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**By: Taylor Pulchinski Daniel Overfelt Whitley Lubeck**

Sine & Cosine Graphs By: Taylor Pulchinski Daniel Overfelt Whitley Lubeck

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**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)

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**Examples y= 4 sin 3(x-2) Finding the Period and Amplitude Amplitude=4**

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

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**Examples of Non-Shifted Graphs**

y = sinx y = cos x

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**Example Given 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

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**Example Given 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

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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

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**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: 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

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**Assessment . y= 2 sin 2x+ 4 A) 2 B) 2x C) 4 y= cos 3(x+1)-4 D) 6**

1&2 Find 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 D) 7 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) left 1 down 4 3 B) left 4 up 1 C) 3 up 1 left 4 D) 1 up 3 left 4

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**Assessment Continued y= 2 sin 6 (x-3)+2 A) 2 down 3 up 2**

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) right 2 down 3 6 C) right 3 up 2 3 D) 6 left 3 up 2

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Assessment Continued 5. Sketch a Graph y= 6sin2x A) B) D) C)

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Assessment Continued 6.Sketch a graph y= -2cos 2(x + )-2 8 A) B) C) D)

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**Assessment Continued A) 4 sin 3x B) 3 sin 4x C) 3 sin 4(x+2)**

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)

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**Assessment Continued A) y=4 sin 3.5(x) +.5 B) y=3.5 sin 4(x) +3.5**

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

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**Assessment Continued A)y=30sin 2 (x-.75)+32 3 B) y=32 sin 3 (x+3)+30 2**

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

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**Answer Key to Assessment**

2. C 3. A 4. C 5. A 6. B 7. B 8. B 9. C 10. A

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**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

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**Tic Tac Toe Answer Key 1.) Amplitude: 5 Period: π Vertical shift: none**

Phase shift: none 2.) Amplitude: 2 Vertical shift: down 2 Phase shift: left π/8 3.) Amplitude: 1 Period: 8π 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 Vertical shift: up 7 Phase shift: none 8.) Amplitude: 4 Vertical shift: down π/2

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**References http://graphsketch.com/**

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