Range restriction with excluded interval (linear/mixed denominator)

Questions where the function has a linear or factored denominator (with vertical asymptotes), requiring proof that y cannot take values in an open interval, typically using discriminant of a quadratic formed by y=k substitution.

6 questions · Challenging +1.0

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CAIE FP1 2016 June Q7
10 marks Standard +0.8
7 A curve \(C\) has equation \(y = \frac { x ^ { 2 } } { x - 2 }\). Find the equations of the asymptotes of \(C\). Show that there are no points on \(C\) for which \(0 < y < 8\). Sketch \(C\), giving the coordinates of the turning points.
AQA FP1 2013 June Q9
14 marks Challenging +1.2
9 A curve has equation $$y = \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 2 } - 2 x - 3 }$$
  1. Find the equations of the three asymptotes of the curve.
    1. Show that if the line \(y = k\) intersects the curve then $$( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0$$
    2. Given that the equation \(( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0\) has real roots, show that $$k ^ { 2 } - k \geqslant 0$$
    3. Hence show that the curve has only one stationary point and find its coordinates.
      (No credit will be given for solutions based on differentiation.)
  3. Sketch the curve and its asymptotes.
OCR FP2 Q5
8 marks Challenging +1.2
5
  1. Find the equations of the asymptotes of the curve with equation $$y = \frac { x ^ { 2 } + 3 x + 3 } { x + 2 }$$
  2. Show that \(y\) cannot take values between - 3 and 1 .
Pre-U Pre-U 9795/1 2012 June Q4
9 marks Standard +0.8
4 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 \(C\).
Pre-U Pre-U 9795/1 2016 June Q3
4 marks Challenging +1.2
3 A curve has equation \(y = \frac { 2 x ^ { 2 } - x - 1 } { 2 x - 3 }\).
  1. Show that the curve meets the line \(y = k\) when \(2 x ^ { 2 } - ( 2 k + 1 ) x + ( 3 k - 1 ) = 0\), and hence show that no part of the curve exists in the interval \(\frac { 1 } { 2 } < y < \frac { 9 } { 2 }\).
  2. Deduce the coordinates of the turning points of this curve.
Pre-U Pre-U 9795/1 Specimen Q7
6 marks Standard +0.8
7 A curve has equation \(y = \frac { 4 x + 11 } { ( x + 3 ) ^ { 2 } }\).
  1. Show that the curve meets the line \(y = k\) if and only if \(k \leq 4\), and deduce the coordinates of the turning point on the curve.
  2. Sketch the curve, stating the coordinates of the points where it cuts the axes, and showing clearly its asymptotes and the turning point.