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Further graph transformations and modelling

Apply transformations (reflections, translations, dilations) to trigonometric, exponential and logarithmic functions; model real periodic and logarithmic phenomena with transformed functions.

Functions · MAV-12-01

Sub-topics

Transformations of trigonometric functions

Apply reflections, translations and dilations to

y=\sin x

y=\cos x

y=\tan x

; determine amplitude, period, phase shift and vertical shift; solve equations and sketch graphs within a specified domain.

Modelling periodic phenomena

Model real periodic situations — tides, temperature, daylight hours — using transformed trigonometric functions; fit amplitude, period and phase shift to data; use a model to find values and solve equations in context.

Logarithmic scales

Recognise when a logarithmic scale is appropriate for large-range data; solve problems using the decibel scale for sound intensity, the Richter scale for earthquakes, the pH scale for acidity, and stellar magnitude for star brightness.

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Further graph transformations and modelling

Functions · MAV-12-01

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Date:

Transformations of trigonometric functions

Q1Straightforward

State the amplitude of

y = 3\sin(2x)

.

Q2Straightforward

Find the period of

y = \cos\!\left(\dfrac{x}{3}\right)

in radians. Give your answer as a multiple of

\pi

, i.e. enter the coefficient

k

where the period is

k\pi

.

Q3Straightforward

What is the period of

y = \sin(3x - \pi)

in radians? Express your answer as a fraction of

\pi

, i.e. enter the value of

k

where the period is

\dfrac{k\pi}{3}

...

Actually: give the period as a decimal rounded to 4 decimal places.

Q4Straightforward

State the amplitude of

y = -4\cos(x/2) + 1

.

Q5Moderate

The function

y = a\sin(b(x - c)) + d

has amplitude 3, period

4\pi

, phase shift

\pi/4

to the right, and vertical shift

-1

.

Find the value of

b

.

Q6Moderate

Find all solutions of

2\sin(x) - 1 = 0

in the interval

[0,\,2\pi]

. Enter the smaller solution in radians, rounded to 4 decimal places.

Q7Moderate

Find all solutions of

2\sin(x) - 1 = 0

in

[0,\,2\pi]

. Enter the **larger** solution in radians, rounded to 4 decimal places.

Q8Moderate

Find all solutions of

2\cos(x) - 1 = 0

in

[0,\,2\pi]

. Enter the smaller solution in radians, rounded to 4 decimal places.

Q9Moderate

How many solutions does

\sin(2x) = \dfrac{\sqrt{3}}{2}

have in the interval

0 \leq x \leq 2\pi

?

Q10Challenging

The graph of

y = A\sin(Bx + C)

passes through the origin, has amplitude 4 and period

\pi

. The graph is increasing at

x=0

.

Given

A > 0

,

B > 0

and

-\pi < C \leq \pi

, find the value of

C

.

Q11Challenging

Find the maximum value of

y = 3\sin\!\left(2x - \dfrac{\pi}{3}\right) + 2

.

Q12Challenging

Solve

\sqrt{3}\tan(x) + 1 = 0

for

x \in [0,\,\pi)

. Give your answer in radians, rounded to 4 decimal places.

Q13Challenging

The function

y = a\cos(b(x - c)) + d

has:
- maximum value

7

- minimum value

-1

- period

3\pi

- a maximum at

x = \pi

Find

a

(the amplitude).

Modelling periodic phenomena

Q14Straightforward

The height of the tide at a harbour is modelled by

H(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 4

, where

H

is the height in metres and

t

is time in hours after midnight.

What is the maximum height of the tide in metres?

Q15Straightforward

The height of the tide is modelled by

H(t) = 3\sin\!\left(\dfrac{\pi t}{6}\right) + 4

, where

t

is in hours.

What is the period of the tide in hours?

Q16Straightforward

The average monthly temperature (°C) in a city is modelled by

T(m) = 12\cos\!\left(\dfrac{2\pi m}{12}\right) + 18

, where

m

is the month number with

m = 0

representing the warmest month.

What is the minimum temperature in °C?

Q17Straightforward

The depth of water (metres) at a pier is modelled by

D(t) = 2.5\sin\!\left(\dfrac{\pi t}{6} - \dfrac{\pi}{3}\right) + 5

, where

t

is time in hours.

State the amplitude of the depth in metres.

Q18Moderate

A Ferris wheel has a maximum height of

30

m and a minimum height of

2

m above the ground. Riders board at the lowest point, and the wheel takes

4

minutes to complete one full revolution.

A model of the form

h(t) = -a\cos(bt) + d

is used, where

t

is time in minutes after boarding.

Find the value of

a

.

Q19Moderate

The temperature

T

(°C) inside a greenhouse follows the model

T(t) = 8\sin\!\left(\dfrac{\pi(t-6)}{12}\right) + 22

, where

t

is hours after midnight,

0 \le t \le 24

.

At what time (hours after midnight) does the temperature first reach its daily maximum? Give an integer answer.

Q20Moderate

The height of the tide is modelled by

H(t) = 1.8\cos\!\left(\dfrac{\pi t}{6}\right) + 3.2

, where

H

is in metres and

t

is hours after midnight.

Find the first time after midnight (to 2 decimal places) when the tide height equals

4

m.

Q21Moderate

The number of hours of daylight in a city in the northern hemisphere is modelled by

D(m) = 3\sin\!\left(\dfrac{\pi m}{6} - \dfrac{\pi}{2}\right) + 12

, where

m

is the month number (January

= 1

, December

= 12

).

Find the number of hours of daylight in April (

m = 4

).

Q22Moderate

Using the daylight model

D(m) = 3\sin\!\left(\dfrac{\pi m}{6} - \dfrac{\pi}{2}\right) + 12

(northern hemisphere, January

= 1

), find the month number

m

in which daylight is longest.

Q23Challenging

Tide measurements at a wharf give the following data:

$t$ (hours after midnight)	$0$	$3$	$6$	$9$	$12$
$H$ (metres)	$5$	$8$	$5$	$2$	$5$

A sine model

H(t) = a\sin(bt) + d

is fitted to the data.

Find the value of

b

, correct to 4 decimal places.

Q24Challenging

The average monthly temperature (°C) in a southern-hemisphere city is modelled by

T(m) = 12\cos\!\left(\dfrac{2\pi(m-1)}{12}\right) + 20

, where

m = 1

is January (the warmest month).

How many months of the year have an average temperature strictly above

26

°C?

Q25Challenging

A trigonometric function of the form

y = a\cos(b(t - c)) + d

models the height (metres) of a buoy above the seabed, with the following properties:
- maximum height:

9.4

m, minimum height:

1.6

m
- period:

10

hours
- first maximum at

t = 2

hours

Find the value of

c

.

Logarithmic scales

Q26Straightforward

The loudness

L

(in decibels) of a sound is given by

L = 10\log_{10}\!\left(\dfrac{I}{I_0}\right)

, where

I

is the sound intensity in W/m² and

I_0 = 10^{-12}

W/m² is the threshold of hearing.

A sound has intensity

I = 10^{-8}

W/m². Find its loudness in decibels.

Q27Straightforward

A jet engine produces a sound intensity of

100

W/m². Using

L = 10\log_{10}\!\left(\dfrac{I}{I_0}\right)

with

I_0 = 10^{-12}

W/m², find the loudness in decibels.

Q28Straightforward

The Richter scale magnitude of an earthquake is

M = \log_{10}\!\left(\dfrac{A}{A_0}\right)

, where

A

is the maximum wave amplitude and

A_0

is a reference amplitude.

Two earthquakes have magnitudes

6

and

8

respectively. How many times greater is the wave amplitude of the larger earthquake compared with the smaller one?

Q29Straightforward

The pH of a solution is defined as

\text{pH} = -\log_{10}[\text{H}^+]

, where

[\text{H}^+]

is the hydrogen ion concentration in mol/L.

Vinegar has pH

3

and drinking water has pH

7

. How many times greater is the

[\text{H}^+]

concentration in vinegar than in water?

Q30Moderate

A sound has loudness

L = 65

dB. Using

L = 10\log_{10}\!\left(\dfrac{I}{I_0}\right)

with

I_0 = 10^{-12}

W/m², find the intensity

I

.

If

I = 10^n

W/m², state the value of

n

.

Q31Moderate

An earthquake has magnitude

7.5

on the Richter scale. A second earthquake produces wave amplitudes

100

times greater. What is the magnitude of the second earthquake?

Q32Moderate

Two identical machines each produce a sound of

70

dB. When both run simultaneously the intensities add. What is the combined loudness in dB, rounded to 1 decimal place?

(Use

\log_{10} 2 \approx 0.3010

.)

Q33Moderate

The apparent magnitude scale for stars is defined so that a difference of

5

magnitudes corresponds to a brightness ratio of exactly

100

. A more precise formula is:

m_1 - m_2 = -2.5\log_{10}\!\left(\frac{F_1}{F_2}\right)

where

F

denotes flux (brightness) and smaller magnitude means brighter.

Star A has apparent magnitude

2.0

and star B has apparent magnitude

4.5

. How many times brighter is star A than star B?

Q34Moderate

A solution has hydrogen ion concentration

[\text{H}^+] = 4.5 \times 10^{-4}

mol/L. Find the pH of the solution, rounded to 2 decimal places.

(Use

\log_{10} 4.5 \approx 0.6532

.)

Q35Challenging

A noise-reduction regulation requires a factory to lower its sound level by

15

dB. By what factor must the sound intensity be reduced?

Round your answer to 2 decimal places.

Q36Challenging

An earthquake in region P has magnitude

6.2

. An earthquake in region Q has magnitude

4.7

. How many times greater is the wave amplitude of the earthquake in P than in Q?

Round your answer to 2 decimal places.

Q37Challenging

Star X is

5

times brighter (greater flux) than star Y. Star Y has apparent magnitude

3.0

. Find the apparent magnitude of star X, rounded to 2 decimal places.

Use the formula

m_1 - m_2 = -2.5\log_{10}(F_1/F_2)

and

\log_{10} 5 \approx 0.6990

.

Worked solutions and answers at openmath.au/year-12/advanced/further-graph-transformations-and-modelling