chapter 8: sinusoidal functions - mr....

Chapter 8: Sinusoidal Functions Section 8.1

181

Chapter 8: Sinusoidal Functions

Section 8.1: Understanding Angles

Terminology:

Radian

The measure of the central angle of a circle subtended by an arc that is the same

length as the radius of the circle.

A radian is denoted by the symbol pi:

𝟏 𝒓𝒂𝒅𝒊𝒂𝒏 = 𝟏𝝅

Converting Between Radian and Degree Measurements

To convert between radians and degrees we must use the conversion that:

𝝅 = 𝟏𝟖𝟎°

Ex. Convert each measurement into radian measure.

(a) 90° (b) 45° (c) 150°

(d) 120° (e) 225° (f) 135°


182

(g) 240° (h) 450° (i) 690°

Ex. Convert each measurement into degrees

(a) 2𝜋 (b) 3.5𝜋 (c) 0.6𝜋

(d) 𝜋

4 (e)

7𝜋

9 (f)

13𝜋

4

Chapter 8: Sinusoidal Functions Section 8.2/8.3

183

Section 8.2: Graphs of Sinusoidal Functions

Terminology:

Periodic Function:

A function whose graph repeats in regular intervals or cycles.

Sinusoidal Function:

A periodic function that is wavelike and resembles the graph of 𝑦 = sin (𝑥).

Properties of a Sinusoidal Function

Midline:

The horizontal line halfway between

the maximum and minimum values of

a periodic function. Note: the midline

is always written as an equation in the form

y = a number.

Amplitude:

The distance from the midline to either

the maximum or minimum values of a

periodic function; the amplitude is always

expressed as a positive number.

Period:

The length of the interval of the domain

to complete one cycle.


184

Sketching a Graph of a Sinusoidal Function

Complete the table below and graph the resulting data.

𝑦 = 𝑆𝑖𝑛(𝑥)

𝑥 0° 45° 90° 135° 180° 225° 270° 315° 360° 𝑦

Features:

Domain: __________________

Range: __________________

Period: __________________

Midline: __________________

Amplitude: __________________

x-intercept(s): __________________

y-intercept(s): __________________

𝑥 405° 450° 495° 540° 585° 630° 675° 720° 𝑦

x- 360 ° - 270 ° - 180 ° - 90 ° 90 ° 180 ° 270 ° 360 ° 450 ° 540 ° 630 ° 720 °

y

- 2

- 1

1

2


185

Complete the table below and graph the resulting data.

𝑦 = 𝐶𝑜𝑠(𝑥)

𝑥 0° 45° 90° 135° 180° 225° 270° 315° 360° 𝑦

Features:

Domain: __________________

Range: __________________

Period: __________________

Midline: __________________

Amplitude: __________________

x-intercept(s): __________________

y-intercept(s): __________________

How are 𝑦 = 𝑆𝑖𝑛(𝑥) and 𝑦 = 𝐶𝑜𝑠(𝑥) similar and how are they different?

𝑥 405° 450° 495° 540° 585° 630° 675° 720° 𝑦

x- 360 ° - 270 ° - 180 ° - 90 ° 90 ° 180 ° 270 ° 360 ° 450 ° 540 ° 630 ° 720 °

y

- 2

- 1

1

2


186

Describing the Graph of a Sinusoidal Function in Degree Measure

Ex1. The graph of a sinusoidal function is shown. Describe this graph by determining its

range, the equation of its midline, its amplitude, and its period.

x120 ° 240 ° 360 ° 480 ° 600 ° 720 °

y

- 4

- 2

2

4

6

8


187



x0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

y

- 5

- 4

- 3

- 2

- 1

1

2

3


188



x120 ° 240 ° 360 ° 480 ° 600 ° 720 °

y

- 4

- 2

2

4

6

8


189



x0.4 0.8 1.2 1.6 2 2.4 2.8

y

- 2.5

- 2

- 1.5

- 1

- 0.5

0.5

1

1.5


190

Connecting a Sinusoidal Function to Oscillating Motion

Ex1. For a physics project, Morgan and Lily had to graph and analyze an example of

simple harmonic motion. Morgan swung on a swing, and Lily used a motion detector to

measure Morgan’s height above ground over time, as she swung back and forth. The

girls then graphed their data as shown, At the end of each cycle, the swing returned to its

initial position, which resulted in a sinusoidal graph.

(a) Interpret the graph

(b) Determine Morgan’s height above the ground at 4 seconds.

x1 2 3 4 5 6 7

y

1

2


191

Ex1. For a physics project, Morgan and Lily had to graph and analyze an example of

simple harmonic motion. Morgan swung on a swing, and Lily used a motion detector to

measure Morgan’s height above ground over time, as she swung back and forth. The

girls then graphed their data as shown, At the end of each cycle, the swing returned to its

initial position, which resulted in a sinusoidal graph.

(a) Interpret the graph

(b) Determine Morgan’s height above the ground at 4 seconds.

x1 2 3 4 5 6 7

y

1

2

3


192

Comparing Two Sinusoidal Graphs

Ex. Alexis and Colin owned a car and a pickup truck. They noticed that the

odometer of the two vehicles gave different values for the same distance. As

part of their investigation into the cause, they put a chalk mark on the outer

edge of a tire on each vehicle. The following graphs show the height of the

tires as they rotated while the vehicles were driven at the same slow,

constant speed. What can you determine about the tires from these graphs?

x0.4 0.8 1.2 1.6 2 2.4 2.8

y

4

8

12

16

20

24

28

32

Car

He

ig

ht

o

f

th

e

Ti

re

(

in

.)

Time (s)

x0.4 0.8 1.2 1.6 2 2.4 2.8

y

4

8

12

16

20

24

28

32

Truck

He

ig

ht

o

f

th

e

Ti

re

(

in

.)

Time (s)


193

Ex2. Alexis installed tires with a lager diameter on the car. She obtained

this graph as she tracked the vertical position of the chalk mark.

(a) Compare this graph with the original graph for the car tire.

(b) A speedometer and an odometer operate based on the number of

revolutions that a wheel makes. Discuss, as a class, how larger tires

might affect the speedometer and the odometer.

x0.4 0.8 1.2 1.6 2 2.4 2.8

y

4

8

12

16

20

24

28

32

Car

He

ig

ht

o

f

th

e

Ti

re

(

in

.)

Time (s)


194

SUMMARY INFORMATION:


195

Section 8.4: The Equations of Sinusoidal Functions

Functional Form of a Sinusoidal Function:

A sinusoidal function can have on of the following functional forms:

𝑦 = 𝑎 𝑆𝑖𝑛 𝑏(𝑥 − 𝑐) + 𝑑

𝑜𝑟

𝑦 = 𝑎 𝐶𝑜𝑠 𝑏(𝑥 − 𝑐) + 𝑑

where a, b, c, and d are real numbers.

The Effects of Each Component on the Graph

First off, let’s revisit the basic graphs of Sin(x) and Cos(x):

𝑦 = 𝑆𝑖𝑛(𝑥) 𝑦 = 𝐶𝑜𝑠(𝑥) Range {𝑦|−1 ≤ 𝑦 ≤ 1, 𝑦𝑒𝑅} {𝑦|−1 ≤ 𝑦 ≤ 1, 𝑦𝑒𝑅}

Midline 𝑦 = 0 𝑦 = 0 Amplitude 1 1

Period 360° 𝑜𝑟 2𝜋 360° 𝑜𝑟 2𝜋

Graph x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1


196

The Effects of a: 𝒚 = 𝒂 𝑺𝒊𝒏(𝒙)

Here is the base graph of 𝑦 = 𝑆𝑖𝑛(𝑥):

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

Here is the graph of 𝑦 = 3 𝑆𝑖𝑛(𝑥):

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1


197

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1


198

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1


199

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

Here is the graph of 𝑦 =1

5 𝑆𝑖𝑛(𝑥):

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 0.8

- 0.6

- 0.4

- 0.2

0.2

0.4

0.6

0.8

What effect does the value of a have on the graph? What feature(s) does it effect?

The Effects of d: 𝒚 = 𝑪𝒐𝒔(𝒙) + 𝒅


200

Here is the base graph of 𝑦 = 𝐶𝑜𝑠(𝑥):

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

Here is the graph of 𝑦 = 𝐶𝑜𝑠(𝑥) + 3:

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 4

- 2

2

4

Here is the graph of 𝑦 = 𝐶𝑜𝑠(𝑥) − 2

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 4

- 2

2

4

What effect does the value of d have on the graph? What feature(s) does it effect?

The Effects of b: 𝒚 = 𝑺𝒊𝒏𝒃(𝒙)


201

Here is the base graph of 𝑦 = 𝑆𝑖𝑛(𝑥):

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

Here is the graph of 𝑦 = 𝑆𝑖𝑛2(𝑥):

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

Here is the graph of 𝑦 = 𝑆𝑖𝑛1

2(𝑥)

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

What effect does the value of b have on the graph? What feature(s) does it effect?

The Effects of c: 𝒚 = 𝑪𝒐𝒔(𝒙 − 𝒄)


202

Here is the base graph of 𝑦 = 𝐶𝑜𝑠(𝑥):

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

Here is the graph of 𝑦 = 𝐶𝑜𝑠(𝑥 − 270):

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

Here is the graph of 𝑦 = 𝐶𝑜𝑠(𝑥 + 90)

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 1

1

What effect does the value of c have on the graph? What feature(s) does it effect?

Determining the Characteristics of Sine and Cosine Functions

Based on their Equations


203

Consider each functions below. For each, determine the amplitude, equation of the

midline, range, period, and any relevant horizontal translations for each.

(a) 𝑦 = 2 𝑆𝑖𝑛 3(𝑥 − 45°) − 1

(b) 𝑦 =1

5𝑆𝑖𝑛 2(𝑥 + 30°) + 4

(c) 𝑦 = 4𝐶𝑜𝑠1

9(𝑥 − 90°) + 3

(d) 𝑦 =1

2𝐶𝑜𝑠 4(𝑥 + 15°) − 10


204

(e) 𝑦 = 𝑆𝑖𝑛 2𝑥 − 1

(f) 𝑦 = 5𝐶𝑜𝑠 (𝑥 + 60°)

(g) ℎ(𝑥) = 6.5 𝑆𝑖𝑛𝜋

6(𝑥 − 1.2) − 1

(h) 𝑗(𝑥) = 2 𝐶𝑜𝑠 4𝜋(𝑥) + 4


205

Compare the graphs of 𝑦 = 3𝑆𝑖𝑛 2 (𝑥 − 45°) and 𝑦 = 3 𝐶𝑜𝑠 2(𝑥 − 45°). How are they

the same and how are they different?

Determining the Equation of a Sinusoidal Function given its

Graph


206

Determine the equation of each graph below. Interpret each in terms of both

𝑦 = 𝑆𝑖𝑛(𝑥) 𝑎𝑛𝑑 𝑦 = 𝐶𝑜𝑠(𝑥) .

(a)

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 4

- 2

2

4

(b)


207

x

- 360 ° - 240 ° - 120 ° 120 ° 240 ° 360 °

y

- 0.5

0.5

1

(c)


208

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 °

y

- 16

- 12

- 8

- 4

4

8

(d)


209

x

- 360 ° - 180 ° 180 ° 360 ° 540 ° 720 ° 900 ° 1080 °

y

- 5

- 4

- 3

- 2

- 1

1

Word Problem Applications


210

Ex1. The Far North is called “The Land of the Midnight Sun” for good reason: during the

summer months, in some locations, the Sun can be visible for 24 hours a day. The

number of hours of daylight in Iqaluit, Nunavut can be expressed by the function:

𝑦 = 8.245 sin 0.0172(𝑥 − 80.988) + 12.585

where x is the day number in the year (1-365)

(a) How many hours of daylight occur in Iqaluit on the following days?

(i) The shortest day of the year?

(ii) The longest day of the year?

(b) In some years, June 21 is the longest day. Suppose that the sun set at 11:01 pm on

June 21, at what time did the sun rise?

(c) In some years, Jan 22 is considered the shortest day. Suppose that sunrise occurs

at 9:22 am, when did the sun set?

(d) What is the period of this sinusoidal function? What does this represent in terms

of the context of this question?


211

𝑦 = 8.245 sin 0.0172(𝑥 − 80.988) + 12.585

(e) What does the c-value of 80.988 represent in terms of the context of this

question?

(f) Determine the number of hours of sunlight on April 10th (the 100th day of the

year)?

Ex2. Ashley boards the Ferris Wheel at the Pacific National Exhibition. When the ride

begins, her position can be modeled by:


212

𝑦 = 43 sin 3.5(𝑥 − 0.9) + 47

Where y represents the height in feet and x represents the time in minutes.

(a) Determine the diameter of the Ferris wheel.

(b) How long does it take for the Ferris wheel to make one complete rotation?

(c) How high above the ground is Ashley at the highest and lowest points on the

Ferris wheel?

Ex3. An apple is attached to a spring. The height of the apple as it oscillates up and

down can be modelled by the equation:


213

ℎ(𝑡) = 4 sin(8𝜋𝑡) + 6.5

where h(t) represents the height of the apple in centimetres and t represents the time

in seconds.

(a) What are the minimum and maximum points that the apple reaches?

(b) What is the period of the function? What does this period tell you about the apple

in this context?

Ex4. A person’s blood pressure, P(t), in millimetres of mercury (mm Hg), can be

modelled by the function:


214

𝑃(𝑡) = −20 cos(8.4𝑡) + 100

Where t is the time in seconds.

(a) What is the period of this function? What does this represent in the context of the

problem?

(b) What is the highest and lowest blood pressure measurements for the person?

(c) What is Jim-Bob’s blood pressure after 3.75 seconds?

Sketching Possible Sinusoidal Functions Given It’s Features


215

Note: This will include one of the rare circumstances in this chapter where the domain

may be restricted. In such cases we will start and end our graph at the outlined

domain value.

Ex. For each situation, sketch the potential sinusoidal graph that has the given features.

(a) Domain:{𝑥| − 180° ≤ 𝑥 ≤ 540°, 𝑥 ∈ 𝑅}

Midline: 𝑦 = −2

Period: 360°

Maximum:(90°, 3)


216

(b) Domain:{𝑥|0° ≤ 𝑥 ≤ 200°, 𝑥 ∈ 𝑅}

Maximum: (50°, 9)

Minimum:(0°, −1)


217

(c) Domain:{𝑥| − 90° ≤ 𝑥 ≤ 90°, 𝑥 ∈ 𝑅}

Maximums: (−30°, 5) & (30°, 5)

Midline: 𝑦 = 3


218

(d) Domain:{𝑥| − 180° ≤ 𝑥 ≤ 1440°, 𝑥 ∈ 𝑅}

Range:{𝑦| − 7 ≤ 𝑦 ≤ −1, 𝑦 ∈ 𝑅}

Period:720°

Y-Intercept:(0°, −4)

chapter 8: sinusoidal functions - mr....

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