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
Period
y= a sin (bx-h)+k
y= 4 sin (bx-h)+k
y= a sin (bx-h) +k
P= 2 / b
P= 2 /3
Examples of Non-Shifted Graphs
y = sinx
y = cos x
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
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
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
1&2 Find the amplitude of the
function
.
A) 2
B) 2x
C) 4
D) 6
y= 2 sin 2x+ 4
y= cos 3(x+1)-4
y= -7 sin 3x-7
4
A) 3
B) -7
C) -7
4
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
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
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)
C)
B)
D)
Assessment Continued
6.Sketch a graph
y= -2cos 2(x + )-2
8
C)
A)
D)
B)
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
2
(x+3)+30
C) y=30 sin 2
3
(x+.75)+32
D)y= 30sin 2
3
(x-.75)-31
Answer Key to Assessment
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.
1.
2.
3.
4.
5.
6.
7.
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
Tic Tac Toe Answer Key
1.) Amplitude: 5
Period: π
Vertical shift: none
Phase shift: none
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
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
8.)
4.) Amplitude: 4
Period: 1/2π
Vertical shift: up 3
Phase shift: right 2
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
http://www.youtube.com/watch?v=9rsJF6lqxaob
The Great Ms. Scarseth