Chapter 8: SINUSODIAL FUNCTIONS · PDF fileChapter 8 Math 3201 4 Worksheet 8.1 Section 1:...

Chapter 8 Math 3201 1

Chapter 8: SINUSODIAL FUNCTIONS

Section 8.1: Understanding Angles p. 484

How can we measure things?

Examples:

► Length - meters (m) or yards (yd.)

►Temperature - degrees Celsius (oC) or Fahrenheit (F)

How can we measure angles?

Up until now we can measure angles using “degrees”.

There is an alternative unit of measurement for measuring angles, that is,

“RADIANS”.

RADIAN:

One radian is the angle made by taking the

radius and wrapping it along the edge (an arc)

of the circle.

INVESTIGATE:

How many pieces of this length do you

think it would take to represent one

complete circumference of the circle?

To help, cut a piece of pipe

cleaner/string to a length equal to radius

CA and bend it around the circle

starting at point A.

1. Approximately, how many radius

lengths are there in one complete

circumference?_________________

2. How many degrees in one complete

circumference?_________________

3. Therefore, approximately, how

degrees are in one

radian?_______________ 1 radian ~ ______ Degrees

r

r


NOTE:1. The size of the radius of a circle has NO effect on the size of 1 radian.

2. The advantage of radians is that it is directly related to the radius of the

circle. This means that the units of the x and y axis is consistent and the

graph of the sine curve will have its true shape, without vertical

exaggeration.

Let’s consider a circle with radius 1 unit (r = 1). (This is called the unit circle!)

1. What is the circumference of a circle? ______________________

2. What is the circumference of the unit circle? _________________

3. How many degrees are there in a complete revolution of a circle?__________

4. Why must the values from #2. and #3. be equal to each other?

__________________________________________________

5. State the proportion:___________________________

6. Divide both sides by 2π to determine the measure of 1 radian:

7. Complete the following and add the equivalent radian measures on the unit

circle below:

a. 2 = _______

b. π = _______

c. 2

_______

d. 4

_______

e. 6

_______

f. 3

_______


Converting Degrees Radians

1. Degrees to Radians:

o To convert from Degrees to Radians multiply by 180o

2. Radians to Degrees:

o To convert from Radians to Degrees multiply by 180o

Examples:

1. Convert to radians:

a. 120o b. 240o c. 450o d. 660o

2. Convert to degrees:

a. 3

4

b.

13

6

c.

4

9

d. 5.3 rad

3. For each pair of angle measures, determine which is greater:

a. 150o, 2rad b. 450o, 7rad c. 3 , 8rad

ASSIGN: p. 489,

#1, 2, 5, 7 - 10


Worksheet 8.1

Section 1: Selected Response

1. What is the best estimate for 136° in radians?

A) 2.4 B) 0.7 C) 2.8 D) 3.1

2. What is 120° expressed in exact radians?

A) 3𝜋

2 B)

3𝜋

4 C)

3𝜋

8 D)

2𝜋

3

3. What is the best estimate for 0.1 radians in degrees?

A) 0.5° B) 1° C) 3° D) 6°

4. What is 5𝜋

8 expressed in degrees?

A) 112.5° B) 288° C) 900° D) 1440°

5. What is the best estimate for the central angle in degrees?

6. What is the best estimate for the central angel in radians?

A) 263° B) 273° C) 283° D) 293°

A) 5𝜋

12 B)

6𝜋

5

C) 5𝜋

4 D)

5𝜋

6


7. Imagine that it is now 2 p.m. What time will it be when the minute

hand has rotated through 300°?

A) 2:40 B) 2:50 C) 3:00 D) 3:10

8. Imagine that it is now 2 p.m. What time will it be when the minute

hand has rotated through 7𝜋

4 radians?

Section 2: Constructed Response

Answer all of the following showing all work.

9. Eddie is facing west. What direction will he be facing if he rotates

235° to his right?

10. For the following pair of angle measures, determine which is greater

75° or 1

2𝜋 ?

ANSWERS

1. A 2. D 3. D 4. A 5. B 6. D 7. B 8. B 9. SE 10. 1

2𝜋

A) 2:20 B) 2:52

C) 3:15 D) 3:45


Section 8.2: Exploring Graphs of Periodic Functions p. 491

Terms to Know:

Periodic

Function

A function whose graph repeats

in regular intervals or cycles.

Midline /

Sinusodial

Axis

The horizontal line halfway

between the maximum and

minimum values of a periodic

function.

Amplitude The distance from the midline to

either the maximum or minimum

value 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.

Sinusodial

Function

Any periodic function whose

graph has the same shape as that

of y = sinx.


“Five Key Points”

x y

Section 8.2 “Exploration” Sine Graph

1. Complete the table of values below for the function siny

(DEGREE MODE)

0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 360°

y

390° 420° 450° 480° 510° 540° 570° 600° 630° 660° 690° 720°

y

2. Sketch the graph of siny on the graph below:

3. Complete the tables below by using the graph. If you wanted to quickly graph

the sine curve, which five points would allow you to easily graph the entire curve?

siny

Period

Sinusoidal Axis

(midline)

Amplitude

Domain

Range

Local Maximums

Local Minimums

x-intercepts

y-intercepts

x90 ° 180 ° 270 ° 360 ° 450 ° 540 ° 630 ° 720 °

y

- 2

- 1

1

2


“Five Key Points”

x y

Section 8.2 “Exploration” Cosine Graph

1. Complete the table of values below for the function cosy

(DEGREE MODE)

0° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 360°

y

390° 420° 450° 480° 510° 540° 570° 600° 630° 660° 690° 720°

y

2. Sketch the graph of cosy on the graph below:

3. Complete the tables below by using the graph. If you wanted to quickly graph

the cosine curve, which five points would allow you to easily graph the entire curve?

cosy

Period

Sinusoidal Axis

(midline)

Amplitude

Domain

Range

Local Maximums

Local Minimums

x-intercepts

y-intercepts

x90 ° 180 ° 270 ° 360 ° 450 ° 540 ° 630 ° 720 °

y

- 2

- 1

1

2


OBSERVATIONS:

For SINE, one complete wave can be seen from an X-INTERCEPT at (0, 0)

to the X-INTERCEPT at (360o, 0).

For COSINE, one complete wave can be seen from the MAXIMUM point

(0, 1) to the next MAXIMUM point at (360o, 1).

The graph of cosy is related to the graph of siny by a shift of 90o to

the left.

Label the following as periodic, sinusoidal or both.

a. _______________________ c.

_________________________

b. ________________________ d. _______________________

ASSIGN: p. 494, #2, 5 - 8

CONCLUSION: All sinusoidal functions are periodic but not all periodic

functions are sinusoidal.


Worksheet 8.2

1. Complete the following table for y = sin 𝜃.

In Degrees or Radians (if

possible)

Period Local Maximum

Sinusoidal Axis

(midline)

Local Minimum

Amplitude x-intercept(s)

Domain y-intercept(s)

Range

Draw the graph of y = sin 𝜃 using the five key points.

x y

2. Draw the graph of y = cos 𝜃 using the five key points.

x y

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

y

-2

-1

1

2

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

y

-2

-1

1

2


Section 8.3: The Graphs of Sinusoidal Functions p. 497

Equation of the midline: is the average of the maximum and minimum values:

𝑦 =𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 + 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒

2

Amplitude: is the positive vertical distance between the midline and either a

maximum or minimum value. It is also half of the vertical distance between a

maximum value and a minimum value.

𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 =𝑚𝑎𝑥𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒 − 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑣𝑎𝑙𝑢𝑒

2

Examples:

1. For the sinusoidal function shown, determine: (Example 1, p. 499)

a. Range:

b. Equation of Midline/Sinusoidal Axis:

c. Amplitude:

d. Period:


2. For the sinusoidal function shown, determine: (Example 1, p. 499)

a. Range:

b. Equation of Midline/Sinusoidal Axis:

c. Amplitude:

d. Period:

3. While riding a Ferris

wheel, Mason’s height

above the ground

in terms of time

can be represented by the

following graph.

a. How far is the Ferris wheel off the ground? __________________________

b. What is the range of the function? What does it represent? _____________

________________________________________________________________

c. What is the height of the Ferris wheel? _____________________________

d. What is the equation of the midline? What does it represent? ____________

________________________________________________________________

e. How long does it take for the Ferris wheel to make one complete

revolution? What characteristic does this correspond to?

y


4. Alexis and Colin own a car and a pickup truck. They noticed that the

odometers 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 characteristics of the tires

from these graphs? (Example 4, p. 504)

ASSIGN: p.

507, #4, 5,

7 – 10, 13 - 15


Worksheet 8.3

1. What is the midline of the following graph?

2. What is the amplitude of the following graph?

3. What is the period of the following graph?

4. What is the range of the following graph?

5. A sinusoidal graph has an amplitude of 10 and a maximum at the

point (18, 5). What is the midline of the graph?

A) y = 0 B) y = -5 C) y = 13 D) y = 8

A) y = 2 B) y = 3

C) y = 4 D) y = 5

A) 2 B) 3

C) 4 D) 5

A) 120° B) 240°

C) 300° D) 360°

A) {𝑦|1 ≤ 𝑦 ≤ 5, 𝑦𝜖𝑅} B) {𝑦| − 2 ≤ 𝑦 ≤2, 𝑦𝜖𝑅}

C) {𝑦|0 ≤ 𝑦 ≤ 4, 𝑦𝜖𝑅} D) {𝑦| 𝑦𝜖𝑅}


x- 90 ° 90 ° 180 ° 270 ° 360 ° 450 ° 540 ° 630 ° 720 ° 810 ° 900 ° 990 ° 1080 °

y

- 6

- 4

- 2

2

4

6

6. A sinusoidal graph has a maximum at the point (4, -8) and the next

minimum is at the point (7, -10). What is the period of the graph?

A) 2 B) 3 C) 4 D) 6

7. Sketch a possible graph of a sinusoidal function with the following set of

characteristics. Explain your decision.

Domain: {𝑥|0 ≤ 𝑥 ≤ 1080°, 𝑥 𝜖 𝑅}

Maximum Value: 5 Minimum Value: -5

Period: 720° y-intercept: 0

8. The graph of a sinusoidal function is shown. Describe the graph by

determining its range, the equation of the midline, its amplitude and its

period. Show any work.


9. Kira is sitting in an inner tube in the wave pool.

The depth of the water below her, in terms of time,

during a series of waves can be represented by

the graph shown.

A) How long does it take for one complete wave to pass?

B) What is the approximate depth of the water below Kira after 4 s?

C) How high is each wave?

D) What is the depth of the water below Kira when no waves are being

generated?

E) What is the depth of the water below Kira at 9s?

Answers: 1A 2A 3B 4A 5B 6D 8.range {𝑦| − 1 ≤ 𝑦 ≤ 5, 𝑦 ∈ 𝑅} midline y = 2 Amplitude 3 period 12

9A) 2seconds B) 2.4m C) 0.8m D) 2m E) 2m


Section 8.4: The Equations of Sinusoidal Functions p. 516

INVESTIGATION:

Using technology, we will explore how the parameters a, b, c and d affect the

graph of sinusoidal functions written in the form:

sin ( )y a b x c d and cos ( )y a b x c d

(A) The Effect of ‘a’ in y = asinx on the graph y = sinx where a > 0

1. Below are the sketches of the graphs of: siny x , 2siny x , 4siny x , 0.5siny x

siny x 2siny x

4siny x 0.5siny x

2. Compare the amplitudes in each graph with its equation.

3. Describe the affect the value of ‘a’ has on the graph of siny x .

4. Will the value of ‘a’ affect the cosine graph in the same way that it affects the

sine graph?

x-360 -270 -180 -90 90 180 270 360

y

-4

-3

-2

-1

1

2

3

4

x-360 -270 -180 -90 90 180 270 360

y

-4

-3

-2

-1

1

2

3

4

x-360 -270 -180 -90 90 180 270 360

y

-4

-3

-2

-1

1

2

3

4

x-360 -270 -180 -90 90 180 270 360

y

-4

-3

-2

-1

1

2

3

4


(B) The Effect of ‘d’ in y = sinx + d on the graph y = sinx

5. Compare the sketches of the graph of: sin 2y x and sin 3y x to the

graph of siny x

sin 2y x sin 3y x

6. How does each graph change when compared to siny x ?

7. How is the value of ‘d’ related to the equation of the midline?

8. Is the shape of the graph or the location of the graph affected by the

parameter ‘d’?

9. Is the period affected by changing the value of ‘d’?

10. Will the value of ‘d’ affect the cosine graph in the same way that it affects

the sine graph?

x-360 -270 -180 -90 90 180 270 360

y

-4

-3

-2

-1

1

2

3

4

x-360 -270 -180 -90 90 180 270 360

y

-4

-3

-2

-1

1

2

3

4


(C) The Effect of ‘b’ in y = sinbx on the graph y = sinx

11. Compare the sketch of the graphs of: sin 2y x and sin 0.5y x to the graph of

siny x

sin 2y x sin 0.5y x

12. What is the period of siny x ? ___________ What is the ‘b’ value? _______

13. What is the period of sin 2y x ? __________ What is the ‘b’ value? _______

14. What is the period of sin 0.5y x ? _________ What is the ‘b’ value? _______

15. What is affected by the value of ‘b’? _____________________________

16. Write an equation that relates the ‘b’ value to the period of the function.

17. Will the value of ‘b’ affect the cosine graph in the same way that it affects

the sine graph?

Period or Period

x-360 -270 -180 -90 90 180 270 360

y

-4

-3

-2

-1

1

2

3

4

x-360 -270 -180 -90 90 180 270 360

y

-4

-3

-2

-1

1

2

3

4


(D) The Effect of ‘c’ in y = sin(x – c) on the graph y = sinx

18. Sketch a graph of: sin( 60 )oy x and sin( 30 )oy x and compare it to the

graph of siny x

sin( 60 )oy x 0sin( 30 ) sin( ( 30 ))oy x x

x-180 -90 90 180 270 360

y

-2

-1

1

2

x-360 -270 -180 -90 90 180 270 360

y

-2

-1

1

2

19. How is the ‘c’ value affecting the graph?

This horizontal shift is also called the phase shift of the graph. In order to

determine the phase shift of the graph, you need to compare a KEY POINT on the

sine graph to determine if it has shifted left or right.

20. If the ‘c’ value is positive, the graph shifts to the __________________.

21. If the ‘c’ value is negative, the graph shifts to the __________________.

22. Will the value of ‘c’ affect the cosine graph in the same way that it affects

the sine graph?

______________________________________________________________

maximum

minimum

maximum

minimum

One key point on the SINE graph is the point (0, 0) which intersects the

midline at x = 0 going from a minimum point to a maximum.


(E) The Effect of ‘c’ in y = cos(x – c) on the graph y = cosx

23. Compare the graphs of: cos( 90 )oy x and cos( 60 )oy x to siny x

cos( 90 )oy x 0cos( 60 ) cos( ( 60 ))oy x x

x-180 -90 90 180 270 360

y

-2

-1

1

2

x-360 -270 -180 -90 90 180 270 360

y

-2

-1

1

2

24. If the ‘c’ value is positive, the graph shifts to the __________________.

25. If the ‘c’ value is negative, the graph shifts to the __________________.

CONCLUSION: For the sinusoidal functions written in the form:

sin ( )y a b x c d and cos ( )y a b x c d

1. a = amplitude = max min

2

2. b affects the period: 360o

periodb

or 2

periodb

3. c = horizontal shift (need to compare a Key point)

4. d = vertical translation, y = d = max min

2

= equation of the

midline/sinusoidal axis

5. maximum value = d + a; minimum value = d – a

6. Range: {min value ≤ y ≤ max value}

maximum maximum

One KEY point on the COSINE graph is the point (0, 1)

which is a maximum point.


NOTE: Beware of the brackets! cos( 2)y x versus cos 2y x

cos( 2)y x results in a horizontal translation

cos 2y x results in a vertical translation

Examples:

1. For the function, 2cos4 1y x , state: (Example 1, p. 518)

a. amplitude:

b. equation of the midline:

c. range:

d. period:

e. phase shift:

2. For the function, 3sin 2( 45 )oy x , state: (Example 2, p.519)

a. amplitude:

b. equation of the midline:

c. range:

d. period:

e. phase shift:

f. How would the graph of 3cos2( 45 )oy x be the same? How would it

be different?


3. Match graph 1 and graph 2 below with the corresponding equation.

(Example 3, p. 522)

i. 4cos( 90 ) 1oy x iii. 5sin3( 60 )oy x

ii. 4sin3( 60 ) 1oy x iv. 4cos3( 60 ) 1oy x


x90 ° 180 ° 270 ° 360 ° 450 ° 540 ° 630 ° 720 °

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

x90 ° 180 ° 270 ° 360 ° 450 ° 540 ° 630 ° 720 °

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

4. Ashley created the following graph for the equation 3sin( 90 ) 2oy x .

Identify her error(s) and construct the correct graph.


5. The temperature of an air-conditioned home on a hot day can be modeled

using the function, 𝑓(𝑥) = 1.5 cos 4𝑥 + 20 ,where x is the time in minutes

after the air conditioner turns on and t(x) is the temperature in degrees

Celsius.

a. What are the maximum and minimum temperatures in the home?

b. What is the temperature 10 minutes after the air conditioner has been

turned on?

c. What is the period of the function? Interpret this value in this

context?


Section 8.5: Modeling Data with Sinusoidal Functions p. 533

When a sinusoidal regression occurs the calculator generates an equation

𝑦 = asin(𝑏𝑥 + 𝑐) = 𝑑 in radian measure.

Example: The average precipitation was

recorded and a sinusoidal

regression was created for the

data which resulted in the following

a. Using the data in the screenshot above, determine the equation of the

sinusoidal regression function that models the change in precipitation.

Use the form 𝑦 = 𝑎𝑠𝑖𝑛(𝑏𝑥 + 𝑐) + 𝑑.

b. Use your equation from Part a to determine what the amount of

precipitation when x = 7

SinReg

y=asin(bx+c)+d

a=4.99

b=.50

c= 1.58

d = 14.91


Unit 8: Sinusoidal Functions

Worksheet 8.4

Section 1: Selected Response

1. Which function below has the greatest amplitude?

A) y = 2 sin 3(x + 90°) + 5 B) y = 3 sin 2(x - 90°) – 3

C) y = 1

3 sin (x + 90°) – 1 D) y = sin 0.5(x - 90°)

2. Which function has the greatest period?

A) y = 2 sin 3(x + 90°) + 5 B) y = 3 sin 2(x - 90°) – 3

C) y = 1

3 sin (x + 90°) – 1 D) y = sin 0.5(x - 90°)

3. Which function has the greatest maximum value?

A) y = 2 sin 3(x + 90°) + 5 B) y = 3 sin 2(x - 90°) – 3

C) y = 1

3 sin (x + 90°) – 1 D) y = sin 0.5(x - 90°)

4. What is the amplitude for the following function: y = 3 sin 2(x + 90°) – 1

A) 2 B) 3 C) 4 D) 5

5. What is the period for the following function: y = cos 1

3x + 12

A) 180° B) 360° C) 720° D) 1080°

6. What is the midline of the following function: y = 0.5 sin(x -2)

A) y = -2 B) y = 0.5 C) y = 0 D) y = 2

7. What is the range of the following function: y = 3 sin 2(x + 90°) – 1

A) {𝑦| − 3 ≤ 𝑦 ≤ 3, 𝑦 𝜖𝑅} B) {𝑦| − 2 ≤ 𝑦 ≤ 4, 𝑦 𝜖𝑅}

C) {𝑦| − 4 ≤ 𝑦 ≤ 2, 𝑦 𝜖𝑅} D) {𝑦| 𝑦 𝜖𝑅}


x2 4 6 8 10 12 14 16 18 20 22 24

y

- 6

- 4

- 2

2

4

6

8. What is the domain of the following function: y = 0.5 sin(x – 2)

A) {𝑥| − 3 ≤ 𝑥 ≤ −1, 𝑥 𝜖𝑅} B) {𝑥| − 0.5 ≤ 𝑥 ≤ 0.5, 𝑥 𝜖𝑅}

C) {𝑥| − 2 ≤ 𝑥 ≤ 2, 𝑥 𝜖𝑅} D) {𝑥| 𝑥 𝜖 𝑅}

Section 2: Constructed Response

1. What is the amplitude of the following function: y = 2

5cos(𝑥 − 𝜋)

2. What is the midline of the following function: y = 5 sin 1.5(x + 60°) – 5

3. What is the range of the following function: y = 10 cos 4(x - 180°) + 2

4. Match each graph with the corresponding equation. Explain your answers:

i) y = 2 cos (x - 120°) + 4

ii) y = 2 cos (x – 60°) + 4

iii) y = 2 cos(x + 60°) + 4

iv) y = 2 cos(x – 120°)

v) y = 2 cos(x - 60°)

vi) y = 2 cos(x + 60°)

5. Describe the graph of the following function by stating the amplitude, equation of the

midline, range and period. y = 1

10sin(2𝑥) + 3.5

6. The following graph represents the rise and fall of sea level in part of the Bay of Fundy,

where t is the time, in hours, and h(t) represents the height relative to the mean sea

level.

A) What is the range of the tide levels?

B) What does the equation of the midline

represent in the graph?

C) What is the period of the graph?

D) The equation of the sinusoidal function

is represented by: ℎ(𝑡) = 6.5 sin𝜋

6𝑡.

Calculate the period from the equation and compare it to your answer in c.


7. The temperature of an air-conditioned home on a hot day can be modelled using the

function 𝑡(𝑥) = 2.5𝑠𝑖𝑛 (𝜋

6𝑥) + 25, where x is the time in hours after the air

conditioner turns on and t(x) is the temperature in degrees Celsius.

A) What are the maximum and minimum temperatures in the home?

B) What is the temperature 20 hours after the air conditioner has been turned on?

C) What is the period of the function? How would you interpret this value in this

context?

Answers: SECTION 1:1B 2D 3A 4B 5D 6C 7C 8D

SECTION 2: 1. 2

5 2. y = -5 3. {𝑦| − 8 ≤ 𝑦 ≤ 12, 𝑦 ∈ 𝑅} 4. i) A v) B 5. Amplitude

1

10 midline y

= 3.5 range {𝑦|3.4 ≤ 𝑦 ≤ 3.6, 𝑦 ∈ 𝑅} period 180o 6.A) {𝑦| − 6.5 ≤ 𝑦 ≤ 6.5, 𝑦 ∈ 𝑅} B) y = 0

when the water is calm. C) 12hours D) period 12 same 7. A) max 27.5 min 22.5 B) 22.8 C) 12


Chapter 8 REVIEW

Section 1: Multiple Choice

1) What is 3

10

radians in degrees?

(A) 30 (B) 54 (C) 108 (D) 600

2) What is 5

3

radians in degrees?

(A) 6 (B) 180 (C) 300 (D) 600

3) What is 2.4 radians in degrees?

(A) 24 (B) 138 (C) 275 (D) 432

4) What is 240 in radians?

(A) 3

2

(B) 2 (C) 3 (D)

4

3

5) What is 230 in radians?

(A) 0.4 (B) 1.3 (C) 4.0 (D) 722.6

6) What is the domain of the function siny x ?

(A) | 0,x x x R (B) | 0,x x x R

(C) |x x R (D) | 1,x x x R

7) What is the domain of the function 2cos 3y x ?

(A) | 2,x x x R (B) | 3,x x x R

(B) | 0,x x x R (D) |x x R

8) What is the range of cosy x ?

(A) | y Ry (B) | 1 y 1,y Ry

(C) | 1 y 1,y Ry (D) | 0y y


x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5- 4- 3- 2- 1

12345

x180 360

y

- 6

- 4

- 2

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5- 4- 3- 2- 1

12345

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5- 4- 3- 2- 1

12345

9) What is the range of siny x ?

(A) | y Ry (B) | 1 y 1,y Ry

(C) | 1 y 1,y Ry (D) | 0y y

10) Which is an intx of cosy x ?

(A) 0 (B) 90 (C) 180 (D) 360

11) Which is an intx of siny x ?

(A) 0 (B) 90 (C) 270 (D) 300

12) How can the graph of cosy x be translated so that we get the graph of

siny x ?

(A) 90 to the left (B) 90 to the right

(C) 45 to the left (D) 45 to the right

13) Which graph is periodic?

(A) (B)

(C) (D)


x60 ° 120 ° 180 ° 240 ° 300 ° 360 °

y

- 8

- 7

- 6

- 5

- 4

- 3

- 2

- 1

1

2

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

x90 ° 180 ° 270 ° 360 °

y

- 1

1

2

3

14) Which graph is sinusoidal?

(A) (B)

(C) (D)

15) What is the equation of the midline for the graph shown?

(A) 2y

(B) 3y

(C) 4y

(D) 6y

16) What is the equation of the midline for the graph shown?

(A) 1.5y

(B) 1y

(C) 0.75y

(D) 0.5y


x60 ° 120 ° 180 ° 240 ° 300 ° 360 °

y

- 8

- 7

- 6

- 5

- 4

- 3

- 2

- 1

1

2

x90 ° 180 ° 270 ° 360 °

y

- 1

1

2

3

x60 ° 120 ° 180 ° 240 ° 300 ° 360 °

y

- 8

- 7

- 6

- 5

- 4

- 3

- 2

- 1

1

2

x90 ° 180 ° 270 ° 360 °

y

- 1

1

2

3

17) What is the amplitude for the graph shown?

(A) -4

(B) -2

(C) 2

(D) 4

18) What is the amplitude for the graph shown?

(A) 0.5

(B) 0.75

(C) 0.5

(D) 1

19) What is the period for the graph shown?

(A) 90

(B) 180

(C) 270

(D) 360

20) What is the period for the graph shown?

(A) 45

(B) 90

(C) 135

(D) 180


While riding a Ferris wheel, Lily’s height above the ground in terms of time can be

represented by the following graph:

Use the graph to answer #21-#25

21) At what height does Lily board the Ferris Wheel?

(A) 1 m (B) 4 m (C) 5 m (D) 9 m

22) What is the maximum height above the ground?

(A) 1 m (B) 4 m (C) 5 m (D) 9 m

23) What is the height of the axle of the Ferris Wheel?

(A) 1 m (B) 4 m (C) 5 m (D) 9 m

24) What is the radius of the Ferris Wheel?

(A) 1 m (B) 4 m (C) 5 m (D) 9 m

25) What length of time does it take for the Ferris Wheel to complete 1 rotation?

(A) 2 seconds (B) 4 seconds (C) 6 seconds (D) 8 seconds

26) What is the amplitude of 3sin4 30 2y x ?

(A) 2 (B) 3 (C) 4 (D) 30

27) What is the amplitude of 0.5cos 3 1.2 4y x

(A) 0.5 (B) 1.2 (C) 3 (D) 4

x2 4 6 8 10 12 14 16 18 20 22 24

y

123456789

10

Height above ground (in m)

Time (seconds)


28) What is the equation of the midline of 4sin2 2.4 5y x ?

(A) 2y (B) 2.4y (C) 4y (D) 5y

29) What is the equation of the midline of 0.75cos0.5 15 2y x ?

(A) 2y (B) 0.5y (C) 0.75 (D) 15y

30) What is the period of 3sin4 2 1y x ?

(A) 4

(B)

2

(C) 4 (D) 8

31) What is the period of 4sin0.5 1y x ?

(A) 90 (B) 180 (C) 360 (D) 720

32) What is the period of 2sin34

y x

?

(A) 4

(B)

2

3

(C) (D) 6

33) What is the phase shift of 4sin2 2.4 5y x ?

(A) 2.4 radians to the right (B) 2.4 radians to the left

(C) 5 radians to the right (D) 5 radians to the left

34) What is the phase shift of 0.75cos0.5 15 2y x ?

(A) 15 to the right (B) 15 to the left

(C) 2 to the right (D) 2 to the left

35) What is the maximum value of 3sin4 30 2y x ?

(A) 1 (B) 5 (C) 6 (D) 7

36) What is the minimum value of 0.75cos0.5 15 2y x ?

(A) -2.75 (B) -2.5 (C) -2 (D) -1.25


37) What is the maximum value of 4cos0.5 1 7y x ?

(A) 3 (B) 4 (C) 8 (D) 11

38) What is the minimum value of 1.5sin5 46

y x

(A) -5.5 (B) -2.5 (C) 1 (D) 6.5

39) What is the range of 2sin4 14

y x

?

(A) | 5 3,y Ry y (B) | 3 1,y Ry y

(C) | 1 1,y Ry y (D) y R

40) What is the range of 2.5cos3 2.4 3.5y x ?

(A) | 1 1,y Ry y (B) |1 6,y Ry y

(C) | 0.5 6.5,y Ry y (D) y R


x0.5 1 1.5 2 2.5 3

y

2

4

6

8

10

12

14

16

18

20

Section 2: Constructed Response- Answer all questions in the space provided. Be sure

to show all workings to receive full credit!

41) The table below shows the height of a bicycle pedal over time:

(a) Graph the function on the grid below:

(b) What is the height of the axle of the bicycle pedal?

(c) What is the length of time taken to complete one rotation?

(d) Suppose another cyclist pedals the same bike at a rate of 1 revolution per

second. How would the graph change?

(e) Write the equation of the function in the form cosy a bx d

Time (s) Height (in.

0 18

0.5 11

1.0 4

1.5 11

2.0 18

2.5 11

3.0 4


42) The height of the water in a harbour since 12 PM is shown in the graph below:

(a) What is the maximum height of the

water?

(b) What is the minimum height of the

water?

(c) What is the length of time between low

tide and high tide?

(d) John wants to go collect mussels (the

tide must be relatively low for this).

He thinks that 6:00 PM would be a good

time to go. Do you agree or disagree

with John? Justify.

43) The height of water in a wave pool oscillated between a maximum of 13 ft and a

minimum of 5 ft. The wave generator pumps 6 waves per minute.

(a) What is the equation of the midline?

(b) What is the amplitude?

(c) What is the period?

(d) Write the equation of the function in the form siny a bx d

5 10 15 20

2

4

6

8

10

12

14

Time (h)

He

ig

ht

(

ft

)


44) The equation 70.5 19.5sin 46

y t

models the average monthly

temperature for Phoenix, Arizona, in degrees Fahrenheit. In this equation, t

denotes the number of months, with 1t representing January. What is the

average monthly temperature for July?

45) A horse on a carousel goes up and down as the carousel goes round and round.

The height of the horse in inches, h , as a function of time (s) is given by

2

9sin 467

h t

(a) Determine the height of the horse after 8 seconds.

(b) The children have been complaining that the carousel is too slow. How

would the equation change the conductor makes the carousel go around

faster?

46) Bob collects data on the average monthly temperatures of his hometown.

He uses his graphing calculator to perform a Sinusoidal Regression.

The screenshot of his calculator is shown below (where y represents the

temperature in degrees Fahrenheit and x represents the month with 1t

representing January:

(a) Write the equation of the curve of best fit.

(b) Use your equation to determine the

temperature in June.


x90 ° 180 ° 270 ° 360 °

y

- 8

- 7

- 6

- 5

- 4

- 3

- 2

- 1

1

2

x180 ° 360 ° 540 ° 720 °

y

- 8

- 7

- 6

- 5

- 4

- 3

- 2

- 1

1

2

x90 ° 180 ° 270 ° 360 °

y

- 4

- 3

- 2

- 1

1

2

3

4

5

6

x90 ° 180 ° 270 ° 360 °

y

- 4

- 3

- 2

- 1

1

2

3

4

5

6

47) Match the graph with its equation:

Graph 1 Graph 2

Graph 3 Graph 4

Equation 1: 2sin2 3y x Equation 2: 3sin 1y x

Equation 3: 2sin0.5 3y x Equation 4: 𝑦 = 3𝑠𝑖𝑛2𝑥 + 1


x0.5 1 1.5 2 2.5 3

y

2

4

6

8

10

12

14

16

18

20

Answers:

1) B 2) C 3) B 4) D 5) C 6) C

7) D 8) B 9) B 10) B 11) A 12) B

13) B 14) C 15) C 16) B 17) C 18) C

19) D 20) D 21) A 22) D 23) C 24) B

25) D 26) B 27) A 28) D 29) A 30) B

31) D 32) B 33) A 34) B 35) B 36) A

37) D 38) A 39) B 40) B

41) (a) (b) Height of axle = midline = 11 in.

(c) Period = time for one

revolution = 2 s

(d) Period would be 1 s. Waves

would be ‘closer’ together. One

complete wave would be shown

from 0-1 second.

(e) 7cos 11y x

42) (a) Maximum height is 13 ft (b) Minimum height is 1 ft

(c) 6 hours (d) No, because the height of water

is over 11 ft.

43) (a) 9y (b) 4 ft (c) 10 s (d) 4sin0.628 9y x

44) 90F 45) (a) 53 in (b) The ‘b’ value would decrease

46) (a) 22.43sin(0.51 2.1) 54.66y x (b) 73F

47) Graph 1 - Equation 1 , Graph 2 - Equation 3

Graph 3 – Equation 2, Graph 4 - Equation 4

Chapter 8: SINUSODIAL FUNCTIONS · PDF fileChapter 8 Math 3201 4 Worksheet 8.1 Section 1:...

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Transcript of Chapter 8: SINUSODIAL FUNCTIONS · PDF fileChapter 8 Math 3201 4 Worksheet 8.1 Section 1:...