Evaluate: Homework and Practice Book 19.1 and...

5
Secondary III 9-4 HW Name: ______________________ Graphs of Trig Functions Book 19.1 and 19.3 For each trigonometric function, identify the vertical stretch or compression and the horizontal stretch or compression. Then, graph the function and identify its period. 1 _ - - - 0 2π -2π π -π π 2 4 1 2 3 5 -4 -5 -1 -2 -3 π 2 3π 2 3π 2 5π 2 5π 2 x y - - - 0 2π 2π π -π π 2 - 2 1 2 1 1 -1 π 2 3π 2 3π 2 5π 2 5π 2 x y 1. y = 4 sin x 2. y = 1 _ 2 cos 2x - - - 0 2π 2π π -π π 2 - 3 1 - 3 2 3 1 3 2 π 2 3π 2 3π 2 5π 2 5π 2 x y - - 0 π -π π 3 - 4 1 - 2 1 4 1 2 1 π 3 2π 3 4π 3 - 2π 3 5π 3 5π 3 - 4π 3 x y 5. y = 1 _ 3 sin 2x 6. y = - 1 _ 4 cos 3x

Transcript of Evaluate: Homework and Practice Book 19.1 and...

Page 1: Evaluate: Homework and Practice Book 19.1 and 19missnobleaf.weebly.com/uploads/1/2/5/8/12582810/hw_9-4.pdf · Secondary III 9-4 HW Name: _____ 2 2 2-2-24 6 0 2 4 6 8 10 x y y x-1-2-3-4-52

Secondary III 9-4 HW Name: ______________________

Graphs of Trig Functions Book 19.1 and 19.3

--- 0 2π-2π π-π π2

4

1

2

3

5

-4

-5

-1

-2

-3

π2

3π2

3π2

5π2

5π2

x

y

--- 0 2π2π π-π π2

- 21

21

1

-1

π2

3π2

3π2

5π2

5π2

x

y

0 3π 6π 9π 12π 15π-3π-6π-9π-12π-15π

1

-1

-2

-3

-4

-5

2

3

4

5

x

y

0 2π 4π 6π 8π 10π-2π-4π-6π-8π-10π

1

-1

-2

-3

-4

-5

2

3

4

5

x

y

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Evaluate: Homework and Practice

For each trigonometric function, identify the vertical stretch or compression and the horizontal stretch or compression. Then, graph the function and identify its period.

1. y = 4 sin x 2. y = 1 _ 2 cos 2x

3. y = -3 sin 1 _ 6 x 4. y = -2 cos 1 _ 3 x

Module 19 969 Lesson 1

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B

--- 0 2π-2π π-π π2

4

1

2

3

5

-4

-5

-1

-2

-3

π2

3π2

3π2

5π2

5π2

x

y

--- 0 2π2π π-π π2

- 21

21

1

-1

π2

3π2

3π2

5π2

5π2

x

y

0 3π 6π 9π 12π 15π-3π-6π-9π-12π-15π

1

-1

-2

-3

-4

-5

2

3

4

5

x

y

0 2π 4π 6π 8π 10π-2π-4π-6π-8π-10π

1

-1

-2

-3

-4

-5

2

3

4

5

x

y

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Evaluate: Homework and Practice

For each trigonometric function, identify the vertical stretch or compression and the horizontal stretch or compression. Then, graph the function and identify its period.

1. y = 4 sin x 2. y = 1 _ 2 cos 2x

3. y = -3 sin 1 _ 6 x 4. y = -2 cos 1 _ 3 x

Module 19 969 Lesson 1

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B

--- 0 2π2π π-π π2

- 31

- 32

31

32

π2

3π2

3π2

5π2

5π2

x

y

-- 0 π-π π3

- 41

- 21

41

21

π3

2π3

4π3

- 2π3

5π3

5π3

-4π3

x

y

0

4

5

1

2

3

1-1 2-2 -1

-2

-3

-4

-5

12-1

232-3

252-5

2

x

y

--- 0 π-π π3

1

-1

π3

2π3

2π3

12

-12

- 4π3

4π3

5π3

5π3

x

y

© H

oughton Mifflin H

arcourt Publishing Company

5. y = 1 _ 3 sin 2x 6. y = - 1 _ 4 cos 3x

Write an equation for each graph.

7. 8.

Module 19 970 Lesson 1

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B

Page 2: Evaluate: Homework and Practice Book 19.1 and 19missnobleaf.weebly.com/uploads/1/2/5/8/12582810/hw_9-4.pdf · Secondary III 9-4 HW Name: _____ 2 2 2-2-24 6 0 2 4 6 8 10 x y y x-1-2-3-4-52

Secondary III 9-4 HW Name: ______________________

--- 0 2π2π π-π π2

- 31

- 32

31

32

π2

3π2

3π2

5π2

5π2

x

y

-- 0 π-π π3

- 41

- 21

41

21

π3

2π3

4π3

- 2π3

5π3

5π3

-4π3

x

y

0

4

5

1

2

3

1-1 2-2 -1

-2

-3

-4

-5

12-1

232-3

252-5

2

x

y

--- 0 π-π π3

1

-1

π3

2π3

2π3

12

-12

- 4π3

4π3

5π3

5π3

x

y

© H

oughton Mifflin H

arcourt Publishing Company

5. y = 1 _ 3 sin 2x 6. y = - 1 _ 4 cos 3x

Write an equation for each graph.

7. 8.

Module 19 970 Lesson 1

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B

-1

-2

1

2

0 0.002 0.004 0.006 0.008 0.010

x

y

-10

-20

10

20

0 2 4 6 8 10 12

-1

-2

-3

-4

-5

2

3

4

5

1

0 2π 3π 4πππ2

3π2

7π2

5π2

x

y

© H

oughton Mifflin H

arcourt Publishing Company

Graph each function, and then find its frequency. What do the frequency, amplitude, and period represent in the context of the problem?

13. Physics Use a sine function to graph a sound wave with a period of 0.003 second and an amplitude of 2 pascals.

14. Physics A mass attached to a spring oscillates up and down every 2 seconds. Draw a graph of the vertical displacement of the mass relative to its position at rest if the spring is stretched to a length of 15 cm before the mass is released.

15. Physics A pendulum is released at a point 5 meters to the right of its position at rest. Graph a cosine function that represents the pendulum's horizontal displacement relative to its position at rest if it completes one back-and-forth swing every π seconds.

Module 19 972 Lesson 1

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B

-1-2-3-4-5

2

3

4

5

1

0 π 2π 3π-2πx

y

π2

3π2

5π2

3π2

π2

-- -π

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Elaborate

8. How can being given the first local maximum and local minimum of a cosine function (a > 0) help you write its equation?

9. Essential Question Check-In How do positive values of h affect the graph of a function in the form ƒ (x) = a sin 1 __ b (x - h) + k? How do negative values of k affect the graph of this function?

For each function, identify the period and the midline of the graph, and where the graph crosses the midline. For a sine or cosine function, identify the amplitude and the maximum and minimum values and where they occur. For a tangent function, identify the asymptotes and the values of the halfway points. Then graph one cycle.

1. ƒ (x) = -3 sin (x + π) + 1

Evaluate: Homework and Practice

Module 19 1003 Lesson 3

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B

0

2

4

-2-2

-4

π-π 2π4π3

5π3

2π3

2π3

π3

π3

--x

y

-1-2-3-4

2

3

4

5

6

1

0 π 2π-2πx

y

© H

oughton Mifflin H

arcourt Publishing Company

2. ƒ (x) = 2 cos3x + 1

3. ƒ (x) = tan 1 __ 2 (x - π) + 3

Module 19 1004 Lesson 3

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B

Page 3: Evaluate: Homework and Practice Book 19.1 and 19missnobleaf.weebly.com/uploads/1/2/5/8/12582810/hw_9-4.pdf · Secondary III 9-4 HW Name: _____ 2 2 2-2-24 6 0 2 4 6 8 10 x y y x-1-2-3-4-52

Secondary III 9-4 HW Name: ______________________

2

-2

-2

4

6

0 8 102 4 6

x

y

y

x

-1-2-3-4-5

2

3

45

1

0 π 2ππ2

3π2

3π2

π2

-- -π-2π

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

4. ƒ (x) = 3sin π __ 2 (x - 2) + 3

Write an equation as indicated for the given graph.

5. a sine function

Module 19 1005 Lesson 3

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B

10

20

30

40

2 4 6 8 10 120t

h(t)

8

16

24

32

40

48

0.5 1 1.5 20t

h(t)

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

For the context described, identify the period, midline, amplitude, and maximum and minimum values of the graph. For one cycle starting from t = 0, find all points where the graph intersects its midline and the coordinates of any local maxima and minima. Interpret these points in the context of the problem, and graph one cycle.

8. Historic Technology Water turns a water wheel at an old mill. The water comes in at the top of the wheel through a wooden chute. The function h (t) = 15 cos π __ 5 t + 20 models the height h in feet above the stream into which the water empties of a point on the wheel where t is the time in seconds.

9. Games A toy is suspended 36 inches above the floor on a spring. A child reaches up, pulls the toy, and releases it. The function h (t) = 10 sinπ (x - 0.5) + 36 models the toy’s height h in inches above the floor after t seconds.

Module 19 1007 Lesson 3

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B

© H

oughton Mifflin H

arcourt Publishing Company

10. Match each sine function with the cosine function that has the same graph.

A. y = sin2x 1. y = cos (x + π _ 2 ) + 1

B. y = 1 _ 2 sin3x 2. y = 1 _ 2 cos4 (x - 16 + π _ 8 ) C. y = 1 _ 2 sin4 (x - 2) 3. y = cos 2 (x - π _ 4 ) D. y = sin (x - π) + 1 4. y = cos (x) + 4

E. y = sin (x + π _ 2 ) + 4 5. y = 1 _ 2 cos3 (x - π _ 6 ) 11. How do h and k affect the key points of the graphs of sine, cosine, and tangent

functions?

H.O.T. Focus on Higher Order Thinking

12. Explain the Error Sage was told to write the equation of a sine function with a period of 2π, an amplitude of 5, a horizontal translation of 3 units right, and a vertical translation of 6 units up. She wrote the equation ƒ (x) = 5 sin 2π (x - 3) + 6. Explain Sage’s error and give the correct equation.

13. Critical Thinking Can any sine or cosine function graph be represented by a sine equation with a positive coefficient a? Explain your answer.

14. Make a Prediction What will the graph of the function ƒ (x) = cos 2 x + sin 2 x look like? Explain your answer and check it on a graphing calculator.

Module 19 1008 Lesson 3

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-B;CA-B

Page 4: Evaluate: Homework and Practice Book 19.1 and 19missnobleaf.weebly.com/uploads/1/2/5/8/12582810/hw_9-4.pdf · Secondary III 9-4 HW Name: _____ 2 2 2-2-24 6 0 2 4 6 8 10 x y y x-1-2-3-4-52

Secondary III 9-4 HW Name: ______________________ Selected Answers: 2.

Period = π 6.

Period = 2π3

8. f x( ) = − 12cos 3x( )

14.

--- 0 2π-2π π-π π2

4

1

2

3

5

-4

-5

-1

-2

-3

π2

3π2

3π2

5π2

5π2

x

y

--- 0 2π2π π-π π2

- 21

21

1

-1

π2

3π2

3π2

5π2

5π2

x

y

0 3π 6π 9π 12π 15π-3π-6π-9π-12π-15π

1

-1

-2

-3

-4

-5

2

3

4

5

x

y

0 2π 4π 6π 8π 10π-2π-4π-6π-8π-10π

1

-1

-2

-3

-4

-5

2

3

4

5

x

y

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Evaluate: Homework and Practice

For each trigonometric function, identify the vertical stretch or compression and the horizontal stretch or compression. Then, graph the function and identify its period.

1. y = 4 sin x 2. y = 1 _ 2 cos 2x

3. y = -3 sin 1 _ 6 x 4. y = -2 cos 1 _ 3 x

vertically stretched by a factor of 4

period: 2π

1_b

= 2 ⇒ b = 1_2

vertically compressed by a factor of 1_2

;

horizontally compressed

by a factor of 1_2

1 _ b

= 1 _ 6

⇒ b = 6

vertically stretched by a factor of 3;

horizontally stretched by a factor of 6;

reflected across the x-axis

period: 2π · 6 = 12π

period: 2π · ⎜b⎟ = 2π · 1 _ 2

= π

1 _ b

= 1 _ 3

⇒ b = 3

vertically stretched by a factor of 2;

horizontally stretched by a factor of 3;

reflected across the x-axis

period: 2π · 3 = 6πModule 19 969 Lesson 1

IN3_MNLESE389892_U7M19L1 969 16/09/14 10:19 AMExercise Depth of Knowledge (D.O.K.)COMMON

CORE Mathematical Practices

1–6 1 Recall of Information MP.7 Using Structure

7–12 2 Skills/Concepts MP.7 Using Structure

13–16 1 Recall of Information MP.4 Modeling

17 1 Recall of Information MP.7 Using Structure

18–19 1 Recall of Information MP.6 Precision

20 1 Recall of Information MP.7 Using Structure

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

Explore 1Graphing the Basic Sine and Cosine Functions

Exercise 18

Explore 2Graphing the Reciprocals of the Basic Sine and Cosine Functions

Exercises 19–22

Example 1Graphing f (x) = asin ( 1 _

b ) x

or f (x) = acos ( 1 _ b

) x

Exercises 1–6, 17

Example 2Writing f (x) = asin (1_

b ) x

or f (x) = acos (1_b ) x

Exercises 7–12

Example 3Modeling with Sine or Cosine Functions

Exercises 13–16

QUESTIONING STRATEGIESHow do the ranges of ƒ (x) = sin x and ƒ(x) = cos x compare? They are the same.

The range for both functions is [–1, 1].

How do the zeros of ƒ(x) = sin x and ƒ (x) = cos x compare? They are different.

The zeros of f (x) = sin x occur at multiples of π. The zeros of f (x) = cos x occur at odd multiples of π _

2 .

969 Lesson 19 . 1

DO NOT EDIT--Changes must be made through “File info”CorrectionKey=NL-B;CA-B

--- 0 2π2π π-π π2

- 31

- 32

31

32

π2

3π2

3π2

5π2

5π2

x

y

-- 0 π-π π3

- 41

- 21

41

21

π3

2π3

4π3

- 2π3

5π3

5π3

-4π3

x

y

0

4

5

1

2

3

1-1 2-2 -1

-2

-3

-4

-5

12-1

232-3

252-5

2

x

y

--- 0 π-π π3

1

-1

π3

2π3

2π3

12

-12

- 4π3

4π3

5π3

5π3

x

y

© H

oughton Mifflin H

arcourt Publishing Company

5. y = 1 _ 3 sin 2x 6. y = - 1 _ 4 cos 3x

Write an equation for each graph.

7. 8.

1 _ b

= 2 ⇒ b = 1 _ 2

vertically compressed by a factor of 1 _ 3 ;

horizontally compressed by a factor of 1 _ 2

1 _ b

= 3 ⇒ b = 1 _ 3

vertically compressed by a factor of 1 _ 4 ;

horizontally compressed by a factor of 1 _ 3 ;

reflected across the x-axis period: 2π · 1_2 = π

period: 2π · 1 _ 3 = 2π __ 3

The graph is a sine function.

a = 3

2πb = 1 ⇒ b = 1 _ 2π ⇒ 1 _ b = 2π

y = 3 sin 2πx

The is a cosine function reflected across the x-axis.

a = 1 _ 2

2πb = 2π _ 3

⇒ b = 1 _ 3 ⇒ 1 _ b = 3

y = - 1 _ 2 cos 3x

Module 19 970 Lesson 1

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B

IN3_MNLESE389892_U7M19L1 970 16/09/14 10:19 AMExercise Depth of Knowledge (D.O.K.)COMMON

CORE Mathematical Practices

21–22 1 Recall of Information MP.2 Reasoning

23–24 2 Skills/Concepts MP.3 Logic

25 3 Strategic Thinking MP.3 Logic

INTEGRATE MATHEMATICAL PRACTICESFocus on Technology

MP.5 Show students how to use a graphing calculator to find the secant or cosecant of a

number, and encourage them to use the calculator to check points on their graphs of these functions.

Stretching, Compressing, and Reflecting Sine and Cosine Graphs 970

DO NOT EDIT--Changes must be made through “File info”CorrectionKey=NL-B;CA-B

-1

-2

1

2

0 0.002 0.004 0.006 0.008 0.010

x

y

-10

-20

10

20

0 2 4 6 8 10 12

-1

-2

-3

-4

-5

2

3

4

5

1

0 2π 3π 4πππ2

3π2

7π2

5π2

x

y

© H

oughton Mifflin H

arcourt Publishing Company

Graph each function, and then find its frequency. What do the frequency, amplitude, and period represent in the context of the problem?

13. Physics Use a sine function to graph a sound wave with a period of 0.003 second and an amplitude of 2 pascals.

14. Physics A mass attached to a spring oscillates up and down every 2 seconds. Draw a graph of the vertical displacement of the mass relative to its position at rest if the spring is stretched to a length of 15 cm before the mass is released.

15. Physics A pendulum is released at a point 5 meters to the right of its position at rest. Graph a cosine function that represents the pendulum's horizontal displacement relative to its position at rest if it completes one back-and-forth swing every π seconds.

frequency = 1 ____ 0.003 ≈ 333 Hz

The frequency represents the number of cycles of the sound wave every second. The amplitude represents the maximum change in air pressure. The period represents the amount of time it takes for the sound wave to repeat.

frequency = 1 _ 2 = 0.5 Hz

The frequency represents the number of oscillations of the spring every second. The amplitude represents the spring’s maximum vertical displacement. The period represents the amount of time that it takes the spring to complete one oscillation.

frequency = 1 __ π ≈ 0.32 Hz

The frequency represents the number of back-and-forth swings every second. The amplitude represents the the pendulum’s maximum horizontal displacement. The period represents the amount of time for the pendulum to complete a back-and-forth swing.

Module 19 972 Lesson 1

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=NL-B;CA-B

IN3_MNLESE389892_U7M19L1.indd 972 16/09/14 9:02 PM

CURRICULUM INTEGRATIONYou may wish to have students reinforce their understanding of the terms amplitude and frequency by investigating their meanings in broadcast media. Radio signals are broadcast on AM (amplitude modulation) and FM (frequency modulation) bands.

Stretching, Compressing, and Reflecting Sine and Cosine Graphs 972

DO NOT EDIT--Changes must be made through “File info”CorrectionKey=NL-B;CA-B

Page 5: Evaluate: Homework and Practice Book 19.1 and 19missnobleaf.weebly.com/uploads/1/2/5/8/12582810/hw_9-4.pdf · Secondary III 9-4 HW Name: _____ 2 2 2-2-24 6 0 2 4 6 8 10 x y y x-1-2-3-4-52

Secondary III 9-4 HW Name: ______________________ 4.

Period = 4 10. A – 3 B – 5 C – 2 D – 1 E – 4

2

-2

-2

4

6

0 8 102 4 6

x

y

y

x

-1-2-3-4-5

2

3

45

1

0 π 2ππ2

3π2

3π2

π2

-- -π-2π

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

4. ƒ (x) = 3sin π __ 2 (x - 2) + 3

Write an equation as indicated for the given graph.

5. a sine function

Period: 1 _ b = π __ 2 , so b = 2 __ π ⇒ 2π · b = 4; Midline: y = k, or y = 3; Amplitude: a = 3

h = 2 ⇒first midline crossing is translated 2 units to the right to (2, 3) ; other midline

crossings are at cycle halfway point, (2 + 2, 3) = (4, 3) , and cycle endpoint, (2 + 4, 3) = (6, 3) .

Maximum: k + a = 3 + 3 = 6; occurs when x = 2 + 4 ____ 2 = 3 at (3, 6) .

Minimum: k - a = 2 - 2 = 0; occurs when x = 4 + 6 ____ 2 = 5 at (5, 6) .

Amplitude: a = 3 - (-1)

______ 2 = 2; Midline: y = 1, so k = 1; Period: 2π; so, 2π · b = 2π ⇒ b = 1.

Graph first crosses midline while rising for x > 0 at x = π ⇒ horizontal translation π units

right, so h = π. An equation is f (x) = 2sin (x - π) + 1. (Note that f (x) = 2sin (x + π) + 1

also represents the graph.)

Module 19 1005 Lesson 3

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A

IN3_MNLESE389892_U7M19L3.indd 1005 4/9/14 2:19 AM

VISUAL CUESWhen they are writing a function rule for a tangent graph, encourage students to circle the point that is the image of the point (0, 0) under the transformation. This may help them remember that the coordinates of this point are h and k, and can help them identify the other key points or features and the parameters needed to write the function rule.

1005 Lesson 19 . 3

DO NOT EDIT--Changes must be made through “File info”CorrectionKey=NL-B;CA-B