Parameter values from curve properties

6. The curve \(C\) has equation \(y = \frac { 4 } { x } + k\), where \(k\) is a positive constant.

1. Sketch a graph of \(C\), stating the equation of the horizontal asymptote and the coordinates of the point of intersection with the \(x\)-axis. The line with equation \(y = 10 - 2 x\) is a tangent to \(C\).
2. Find the possible values for \(k\).
   \(\_\_\_\_\) -

The curve \(C\) has equation $$y = \frac { x ^ { 2 } + 2 \lambda x } { x ^ { 2 } - 2 x + \lambda }$$ where \(\lambda\) is a constant and \(\lambda \neq - 1\).

1. Show that \(C\) has at most two stationary points.
2. Show that if \(C\) has exactly two stationary points then \(\lambda > - \frac { 5 } { 4 }\).
3. Find the set of values of \(\lambda\) such that \(C\) has two vertical asymptotes.
4. Find the \(x\)-coordinates of the points of intersection of \(C\) with
   (a) the \(x\)-axis,
   (b) the horizontal asymptote.
5. Sketch \(C\) in each of the cases
   (a) \(\lambda < - 2\),
   (b) \(\lambda > 2\).

3. The function f is even and has domain \(\mathbb { R }\). For \(x \geq 0 , \mathrm { f } ( x ) = x ^ { 2 } - 4 a x\), where \(a\) is a positive constant.

1. In the space below, sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of all the points at which the curve meets the axes.
2. Find, in terms of \(a\), the value of \(\mathrm { f } ( 2 a )\) and the value of \(\mathrm { f } ( - 2 a )\). Given that \(a = 3\),
3. use algebra to find the values of \(x\) for which \(\mathrm { f } ( x ) = 45\).

1 A family of functions is defined as $$f ( x ) = a x + \frac { x ^ { 2 } } { 1 + x } , \quad x \neq - 1$$ where the parameter \(a\) is a real number. You may find it helpful to use a slider (for \(a\) ) to investigate the family of curves \(y = f ( x )\).

1. 1. On the axes in the Printed Answer Booklet, sketch the curve \(y = f ( x )\) in each of the following cases.
      • \(a = - 2\)
2. \(a = - 1\)
3. \(a = 0\)
   (ii) State a feature which is common to the curve in all three cases, \(a = - 2\), \(a = - 1\) and \(a = 0\).
   (iii) State a feature of the curve for the cases \(a = - 2 , a = - 1\) that is not a feature of the curve in the case \(a = 0\).
4. 1. Determine the equation of the oblique asymptote to the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\) in terms of \(a\).
   2. For \(b \neq - 1,0,1\) let \(A\) be the point with coordinates ( \(- b , \mathrm { f } ( - b )\) ) and let \(B\) be the point with coordinates ( \(b , \mathrm { f } ( b )\) ).
Show that the \(y\)-coordinate of the point at which the chord to the curve \(y = f ( x )\) between \(A\) and \(B\) meets the \(y\)-axis is independent of \(a\).
(iii) With \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), determine the range of values of \(a\) for which
• \(y \geqslant 0\) for all \(x \geqslant 0\)
• \(y \leqslant 0\) for all \(x \geqslant 0\)
• In the case of \(a = 0\), the curve \(\mathrm { y } = \sqrt [ 4 ] { \mathrm { f } ( \mathrm { x } ) }\) has a cusp.
Find its coordinates and fully justify that it is a cusp.

1 A family of curves is given by the equation $$y = \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 }$$ where the parameter \(a\) is a real number.

1. 1. On the axes in the Printed Answer Booklet, sketch the curve in each of these cases.
      • \(a = - 0.5\)
2. \(a = - 0.1\)
3. \(a = 0.5\)
   (ii) State one feature of the curve for the cases \(a = - 0.5\) and \(a = - 0.1\) that is not a feature of the curve in the case \(a = 0.5\).
   (iii) By using a slider for \(a\), or otherwise, write down the non-zero value of \(a\) for which the points on the curve (\textit{) all lie on a straight line.
   (iv) Write down the equation of the vertical asymptote of the curve (}).
The equation of the curve (*) can be written in the form \(y = x + A + \frac { a ^ { 2 } - a } { x - 1 }\), where \(A\) is a constant.
(v) Show that \(A = 0\).
(vi) Hence, or otherwise, find the value of $$\lim _ { x \rightarrow \infty } \left( \frac { x ^ { 2 } - x + a ^ { 2 } - a } { x - 1 } - x \right) .$$ (vii) Explain the significance of the result in part (a)(vi) in terms of a feature of the curve (*).
4. In this part of the question the value of the parameter \(a\) satisfies \(0 < a < 1\). For values of \(a\) in this range the curve intersects the \(x\)-axis at points X and Y . The point Z has coordinates \(( 0 , - 1 )\). These three points form a triangle XYZ.
   1. Determine, in terms of \(a\), the area of the triangle XYZ.
   2. Find the maximum area of the triangle XYZ.