Trigonometric Graphs and Transformations
You're viewing free preview questions — 4 questions previewing here — subscribe to unlock 63 more, plus the Full Question Bank, Mock Papers & Worksheets.Upgrade

Trigonometric Graphs and Transformations — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionSketching Graphs of y = sin x cos x tan xConcept Practice
2 marks~3 minCriterion A
A coastal monitoring buoy records wave height. Engineers model the displacement (in metres) of the buoy from its rest position using y=sin⁡xy = \sin x, where xx represents time in degrees over one complete cycle from 0°0° to 360°360°.
a
State the amplitude and period of y=sin⁡xy = \sin x, and identify the coordinates of the maximum, minimum, and all intercepts over this cycle. [1]
b
Explain what the amplitude and the minimum value tell an engineer about the buoy's motion in this context. [1]
Question diagram

Solutions

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]
Question diagram

Solutions

3QuestionGraphing Real-Life Periodic PhenomenaAssessment Practice
8 marks~12 minCriterion D
The water height at a harbour is modelled by

h(t)=3.5sin⁡(0.52t−1.57)+4.2h(t) = 3.5\sin(0.52t - 1.57) + 4.2

where hh is in metres and tt is hours after midnight.

Actual measurements:

tt (hours after midnight): 2, 8, 14

Actual hh (m): 2.1, 6.3, 2.3
a
Calculate the predicted water height at t=2t = 2, t=8t = 8, and t=14t = 14. Show all substitutions clearly. [3]
b
Calculate the percentage error at each time using

percentage error=∣predicted−actual∣actual×100%\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%

Show your working for each value. [3]
c
A fishing vessel requires a minimum water height of 3.8 m to enter the harbour safely. Advise the harbour master whether the model alone is sufficient to determine safe entry times, justifying your answer using the percentage errors from part (b). [2]
Question diagram

Solutions

4QuestionGraphing Real-Life Periodic PhenomenaAssessment Practice
6 marks~9 minCriterion B
The water depth (in metres) at a harbour is modelled over a 24-hour period. Measurements are taken every 2 hours.

Time (h)0246810121416182022
Depth (m)3.25.67.05.63.21.83.25.67.05.63.21.8


The depth is modelled by h(t)=Asin⁡(Bt)+Dh(t) = A\sin(Bt) + D, where tt is time in hours, AA is the amplitude, BB is related to the period by period=2πB\text{period} = \dfrac{2\pi}{B}, and DD is the vertical shift.
a
Justify that the period of the tidal cycle is 12 hours. [2]
b
Deduce the values of AA, BB, and DD, and write the complete model for h(t)h(t). [2]
c
A vessel requires a minimum depth of 5.0 m to enter the harbour safely. Using your model, advise the harbour master whether the vessel can enter safely at t=3t = 3 hours, justifying your answer with a calculation. [2]

Solutions

Questions 5-67

Unlock these questions with Premium

Sign Up →

That's every free question here — 4 of 67.

+63 more questions, plus Mock Papers & Worksheets, are waiting in Premium.

Upgrade