Range Restrictions of Rational Functions

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

1. Show that \(y\) can never be zero.
2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
3. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your description.
4. Sketch the curve.
5. Solve the inequality \(\frac { 3 + x ^ { 2 } } { 4 - x ^ { 2 } } \leqslant - 2\).

6 The equation of a curve is \(y = \frac { 2 x ^ { 2 } - 11 x - 6 } { x - 1 }\).

1. Find the equations of the asymptotes of the curve.
2. Show that \(y\) takes all real values.

9 It is given that the equation of a curve is $$y = \frac { x ^ { 2 } - 2 a x } { x - a }$$ where \(a\) is a positive constant.

1. Find the equations of the asymptotes of the curve.
2. Show that \(y\) takes all real values.
3. Sketch the curve \(y = \frac { x ^ { 2 } - 2 a x } { x - a }\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. 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 A curve has equation \(y = \frac { x ^ { 2 } - 6 x - 5 } { x - 2 }\).

1. Find the equations of the asymptotes.
2. Show that \(y\) can take all real values.

7. (a) Find the set of values of \(k\) for which the equation $$\frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 } = k$$ has no real roots. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 }\) The curve has asymptotes \(x = a , x = b\) and \(y = c\), where \(a , b\) and \(c\) are integers.
(b) Find the value of \(a\), the value of \(b\) and the value of \(c\).
(c) Find the coordinates of the points of intersection of \(C _ { 1 }\) with the line \(y = 2\) (d) Find all the integer pairs \(( r , s )\) that satisfy \(s = \frac { r ^ { 2 } + 3 r + 8 } { r ^ { 2 } + r - 2 }\) The curve \(C _ { 2 }\) has equation \(y = \mathrm { g } ( x )\) where \(\mathrm { g } ( x ) = \frac { 2 x ^ { 2 } - 4 x + 6 } { x ^ { 2 } - 3 x }\) (e) Show that, for suitable integers \(m\) and \(n , \mathrm {~g} ( x )\) can be written in the form \(\mathrm { f } ( x + m ) + n\).
(f) Sketch \(C _ { 2 }\) showing any asymptotes and stating their equations.

5 A curve has equation \(y = \frac { x ^ { 2 } - 8 } { x - 3 }\).

1. Find the equations of the asymptotes of the curve.
2. Prove that there are no points on the curve for which \(4 < y < 8\).
3. Sketch the curve. Indicate the asymptotes in your sketch.

9 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 2 x + \lambda } { x + 1 }$$ where \(\lambda\) is a constant. Show that the equations of the asymptotes of \(C\) are independent of \(\lambda\). Find the value of \(\lambda\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case. Sketch \(C\) in the case \(\lambda = - 4\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.

The curve \(C\) has equation $$y = \frac { a x ^ { 2 } + b x + c } { x + d }$$ where \(a , b , c\) and \(d\) are constants. The curve cuts the \(y\)-axis at \(( 0 , - 2 )\) and has asymptotes \(x = 2\) and \(y = x + 1\).

1. Write down the value of \(d\).
2. Determine the values of \(a , b\) and \(c\).
3. Show that, at all points on \(C\), either \(y \leqslant 3 - 2 \sqrt { 6 }\) or \(y \geqslant 3 + 2 \sqrt { 6 }\).

The curve \(C\) has equation $$y = \frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } ,$$ where \(q\) is a positive constant.

1. Obtain the equations of the asymptotes of \(C\).
2. Find the value of \(q\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
3. Sketch \(C\) for the case \(q = 3\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.
4. It is given that, for all values of the constant \(\lambda\), the line $$y = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ passes through the point of intersection of the asymptotes of \(C\). Use this result, with the diagrams you have drawn, to show that if \(\lambda < \frac { 1 } { 2 }\) then the equation $$\frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ has no real solution if \(q\) has the value found in part (ii), but has 2 real distinct solutions if \(q = 3\).

4 A curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + x - 1 } { x - 1 }\). Find the equations of the asymptotes of \(C\). Show that there is no point on \(C\) for which \(1 < y < 9\).

9 The curve \(C\) has equation $$y = \frac { 5 x ^ { 2 } + 5 x + 1 } { x ^ { 2 } + x + 1 } .$$

1. Find the equation of the asymptote of \(C\).
2. Show that, for all real values of \(x , - \frac { 1 } { 3 } \leqslant y < 5\).
3. Find the coordinates of any stationary points of \(C\).
4. Sketch \(C\), stating the coordinates of any intersections with the \(y\)-axis.

7 The curve \(C\) has equation \(y = \frac { x ^ { 2 } + p x + 1 } { x - 2 }\), where \(p\) is a constant. Given that \(C\) has two asymptotes, find the equation of each asymptote. Find the set of values of \(p\) for which \(C\) has two distinct turning points. Sketch \(C\) in the case \(p = - 1\). Your sketch should indicate the coordinates of any intersections with the axes, but need not show the coordinates of any turning points.

9 The curve \(C\) has equation \(y = \frac { x ^ { 2 } - 3 x + 3 } { x - 2 }\). Find the equations of the asymptotes of \(C\). Show that there are no points on \(C\) for which \(- 1 < y < 3\). Find the coordinates of the turning points of \(C\). Sketch \(C\).

8 For a particular value of \(a\), the curve \(\mathrm { y } = \frac { \mathrm { a } } { \mathrm { x } ^ { 2 } }\) passes through the point \(( 3,1 )\).
Find the coordinates of all the other points on the curve where both the \(x\)-coordinate and the \(y\)-coordinate are integers.