Section 4.4 Notes

Graphs of SINE and COSINE Functions

[ 2 ,2 ] by [ 2,2]

cosy x

Graphs of SIN and COS Functions

[ 2 ,2 ] by [ 2,2]

siny x[ 2 , 2 ] by [ 2,2]

2 2

Stretches – Translations of Sin and Cos Graphs

sin( )y x

sin( ) 1y x

sin 12

y x

Can you identify this sinusoid?

Graph of sin(x) shifted left by ¼ of period…

sin2

y x

OR…

cos( )y x2cos( )y x

General Form for Sinusoidal Functions

( ) sin( ) orA function is a

( ) sinusoid if it can be written in the form

where , , and are constants and neither norcos( )

is 0.f x a bx c d f x a bx c

a b c d a bd

Amplitude = a

2Period = b

Period is the length of one full cycle of the wave.

Amplitude is half of the total height of the wave.

1 | |Frequency = 22

or the reciprocal of the abs val of period

b

b

The graphs of sin( ( )) and cos( ( ))where and , have the following characteristi0 :0 cs

a aa

y x yh xb bk kb

h

amplitude =|a|

2period = |b| |b|frequency =

2which implies…

When compared to the graphs of sin( ) and cos( ),they also have the following characteristics:

y x y x

a phase shift (horizontal translation) of unitsh

a vertical translation of unitskNote: has been "factored out" of the argumentb

Setting Your Viewing Windowx-min : periodx-max: + period

periodx-scl: 4

y-min : min 1y-max : max 1y-scl : 1?? (doesn't matter too much)

max minAmplitude 2

a

max minVertical shift is 2

k

2period b

Tips to get started…

( ) 2sin 5 32

y f x x

10( ) sin 5 32y f x x

5

103

2b

k

h

a

2 25

Periodb

| | 2Amp a

25

5

10 3

10

Basic SINE curve starts at ORIGIN, on its way UP

10

10

RIGHT 3DOWN

A roller coaster does a 360o loop. The bottom of the loop is 20 off the ground and the loop has a diameter of 100 feet. If it takes the coaster 4 seconds to go around the loop, write a sinusoidal function to determine h(t), the height of the coaster after t seconds.

20 '

100 '

t = time (sec)

h(t)

Height in ft

20

120

42

We could think of the sinusoid we created in two different ways: sine shifted right or the opposite of a cosine curve. We will use the opposite of the cosine curve.

Max - min Max + min 120 20 120 20Amplitude is , Vertical shift is , so 50, 702 2 2 2

a k

70

2We know the period is 4 seconds as defined in the problem. So 4 = , or b= b 2

So a sinusoidal function is given by: ( ) 50cos 702th t

Tarzan is swinging back and forth on a vine. As he swings, he goes back and forth across the river bank below, going alternately over land and water. Jane decides to provide a mathematical model for his horizontal motion and starts her stopwatch. Let t be the number of seconds that the stopwatch reads and y be the number of meters that Tarzan is from the river bank. Assume that y varies sinusoidally with t, and that y is positive when Tarzan is over the water and negative when he is over the land.

Jane finds that when t = 2, Tarzan is at one end of his swing, where y = -23. She finds that when t = 5, he reaches the other end of his swing and y = 17.

a) Sketch a graph of the sinusoidal function.

b) Write an equation expressing Tarzan’s distance from the river bank in terms of t.

c) Predict y when t = 2.8 and t = 15

d) Where was Tarzan when Jane started the stopwatch?

a) When t = 0.

e) Find the least positive value of t for which Tarzan is directly over the riverbank.

a) When y = 0.

Using as POSITIVE sine function (a > 0), write a sinusoidal function 2 7that would have its 1st MAX at ,5 and its 1st MIN at , 1 .9 9

7 , 19

2 ,59

max min 5 ( 1) 32 2

a

max min 5 ( 1) 22 2

k

sin( ( ))y a b x h k

1 7 2 5 2 10 9 so so 2 9 9 9 9 5per b

b

93sin ( ) 25

y x h

59

Using as POSITIVE sine function (a > 0), write a sinusoidal function 2 7that would have its 1st MAX at ,5 and its 1st MIN at , 1 .9 9

7 , 19

2 ,59

93sin ( ) 25

y x h

518

518

So the coordinates of this point 2 5 4 5 19 18

are fo

18

und b :

18

y

18

1,2

18

1So the horizontal shift is left :18

9 13sin 25 18

y x 2y