Trigonometric Graphs and Transformations

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

1QuestionSketching Graphs of y = sin x cos x tan xConcept Practice

2 marks~3 minCriterion A

A buoy bobs in the ocean. Its vertical displacement,

d

metres, from its rest position is modelled by

d = \sin x

, where

x

is measured in degrees and

0° \leq x \leq 360°

represents one tidal cycle.

Construct a sketch of

y = \sin x

for

-360° \leq x \leq 360°

, plotting key points at multiples of

90°

and drawing a smooth curve through them. [1]

The buoy is considered "active" when

d > 0

. Interpret how many degrees of each

360°

cycle the buoy spends above its rest position, and explain what this reveals about the symmetry of the model. [1]

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 x

, where

x

represents time in degrees over one complete cycle (

0° \leq x \leq 360°

).

The function passes through

(0°, 0)

(90°, 1)

(180°, 0)

(270°, -1)

, and

(360°, 0)

Calculate

\sin 45°

, giving your answer in exact form. [1]

Deduce the values of

\sin 135°

and

\sin 225°

, using symmetry properties of the sine function. [1]

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°

, justify whether this statement is correct. [1]

Solutions

3QuestionSketching Graphs of y = sin x cos x tan xAssessment Practice

6 marks~9 minCriterion D

The tide height at a coastal station is modelled by

h(t) = 3.2\sin(30t)^\circ + 4.5

where

h

is the height in metres and

t

is the time in hours after midnight. Observed heights are recorded below.

Time (h)	0	2	4	6	8	10	12
Height (m)	4.5	7.2	5.8	1.3	3.8	6.9	4.5

Calculate the predicted tide heights at

t = 2

h and

t = 8

h. [2]

Calculate the percentage error between the predicted and observed heights at

t = 8

h, using

\text{percentage error} = \frac{|\text{predicted} - \text{observed}|}{\text{observed}} \times 100\%.

[2]

A harbour authority requires model predictions to be within 15% of observed heights at all recorded times to approve the model for operational use. Advise the harbour authority whether this model should be approved, justifying your answer using your results. [2]

Solutions

4QuestionAmplitude Period and Phase ShiftAssessment Practice

8 marks~12 minCriterion B

A coastal engineer models ocean wave height using functions of the form

y = a\sin(bx) + d

, where

b > 0

x

is time in seconds, and

y

is height in metres. Three recorded wave patterns are:

y = 2\sin(3x) + 1

: amplitude

= 2

, period

= \dfrac{2\pi}{3}

, vertical shift

= 1

y = 5\sin(2x) - 4

: amplitude

= 5

, period

= \pi

, vertical shift

= -4

y = \dfrac{1}{2}\sin(4x) + 3

: amplitude

= \dfrac{1}{2}

, period

= \dfrac{\pi}{2}

, vertical shift

= 3

Deduce the amplitude of

y = a\sin(bx) + d

in terms of

a

. [2]

Deduce the period of

y = a\sin(bx) + d

in terms of

b

. [2]

A new wave is modelled by

y = -4\sin\!\left(\dfrac{1}{2}x\right) + 6

. Determine the amplitude, period, and vertical shift of this wave. [2]

The engineer states that a buoy rated to operate within

2

metres of a

6

-metre mean water level can safely operate in this wave pattern. Advise the engineer whether this claim is correct, justifying your answer with reference to the wave's range of heights. [2]

Solutions

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