Sketch reciprocal function graphs

5. \includegraphics[max width=\textwidth, alt={}, center]{5dd332a5-56d9-407a-8ff6-fa59294b358d-2_520_787_246_479} The diagram shows the graph of \(y = \mathrm { f } ( x )\). The graph has a minimum at \(\left( \frac { \pi } { 2 } , - 1 \right)\), a maximum at \(\left( \frac { 3 \pi } { 2 } , - 5 \right)\) and an asymptote with equation \(x = \pi\).

1. Showing the coordinates of any stationary points, sketch the graph of \(y = | \mathrm { f } ( x ) |\). Given that $$\mathrm { f } : x \rightarrow a + b \operatorname { cosec } x , \quad x \in \mathbb { R } , \quad 0 < x < 2 \pi , \quad x \neq \pi$$
2. find the values of the constants \(a\) and \(b\),
3. find, to 2 decimal places, the \(x\)-coordinates of the points where the graph of \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.

1. (i) Sketch the graph of \(y = 2 + \sec \left( x - \frac { \pi } { 6 } \right)\) for \(x\) in the interval \(0 \leq x \leq 2 \pi\).

Show on your sketch the coordinates of any turning points and the equations of any asymptotes.
(ii) Find, in terms of \(\pi\), the \(x\)-coordinates of the points where the graph crosses the \(x\)-axis.

7

1. Sketch the graph of \(y = \sec x\) for \(0 \leqslant x \leqslant 2 \pi\).
2. Solve the equation \(\sec x = 3\) for \(0 \leqslant x \leqslant 2 \pi\), giving the roots correct to 3 significant figures.
3. Solve the equation \(\sec \theta = 5 \operatorname { cosec } \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\), giving the roots correct to 3 significant figures.

8

1. 1. Sketch the graph of \(y = \operatorname { cosec } x\) for \(0 < x < 4 \pi\).
   2. It is given that \(\operatorname { cosec } \alpha = \operatorname { cosec } \beta\), where \(\frac { 1 } { 2 } \pi < \alpha < \pi\) and \(2 \pi < \beta < \frac { 5 } { 2 } \pi\). By using your sketch, or otherwise, express \(\beta\) in terms of \(\alpha\).
2. 1. Write down the identity giving \(\tan 2 \theta\) in terms of \(\tan \theta\).
   2. Given that \(\cot \phi = 4\), find the exact value of \(\tan \phi \cot 2 \phi \tan 4 \phi\), showing all your working.

6

1. The diagram shows part of the graph of \(\mathrm { y } = \operatorname { cosec } \mathrm { x }\), where \(x\) is in radians. State the equations of the three vertical asymptotes that can be seen. \includegraphics[max width=\textwidth, alt={}, center]{4fac72cb-85cb-48d9-8817-899ef3f80a0f-06_734_672_603_324} The tangent to the graph at the point P with \(x\)-coordinate \(\frac { \pi } { 3 }\) meets the \(x\)-axis at Q .
2. Show that the \(x\)-coordinate of Q is \(\frac { \pi } { 3 } + \sqrt { 3 }\). (You may use without proof the result that the derivative of \(\operatorname { cosec } x\) is \(- \operatorname { cosec } x \cot x\).)

5 Fig. 5 shows part of the curve \(y = \operatorname { cosec } x\) together with the \(x\) - and \(y\)-axes. \begin{figure}[h] \end{figure}

1. For the section of the curve which is shown in Fig. 5, write down
   1. the equations of the two vertical asymptotes,
   2. the coordinates of the minimum point.
2. Show that the equation \(x = \operatorname { cosec } x\) has a root which lies between \(x = 1\) and \(x = 2\).
3. Use the iteration \(\mathrm { x } _ { \mathrm { n } + 1 } = \operatorname { cosec } \left( \mathrm { x } _ { \mathrm { n } } \right)\), with \(x _ { 0 } = 1\), to find
   1. the values of \(x _ { 1 }\) and \(x _ { 2 }\), correct to 5 decimal places,
   2. this root of the equation, correct to 3 decimal places.
4. There is another root of \(x = \operatorname { cosec } x\) which lies between \(x = 2\) and \(x = 3\). Determine whether the iteration \(\mathrm { x } _ { \mathrm { n } + 1 } = \operatorname { cosec } \left( \mathrm { x } _ { \mathrm { n } } \right)\) with \(x _ { 0 } = 2.5\) converges to this root.
5. Sketch the staircase or cobweb diagram for the iteration, starting with \(x _ { 0 } = 2.5\), on the diagram in the Printed Answer Booklet.

6

1. Sketch the curve with equation \(y = \operatorname { cosec } x\) for \(0 < x < \pi\).
2. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 0.1 } ^ { 0.5 } \operatorname { cosec } x \mathrm {~d} x\), giving your answer to three significant figures.

2

1. The diagram shows the graph of \(y = \sec x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   
   [diagram]
   1. The point \(A\) on the curve is where \(x = 0\). State the \(y\)-coordinate of \(A\).
   2. Sketch, on the axes given on page 3, the graph of \(y = | \sec 2 x |\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Solve the equation \(\sec x = 2\), giving all values of \(x\) in degrees in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
3. Solve the equation \(\left| \sec \left( 2 x - 10 ^ { \circ } \right) \right| = 2\), giving all values of \(x\) in degrees in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

5. (a) Sketch the graph of \(y = 2 + \sec \left( x - \frac { \pi } { 6 } \right)\) for \(x\) in the interval \(0 \leq x \leq 2 \pi\). Show on your sketch the coordinates of any turning points and the equations of any asymptotes.
(b) Find, in terms of \(\pi\), the \(x\)-coordinates of the points where the graph crosses the \(x\)-axis.

4. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\). The graph has a minimum at \(\left( \frac { \pi } { 2 } , - 1 \right)\), a maximum at \(\left( \frac { 3 \pi } { 2 } , - 5 \right)\) and an asymptote with equation \(x = \pi\).

1. Showing the coordinates of any stationary points, sketch the graph of \(y = | \mathrm { f } ( x ) |\). Given that $$f : x \rightarrow a + b \operatorname { cosec } x , \quad x \in \mathbb { R } , \quad 0 < x < 2 \pi , \quad x \neq \pi ,$$
2. find the values of the constants \(a\) and \(b\),
3. find, to 2 decimal places, the \(x\)-coordinates of the points where the graph of \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.

6 Use the Insert provided for this question. The graph of \(y = \tan x\) is given on the Insert.
On this graph sketch the graph of \(y = \operatorname { cotx }\).
Show clearly where your graph crosses the graph of \(y = \tan x\) and indicate the asymptotes.

4

1. Sketch the graph of \(y = \sec \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
2. Solve \(\sec \theta = \operatorname { cosec } \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).