1.05f Trigonometric function graphs: symmetries and periodicities

162 questions

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AQA AS Paper 1 2018 June Q3
2 marks Easy -1.2
State the interval for which \(\sin x\) is a decreasing function for \(0° \leq x \leq 360°\) [2 marks]
AQA AS Paper 1 2022 June Q2
1 marks Easy -1.8
The graph of the function \(y = \cos \frac{1}{2}x\) for \(0° \leq x \leq 360°\) is one of the graphs shown below. Identify the correct graph. Tick (✓) one box. [1 mark] \includegraphics{figure_2}
AQA AS Paper 1 2024 June Q12
6 marks Moderate -0.8
The monthly mean temperature of a city, \(T\) degrees Celsius, may be modelled by the equation $$T = 15 + 8 \sin (30m - 120)^\circ$$ where \(m\) is the month number, counting January = 1, February = 2, through to December = 12
  1. Using this model, calculate the monthly mean temperature of the city for May, the fifth month. [2 marks]
  2. Using this model, find the month with the highest mean temperature. [2 marks]
  3. Climate change may affect the parameters, 8, 30, 120 and 15, used in this model.
    1. State, with a reason, which parameter would be increased because of an overall rise in temperatures. [1 mark]
    2. State, with a reason, which parameter would be increased because of the occurrence of more extreme temperatures. [1 mark]
AQA Paper 2 2018 June Q8
10 marks Standard +0.8
  1. Determine a sequence of transformations which maps the graph of \(y = \sin x\) onto the graph of \(y = \sqrt{3} \sin x - 3 \cos x + 4\) Fully justify your answer. [7 marks]
    1. Show that the least value of \(\frac{1}{\sqrt{3} \sin x - 3 \cos x + 4}\) is \(\frac{2 - \sqrt{3}}{2}\) [2 marks]
    2. Find the greatest value of \(\frac{1}{\sqrt{3} \sin x - 3 \cos x + 4}\) [1 mark]
AQA Paper 2 Specimen Q5
9 marks Standard +0.3
  1. Determine a sequence of transformations which maps the graph of \(y = \cos \theta\) onto the graph of \(y = 3\cos \theta + 3\sin \theta\) Fully justify your answer. [6 marks]
  2. Hence or otherwise find the least value and greatest value of $$4 + (3\cos \theta + 3\sin \theta)^2$$ Fully justify your answer. [3 marks]
AQA Paper 3 2022 June Q5
3 marks Easy -1.2
  1. Sketch the graph of $$y = \sin 2x$$ for \(0° \leq x \leq 360°\) \includegraphics{figure_5a} [2 marks]
  2. The equation $$\sin 2x = A$$ has exactly two solutions for \(0° \leq x \leq 360°\) State the possible values of \(A\). [1 mark]
SPS SPS FM 2020 December Q4
6 marks Moderate -0.8
The following diagram shows the curve \(y = a \sin(b(x + c)) + d\), where \(a, b, c\) and \(d\) are all positive constants and \(x\) is measured in radians. The curve has a maximum point at \((1, 3.5)\) and a minimum point at \((2, 0.5)\). \includegraphics{figure_4}
  1. Write down the value of \(a\) and the value of \(d\). [2]
  2. Find the value of \(b\). [2]
  3. Find the smallest possible value of \(c\), given that \(c > 0\). [2]
SPS SPS SM Pure 2022 June Q10
6 marks Moderate -0.8
  1. Sketch the graph of \(y = \cos x\) in the interval \(0 \leq x \leq 2\pi\). State the values of the intercepts with the coordinate axes. [2 marks]
    1. Given that $$\sin^2 \theta = \cos \theta(2 - \cos \theta)$$ prove that \(\cos \theta = \frac{1}{2}\). [2 marks]
    2. Hence solve the equation $$\sin^2 2x = \cos 2x(2 - \cos 2x)$$ in the interval \(0 \leq x \leq \pi\) [2 marks]
SPS SPS SM Pure 2023 June Q16
8 marks Standard +0.3
\includegraphics{figure_5} A horizontal path connects an island to the mainland. On a particular morning, the height of the sea relative to the path, \(H\) m, is modelled by the equation $$H = 0.8 + k \cos(30t - 70)°$$ where \(k\) is a constant and \(t\) is number of hours after midnight. Figure 5 shows a sketch of the graph of \(H\) against \(t\). Use the equation of the model to answer parts (a), (b) and (c).
  1. Find the time of day at which the height of the sea is at its maximum. [2] Given that the maximum height of the sea relative to the path is 2 m,
    1. find a complete equation for the model,
    2. state the minimum height of the sea relative to the path.
    [2] It is safe to use the path when the sea is 10 centimetres or more below the path.
  3. Find the times between which it is safe to use the path. (Solutions relying entirely on calculator technology are not acceptable.) [4]
OCR Further Pure Core 2 2021 June Q3
6 marks Standard +0.3
\(A\) is a fixed point on a smooth horizontal surface. A particle \(P\) is initially held at \(A\) and released from rest. It subsequently performs simple harmonic motion in a straight line on the surface. After its release it is next at rest after \(0.2\) seconds at point \(B\) whose displacement is \(0.2\) m from \(A\). The point \(M\) is halfway between \(A\) and \(B\). The displacement of \(P\) from \(M\) at time \(t\) seconds after release is denoted by \(x\) m.
  1. Sketch a graph of \(x\) against \(t\) for \(0 \leq t \leq 0.4\). [4]
  2. Find the displacement of \(P\) from \(M\) at \(0.75\) seconds after release. [2]
Pre-U Pre-U 9794/2 2011 June Q4
9 marks Standard +0.3
  1. On the same diagram, sketch the graphs of \(y = 2 \sec x\) and \(y = 1 + 3 \cos x\), for \(0 \leqslant x \leqslant \pi\). [4]
  2. Solve the equation \(2 \sec x = 1 + 3 \cos x\), where \(0 \leqslant x \leqslant \pi\). [5]
Pre-U Pre-U 9794/2 2012 June Q5
3 marks Easy -2.0
Sketch, on separate diagrams, the graphs of the following functions for \(0 \leqslant x \leqslant 2\pi\) giving the coordinates of all points of intersection with the axes.
  1. \(y = \sin x\). [1]
  2. \(y = \sin\left(x + \frac{1}{6}\pi\right)\). [2]