Transformation of specific function type

1. (a) Sketch the graph of the curve \(C\) with equation

$$y = \frac { 4 } { x - k }$$ where \(k\) is a positive constant.
Show on your sketch

• the coordinates of any points where \(C\) cuts the coordinate axes
• the equation of the vertical asymptote to \(C\)

Given that the straight line with equation \(y = 9 - x\) does not cross or touch \(C\)
(b) find the range of values of \(k\).

10

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{450c1c3a-9290-4afa-a051-112b60cf19c0-3_753_775_360_726} \captionsetup{labelformat=empty} \caption{Fig. 10}
   \end{figure} Fig. 10 shows a sketch of the graph of \(y = \frac { 1 } { x }\).
   Sketch the graph of \(y = \frac { 1 } { x - 2 }\), showing clearly the coordinates of any points where it crosses the axes.
2. Find the value of \(x\) for which \(\frac { 1 } { x - 2 } = 5\).
3. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = x\) and \(y = \frac { 1 } { x - 2 }\). Give your answers in the form \(a \pm \sqrt { b }\). Show the position of these points on your graph in part (i).

11.

1. Sketch the curve with equation $$y = k - \frac { 1 } { 2 x } \quad \text { where } k \text { is a positive constant }$$ State, in terms of \(k\), the coordinates of any points of intersection with the coordinate axes and the equation of the horizontal asymptote. The straight line \(l\) has equation \(y = 2 x + 3\)
   Given that \(l\) cuts the curve in two distinct places,
2. find the range of values of \(k\), writing your answer in set notation.

2. Given that $$f ( x ) = \ln x , x > 0$$ Sketch on separate axes the graphs of
i) \(y = f ( x )\)
ii) \(\quad y = f ( x - 4 )\) Show on each diagram, the point where the graph meets or crosses the \(x\)-axis. In each case, state the equation of the asymptote.

7

1. The sketch shows the graph of \(y = \sin ^ { - 1 } x\).
   \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-006_819_824_456_591} Write down the coordinates of the points \(P\) and \(Q\), the end-points of the graph.
2. Sketch the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\).

7

1. The sketch shows the graph of \(y = \sin ^ { - 1 } x\).
   \includegraphics[max width=\textwidth, alt={}, center]{908f530c-076d-47b1-90dd-38dbfe44f898-05_835_834_447_587} Write down the coordinates of the points \(P\) and \(Q\), the end-points of the graph.
2. Sketch the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\).