Electricity
 EKG
 Pendulum
 Respiratory cycles
 Tuning fork
 Seismograph


Amplitude – how high and low the graph goes
› think amps on a speaker

Period – how long the graph takes to repeat
itself
› think about periods in history or fads

Phase shift – horizontal shifts in the graph
› Think about how you go through phases, you shift or
change

Vertical shift – a shift up or down in the graph

Standard form
y = A sin Bθ (tomorrow we will add more to this
standard form)
 A = amplitude
 Period = 2π/B or 360/B
 In one period, there are
› Begins at (0,0)
› Goes up to +A
› Comes back to x-axis
› Goes down to –A
› Comes back to x-axis

Repeats itself
4 equal intervals

Standard form
y = A cos Bθ
= amplitude
 Period = 2π/B or 360/B
 In one period, there are 4 equal intervals
 A
›
›
›
›
›

Begins at (0,+A)
Goes down x-axis
Goes down to -A
Comes back to x-axis
Goes back to +A
Repeats itself
Example: Sketch the graph of y = 3 cos θ
Amplitude = 3, Period = 2π
Break the interval into four equal parts. 2π/4 = π/2
θ
y = 3 cos θ
y

(0, 3)
2
1

0
3
2
0
max

-3
x-int min
3
2
0
2
3
x-int
max
(2, 3)

1 
( , 0)
2
2
3
( , –3)
2
( 3 , 0)
2
3
4 θ
Example: Sketch the graph of y = 2 sin 4θ
Amplitude = 2, Period = 2π/4 = π/2
Break the interval into four equal parts. π/2/4 = π/8
θ
y = 2cos 4θ
y
 / 8
2
1
1
2
3

0
0
8
2
x-int
/4
0
max x-int
 /8
 /4
3
8
-2
/2
0
min
x-int
3 / 8
 /2 θ
Example: Sketch the graph of y = -2 sin 3θ.


amplitude: |a| = |–2| = 2
period: 2 = 2
3
k
Calculate the five key points.


interval:2 =
6
12
θ
0
y = –2 sin 3θ
0
y



6
3
2
2
3
–2
0
2
0
(  , 2)
2
6


6
3

(6
2
2
3

2
(  , 0) 2
3
( , 0)
(0, 0)
2

,-2)
3
5
6

θ
Example: Sketch the graph of y = -4 cos (3/2 θ)
Amplitude = 4, Period = 2π/(3/2) = 4π/3
Break the interval into four equal parts. 4π/3/4 = 4π/12= π/3
θ
y=-4 cos 3/2θ
y
 / 3
2
1
1
2
3
0
-4
min
 /3

3
0
2/3
4

0
4/3
-4
x-int max x-int
2 / 3
min

4 / 3 θ
Example: Sketch the graph of y = sin (1/3 θ)
Amplitude = 1, Period = 2π/(1/3) = 6π
Break the interval into four equal parts. 6π/4 = 3π/2

Pretend to write a letter to a friend and
explain how to graph sin and cos. Use 1
or 2 examples to help explain.