Simple rational function analysis

6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).

1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote. [2]
2. Find the coordinates of any stationary points on \(C\).
   \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-12_2715_35_144_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-13_2718_33_141_23}
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch \(y ^ { 2 } = \frac { x + 1 } { x ^ { 2 } + 3 }\), stating the coordinates of the stationary points and the intersections with the axes.

9 The curve \(C\) has equation $$y = \frac { 2 x ^ { 2 } + 2 x + 3 } { x ^ { 2 } + 2 }$$ Show that, for all \(x , 1 \leqslant y \leqslant \frac { 5 } { 2 }\). Find the coordinates of the turning points on \(C\). Find the equation of the asymptote of \(C\). Sketch the graph of \(C\), stating the coordinates of any intersections with the \(y\)-axis and the asymptote.

9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 4 x } { x ^ { 2 } + 9 }$$

1. 1. The graph of \(y = \mathrm { f } ( x )\) has an asymptote which is parallel to the \(x\)-axis. Find the equation of this asymptote.
   2. Explain why the graph of \(y = \mathrm { f } ( x )\) has no asymptotes parallel to the \(y\)-axis.
2. Show that the equation \(\mathrm { f } ( x ) = k\) has two equal roots if \(9 k ^ { 2 } - 9 k - 4 = 0\).
3. Hence find the coordinates of the two stationary points on the graph of \(y = \mathrm { f } ( x )\).

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   General Certificate of Education
   June 2005
   Advanced Subsidiary Examination MATHEMATICS
   MFP1
   Unit Further Pure 1 ASSESSMENT and
   QUALIFICATIONS
   ALLIANCE Wednesday 22 June 2005 Afternoon Session Insert for use in Question 7.
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9 A curve \(C\) has equation $$y = \frac { ( x + 1 ) ( x - 3 ) } { x ( x - 2 ) }$$

1. 1. Write down the coordinates of the points where \(C\) intersects the \(x\)-axis. (2 marks)
   2. Write down the equations of all the asymptotes of \(C\).
2. 1. Show that, if the line \(y = k\) intersects \(C\), then $$( k - 1 ) ( k - 4 ) \geqslant 0$$
   2. Given that there is only one stationary point on \(C\), find the coordinates of this stationary point.
      (No credit will be given for solutions based on differentiation.)
3. Sketch the curve \(C\).

2 Curve \(C\) has equation \(y = \frac { 1 } { ( x - 1 ) ^ { 2 } }\)
State the equations of the asymptotes to curve \(C\)
Tick ( ✓ ) one box.
[0pt] [1 mark]
\(x = 0\) and \(y = 0\)
\includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-02_111_113_1975_735}
\(x = 0\) and \(y = 1\)
\includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-02_113_113_2126_735}
\(x = 1\) and \(y = 0\)
\includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-02_111_113_2279_735}
\(x = 1\) and \(y = 1\)
\includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-02_117_113_2426_735}

11

1. Curve \(C\) has equation $$y = \frac { x ^ { 2 } + p x - q } { x ^ { 2 } - r }$$ where \(p , q\) and \(r\) are positive constants.
   Write down the equations of its asymptotes.

2 Which of the straight lines given below is an asymptote to the curve $$y = \frac { a x ^ { 2 } } { x - 1 }$$ where \(a\) is a non-zero constant? Circle your answer.
[0pt] [1 mark]
\(y = a x + a\)
\(y = a x\)
\(y = a x - a\)
\(y = a\)