Sine &amp; Cosine Graphs

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# Sine &amp; Cosine Graphs - PowerPoint PPT Presentation

Sine &amp; Cosine Graphs. By: Taylor Pulchinski Daniel Overfelt Whitley Lubeck. http://www.youtube.com/watch?v=9rsJF6lqxao. Equations. y = a sin (bx-h)+ k y = a cos (bx-h)+k a = Amplitude (height of the wave) 2( )/b = Period (time it take to complete one trip around)

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### Sine & Cosine Graphs

By: Taylor Pulchinski

Daniel Overfelt

Whitley Lubeck

Equations

y = a sin (bx-h)+ k

y = a cos (bx-h)+k

a = Amplitude (height of the wave)

2( )/b = Period (time it take to complete one trip around)

h = Phase Shift (left or right movement)

k = Vertical Shift (up or down movement)

Examples

Finding the Period and Amplitude

y= 4 sin 3(x-2)

Amplitude=4

Period=2 /3

Amplitude y= a sin (bx-h)+k

y= 4 sin (bx-h)+k

Period y= a sin (bx-h) +k

P= 2 / b

P= 2 /3

ExampleGiven the equation: Graph

y = -2 sin (x-π/4) +1

amplitude = 2

period = 2π (2π/b = 2π/1 = 2π)

phase shift = right π/4

vertical shift = up 1

ExampleGiven the Graph: write equation

(graph goes by

incriments of one)

1)Find what we know

amplitude: = 4

Period = π/2 (2π/4 = 1π/2)

phase shift = left π

vertical shift = down 3

2) Plug into equation

y = __cos__(x__)___

y = 4 cos 4 (x+π) -3

Example Story Problem

A Ferris Wheel with a diameter of 60 feet completes one revolution every 5 minutes. The closest a chair gets to the ground is 2 feet. Write a cosine function for the height of the person above ground x minutes after boarding.

1) Find what we know

Amplitude: 30

Vertical Shift: 32

Period: 2π/b = 5so 2π = 5bso 2π/5 = b

phase shift: none

2) plug into equation

y = ___ cos __(x___)____

y = 30cos(2π/5) x +32

graph start at 0 and goes to 62 on the y axis; graph starts at 0 and goes to 5 on x axis

Story Problem Continued

For the same problem, now write a sine function for the height of the person above ground x minutes after boarding

1) Find what we know

amplitude: 30

Vertical Shift: 32

Period: 2π/b = 5 so 2π = 5b so 2π/5 = b

Phase Shift: 1.25 left (sine graph starts halfway between the starting point and middle...so 5/2 = 2.5/2 = 1.5)

2) Plug into equation

y = __ sin____(x____)____

y = 30sin(2π/5)(x-1.25)+32

Graph tarts at 0 and goes to 62 on the y axis; graph starts at 1.25 and goes to 6 on the x axis

Assessment

3. Find the Period of the function and use the language of transformations to describe the graph of the function related to y= cosx

y= cos 3(x+1)-4

A) 2 left 1 down 4

3

B) 2 left 4 up 1

3

C) 3 up 1 left 4

D) 1 up 3 left 4

1&2Find the amplitude of the

function

. y= 2 sin 2x+ 4

A) 2

B) 2x

C) 4

D) 6

y= -7 sin 3x-7

4

A) 3

B) -7

C) -7

4

D) 7

Assessment Continued

4. Find the Period of the function and use the language of transformations to describe how the graph of the function related to the graph y= cosx

y= 2 sin 6 (x-3)+2

A) 2 down 3 up 2

B) 2 right 2 down 3

6

C) 1 right 3 up 2

3

D) 6 left 3 up 2

Assessment Continued

5. Sketch a Graph

y= 6sin2x

A)

B)

D)

C)

Assessment Continued

6.Sketch a graph

y= -2cos 2(x + )-2

8

A)

B)

C)

D)

Assessment Continued

7&8 Write a Sin equation from the given graph. Then write a Cos equation

A) 4 sin 3x

B) 3 sin 4x

C) 3 sin 4(x+2)

D) 4 sin 3(x+2)

A) 3 cos 4 (x+3)

B) 3 cos 4 (x+2)

C) 4 cos 4 (x+2)

D) 4 cos 2 (x+3)

Assessment Continued

9. Write a Sin equation for the graph below.

A) y=4 sin 3.5(x) +.5

B) y=3.5 sin 4(x) +3.5

C) y=3.5 sin 4(x) +.5

D) y=4 sin 4(x) +3.5

Assessment Continued

10. Write a Sin equation when the diameter of a ferris wheel is 60 feet and it takes 3 minutes to make one round. The elevation is 2 feet off the ground.

A)y=30sin 2 (x-.75)+32

3

B) y=32 sin 3 (x+3)+30

2

C) y=30 sin 2 (x+.75)+32

3

D)y= 30sin 2 (x-.75)-31

3

1. A

2. C

3. A

4. C

5. A

6. B

7. B

8. B

9. C

10. A

Supplement Activity Tic Tac Toe

Directions: Two teams will play against one another. If you get a problem correct you can play an “x” or “o” depending on which team you’re on. First team to get three in a row wins.

Problems: State the amplitude is, vertical and horizontal shifts, and what the period is.

• y=5sin(2x)
• y=2cos2(x+π/8)-2
• y=cos(x/4)
• y=4sin4(x-2)+3
• y=3sin2(x+4)
• y=sin(x-π/4)+1
• y=2cos(x)+7

8. y=4sin2(x)-π/2

1.)Amplitude: 5

Period: π

Vertical shift: none

Phase shift: none

2.) Amplitude: 2

Period: π

Vertical shift: down 2

Phase shift: left π/8

3.) Amplitude: 1

Period: 8π

Vertical shift: none

Phase shift: none

4.) Amplitude: 4

Period: 1/2π

Vertical shift: up 3

Phase shift: right 2

5.) Amplitude: 3

Period: π

Vertical shift: none

Phase shift: left 4

6.) Amplitude: 1

Period: 2π

Vertical shift: up 1

Phase shift: right π/4

7.) Amplitude: 2

Period: 2π

Vertical shift: up 7

Phase shift: none

8.) Amplitude: 4

Period: 2π

Vertical shift: down π/2

Phase shift: none

References

http://graphsketch.com/

http://mouserunner.com/MozllaTicTacToe/Mozilla_Tic_Tac_Toe.htm