch the graph of \(y = 9 - x ^ { 2 }\).

5 In this question, you are required to investigate the curve with equation $$y = x ^ { m } ( 1 - x ) ^ { n } , \quad 0 \leqslant x \leqslant 1 ,$$ for various positive values of \(m\) and \(n\).

1. On separate diagrams, sketch the curve in each of the following cases.
   (A) \(m = 1 , n = 1\),
   (B) \(m = 2 , n = 2\),
   (C) \(m = 2 , n = 4\),
   (D) \(m = 4 , n = 2\).
2. What feature does the curve have when \(m = n\) ? What is the effect on the curve of interchanging \(m\) and \(n\) when \(m \neq n\) ?
3. Describe how the \(x\)-coordinate of the maximum on the curve varies as \(m\) and \(n\) vary. Use calculus to determine the \(x\)-coordinate of the maximum.
4. Find the condition on \(m\) for the gradient to be zero when \(x = 0\). State a corresponding result for the gradient to be zero when \(x = 1\).
5. Use your calculator to investigate the shape of the curve for large values of \(m\) and \(n\). Hence conjecture what happens to the value of the integral \(\int _ { 0 } ^ { 1 } x ^ { m } ( 1 - x ) ^ { n } \mathrm {~d} x\) as \(m\) and \(n\) tend to infinity.
6. Use your calculator to investigate the shape of the curve for small values of \(m\) and \(n\). Hence conjecture what happens to the shape of the curve as \(m\) and \(n\) tend to zero. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
   If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1 GE.
   OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

2 Sketch the graph of \(y = 9 - x ^ { 2 }\).

6 Sketch the following curves.

1. \(y = \frac { 2 } { x }\)
2. \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\)
   \(7 \quad O A B C\) is a parallelogram with \(\overrightarrow { O A } = \mathbf { a }\) and \(\overrightarrow { O C } = \mathbf { c } . P\) is the midpoint of \(A C\).
   \includegraphics[max width=\textwidth, alt={}, center]{3fdf2b2d-20a9-4d68-b760-57ec529b5893-4_298_735_383_657}
3. Find the following in terms of \(\mathbf { a }\) and \(\mathbf { c }\), simplifying your answers.

6. Sketch the graph of $$y = ( x - k ) ^ { 2 } ( x + 2 k )$$ where \(k\) is a positive constant.
Label the coordinates of the points where the graph meets the axes.
\includegraphics[max width=\textwidth, alt={}, center]{2fa9e78c-8210-456c-9b70-5378609ac47d-10_1253_1207_596_395}
[0pt] [BLANK PAGE]

14. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),

1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).