Sketch transformed/compound trig graph and identify features

Questions that ask for a sketch of a trig function with a phase shift or compound argument (e.g. sin(x + π/6), cos 2x, f(x) = a - b cos x) and require stating coordinates of intercepts, maxima, minima, or period, with equation-solving as a secondary part.

5 questions · Moderate -0.8

1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals
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CAIE P1 2008 November Q5
8 marks Moderate -0.8
5 The function f is such that \(\mathrm { f } ( x ) = a - b \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), where \(a\) and \(b\) are positive constants. The maximum value of \(\mathrm { f } ( x )\) is 10 and the minimum value is - 2 .
  1. Find the values of \(a\) and \(b\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\).
  3. Sketch the graph of \(y = \mathrm { f } ( x )\).
Edexcel C12 2014 January Q10
9 marks Moderate -0.8
10. The curve \(C\) has equation \(y = \cos \left( x - \frac { \pi } { 3 } \right) , 0 \leqslant x \leqslant 2 \pi\)
  1. In the space below, sketch the curve \(C\).
  2. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes.
  3. Solve, for \(x\) in the interval \(0 \leqslant x \leqslant 2 \pi\), $$\cos \left( x - \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 2 } }$$ giving your answers in the form \(k \pi\), where \(k\) is a rational number.
Edexcel C12 2015 January Q11
8 marks Moderate -0.8
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-16_608_952_267_495} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve \(C\) with equation \(y = \sin \left( x - 60 ^ { \circ } \right) , - 360 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\)
  1. Write down the exact coordinates of the points at which \(C\) meets the two coordinate axes.
  2. Solve, for \(- 360 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), $$4 \sin \left( x - 60 ^ { \circ } \right) = \sqrt { 6 } - \sqrt { 2 }$$ showing each stage of your working.
Edexcel C12 2016 June Q10
9 marks Moderate -0.8
10. The curve \(C\) has equation \(y = \sin \left( x + \frac { \pi } { 4 } \right) , \quad 0 \leqslant x \leqslant 2 \pi\)
  1. On the axes below, sketch the curve \(C\).
  2. Write down the exact coordinates of all the points at which the curve \(C\) meets or intersects the \(x\)-axis and the \(y\)-axis.
  3. Solve, for \(0 \leqslant x \leqslant 2 \pi\), the equation $$\sin \left( x + \frac { \pi } { 4 } \right) = \frac { \sqrt { 3 } } { 2 }$$ giving your answers in the form \(k \pi\), where \(k\) is a rational number. \includegraphics[max width=\textwidth, alt={}, center]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-14_677_1031_1446_445}
Edexcel C2 2007 June Q9
10 marks Moderate -0.8
9. (a) Sketch, for \(0 \leqslant x \leqslant 2 \pi\), the graph of \(y = \sin \left( x + \frac { \pi } { 6 } \right)\).
(b) Write down the exact coordinates of the points where the graph meets the coordinate axes.
(c) Solve, for \(0 \leqslant x \leqslant 2 \pi\), the equation $$\sin \left( x + \frac { \pi } { 6 } \right) = 0.65$$ giving your answers in radians to 2 decimal places.