Trigonometric Graphs and Transformations
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Trigonometric Graphs and Transformations — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionSketching Graphs of y = sin x cos x tan xConcept Practice
2 marks~3 minCriterion B
The diagram shows the graph of y=tan⁡xy = \tan x for −π<x<π-\pi < x < \pi, with asymptotes at x=−π2x = -\frac{\pi}{2} and x=π2x = \frac{\pi}{2} clearly marked. Below it is an incomplete sketch of y=tan⁡(2x)y = \tan(2x) for the same domain, with only one asymptote drawn at x=−π4x = -\frac{\pi}{4}. Outline the pattern in the spacing of the asymptotes as the coefficient inside the tangent function changes. Use this pattern to predict the x-coordinate of the next asymptote for y=tan⁡(2x)y = \tan(2x) to the right of the origin. State a general rule linking the coefficient kk in y=tan⁡(kx)y = \tan(kx) to the distance between consecutive asymptotes.
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2QuestionSketching Graphs of y = sin x cos x tan xAssessment Practice
3 marks~5 minCriterion C
A sound engineer models the pressure wave of a pure musical tone using y=sin⁡xy = \sin x, where xx represents time in degrees over one complete cycle (0°≤x≤360°0° \leq x \leq 360°).

The function passes through (0°,0)(0°, 0), (90°,1)(90°, 1), (180°,0)(180°, 0), (270°,−1)(270°, -1), and (360°,0)(360°, 0).
a
Calculate sin⁡45°\sin 45°, giving your answer in exact form. [1]
b
Deduce the values of sin⁡135°\sin 135° and sin⁡225°\sin 225°, using symmetry properties of the sine function. [1]
c
The engineer states: "The pressure wave is perfectly balanced — positive and negative peaks are equal in magnitude throughout the cycle." Using your results from (a) and (b), and the value sin⁡315°\sin 315°, justify whether this statement is correct. [1]
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3QuestionSketching Graphs of y = sin x cos x tan xConcept Practice
2 marks~3 minCriterion A
A buoy bobs in the ocean. Its vertical displacement, dd metres, from its rest position is modelled by d=sin⁡xd = \sin x, where xx is measured in degrees and 0°≤x≤360°0° \leq x \leq 360° represents one tidal cycle.
a
Construct a sketch of y=sin⁡xy = \sin x for −360°≤x≤360°-360° \leq x \leq 360°, plotting key points at multiples of 90°90° and drawing a smooth curve through them. [1]
b
The buoy is considered "active" when d>0d > 0. Interpret how many degrees of each 360°360° cycle the buoy spends above its rest position, and explain what this reveals about the symmetry of the model. [1]
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4QuestionSketching Graphs of y = sin x cos x tan xAssessment Practice
7 marks~11 minCriterion D
A Ferris wheel has a diameter of 20 m. Its lowest point is 2 m above the ground. The height hh (in metres) of a passenger above the ground after tt seconds is modelled by

h(t)=10sin⁡(0.5t)+12,h(t) = 10\sin(0.5t) + 12,

where the wheel rotates at constant speed.
a
Calculate the height of the passenger after 3 seconds. Give your answer to one decimal place. [2]
b
Deduce the first time, to one decimal place, at which the passenger reaches a height of 17 m. [2]
c
The Ferris wheel operator uses this model to plan safe stopping positions. Advise the operator whether this model alone is sufficient for that purpose, referring to its assumptions and to real-world conditions. [3]
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