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
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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 \(C\).

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

1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
2. Deduce the coordinates of the turning points on \(C\).
3. Sketch \(C\).

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

1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
2. Deduce the coordinates of the turning points on \(C\).
3. Sketch \(C\).

A curve has equation \(y = \frac{x^2}{(x-2)(x+1)}\).

1. Write down the equations of the three asymptotes. [3]
2. Determine whether the curve approaches the horizontal asymptote from above or from below for
   1. large positive values of \(x\),
   2. large negative values of \(x\). [3]
3. Sketch the curve. [4]
4. Solve the inequality \(\frac{x^2}{(x-2)(x+1)} > 0\). [3]

A curve has equation \(y = \frac{x^2 - 4}{(x-3)(x+1)(x-1)}\).

1. Write down the coordinates of the points where the curve crosses the axes. [3]
2. Write down the equations of the three vertical asymptotes and the one horizontal asymptote. [4]
3. Determine whether the curve approaches the horizontal asymptote from above or below for
   1. large positive values of \(x\),
   2. large negative values of \(x\). [3]
4. Sketch the curve. [4]

Curve \(C\) has equation \(y = \frac{1}{(x-1)^2}\) State the equations of the asymptotes to curve \(C\) Tick (\(\checkmark\)) one box. [1 mark] \(x = 0\) and \(y = 0\) \qquad \(\square\) \(x = 0\) and \(y = 1\) \qquad \(\square\) \(x = 1\) and \(y = 0\) \qquad \(\square\) \(x = 1\) and \(y = 1\) \qquad \(\square\)

% Figure 2 shows curve with vertical asymptotes at x = -2 and x = 2, horizontal asymptote at y = 1, with U-shaped region between asymptotes \includegraphics{figure_2} Figure 2 Figure 2 shows a sketch of the curve \(C\) with equation \(y = \frac{x^2 - 2}{x^2 - 4}\) and \(x \neq \pm 2\). The curve cuts the \(y\)-axis at \(U\).

1. Write down the coordinates of the point \(U\). [1]

The point \(P\) with \(x\)-coordinate \(a\) (\(a \neq 0\)) lies on \(C\).

1. Show that the normal to \(C\) at \(P\) cuts the \(y\)-axis at the point $$\left(0, \frac{a^2 - 2}{a^2 - 4} - \frac{(a^2 - 4)^2}{4}\right)$$ [6]

The circle \(E\), with centre on the \(y\)-axis, touches all three branches of \(C\).

1. 1. Show that $$\frac{a^2}{2(a^2-4)} - \frac{(a^2-4)^2}{4} = a^2 + \frac{(a^2-4)^4}{16}$$
   2. Hence, show that $$(a^2 - 4)^2 = 1$$
   3. Find the centre and radius of \(E\).
   [10]

[Total 17 marks]