4.5 Graphs of Sine and Cosine Functions Graphing Sine

The Graph of y = s in x

1

0.5 /

-0.5 •̂ -1 i

V ' S{« 1, 0 s ,v s 2vr

-2iT -3IT/2 -TT -TT/Z 0 Tr/2 n 3n/Z Zn -270° -180° -90' 90° ISO" 27'0° 360° -Pemd'2rr-

3 2 3 57T 47T 6

V3 V 3 V 3 -1 - V3

t(m 0 t» t.

As X 'mtmm

fidB 1 its 0. 0 f» - I.

At X iMttmi

y incrtim }»«! - I le 0.

Properties of the Graph of y = sin x

m The domain is (-^.'"•'-^)

• The range is

• The period is t • The function is f i-'j because s i n ( - x ) = - s i n x

J

^rx I-

Tk« rs«|« Is - -~/ \

J «tin ,v

4 /

? \ ' i

1 cvck — 1 . — * ^

1 r ! ~

1 cvcle "— —^

^ \

I CVCk:

4 IT

Key Points on the Graph of y = s in x

X y 1, y ~ mnS for 0 s 8ss 2s

0 \ - * \ 0 i A , *!= / 2 -

0 -1 •

0 n t

T

t

Transformations of the Graph of y = sin Jt y = cfix) c> I

y = cf(A) 0 < (• < 1 y = /(cjf) r > 1 y = /(cx) 0 < r < 1

Vertical stretch by a factor of c Vertical shrink by a factor of r Horizontal shrink by a factor of c Horizontal stretch bv a factor of c 5 ̂ ^ ^ ^

\ - A s m 6 x

c ^

Amplitudes and Periods

The graph of y = / I sin Bx. B > 0. has

amplitude = I i V = ,4 i l l E l -

period

Example: Graphing y = A s in Bx

Determine the amplitude and period of y = 2 s i n - x . Then graph one period of the function.

Amplitude: A \ 2,^"- j^l \

Period: ^ . , Tji: . 2^ • 2

...

Transformations:

s-lrtVr'h h "I

X y

0 0

1 2

0

-1 2

27r a

a

X V X

If " . F̂ .

Y

0 0 1

T |

0 "2

4f 0

2 '

TT

Example: Graphing y = A s in Bx Determine the amplitude and period of y = 3 s i n x . Then graph one period of the function.

Amp'hi X 1

0

i 1 0

-'1 2

D

31 X V

D 0 f z rf 0 1 -2-

2 ^

The Graph ofy = A sin^Bx - Q

Tlie graph of y = A <in(B.\ C), B > 0. is obtained by horizontally shifting the graph of y = A sin Bx ->o that the starting point of the cycle is shifted from x = 0

to X = I f 4- > 0. the shift is t o the B C

right. I f -C .

B

0. the shift is to the left. The

number -r- is called the p h a s e s h i f t . B

ampHtiide = \A

v = A m iHx~C]

period = B

c p h a s e s h i f t = " g

The Graph of y = A sin(^;c - C)

y = f{x + b)

y = fQ-b)^

Shifted left b units Shifted right b units

Note' 6 «s 1 I f % |io|. l^^^g 1̂ 1̂̂ 1- %

Mini Example: Find the horizontal shift (aka phase shift). y = - 2 s i n ( 2 x - - j

TT 3

2r 3

steps to Graphing y = A svaiBx - C) 1. Find amplitude, period, phase shift. 2. Determine transformations. 3. Make starting table. 4. Transform the coordinates in the correct order (Transformation Yoga). 5. Scale the X-axis and y-axis, plot the points, and connect them with a smooth curve.

Example: Graphing y = A sin(Bx - C) Determine the amplitude, period, and phase shift of y = - 3 sin (ix - T h e n graph one period of the function.

. s h i f t n g h t | Q

0 "r z

I ? 1

2^

0

\

0 -\

0

TT 3

•3

0

I

• *

2. _X V X 0

\ a

0

\

0 HIT

\ - ) - )

Ttr ID 0

0

3

0

l i lT

0

0

Example: Graphing y = i4sin(FA: - C) -3 Determine the amplitude, period, and phase shift of y — 4 s in (3x — Then graph one period of the function

XT 2. i

"'0 0 \

D

'2"R b

0, X

^ 1 1 0 0

1 \

0 0

1 0 % 0

0

0

4.5 Graphs of Sine and Cosine Functions Graphing Cosine

The Graph of y = cos x

0.5 0

-0.5 -1

cos{x)^'

X

1 V i y i

TJM f<t)l II X -1 s » s t.

-2Tt - S i r / 2 -IT - T r / 2 0 IT/2 TT 3TT/2 ZTC -360" -270° -180° -90° 90' 180"" 270=- 360'

r " f T n 2w 2 3

51T Jf

ft

7-jr 6

4ir >

3lT > 3

11 TT 6

• .V 1 1 "I " 4

\ ' 3 t (1 1 V'3

2

As I isemttt

(mm On J ,

y ii<c«it«i

At I (Mr-eases (ttm t« y <i«efesje»

ffe« 0 t» - J .

A« A ttef«ss»s frem ,T t» .

ffsra -1 It 0.

Properties of the Graph of y = cos x

m The domain is j-^yO^^

• The range is L T ^ ) 0

• The period is_

• The function is because cos(—x) = cosx

Word of the Day: S l N D S O J D A L

hm in isr, y intnmi

ff«« 0 <« t.

1 0.5

0 -0.5

-1 /

-2TT -3Tr/2 TT -TT/2 0 Tr/2 TT 3IT/2 Sir -360 -2. J -180 -9;i 90" 180 ,.'0 MiC

1

0.5 0

-0.5 -1

€OS(x)

- 2 i T -3TT/2 -TT -TT/2 0 TT/2 TT 3IT/2 2TT -360" -270'- -180' -90= 90= 180 > 270° 360'=

The graphs of sine and cosine functions are called S ^ n U S O l d ^ l graphs.

The graph of y = cos x is the graph of y = sin x with

phase shift of 'X

1 )

Key Points X

on the Graph of y = cos x

y

0 i : 2

T

2

2 T I

4

0

\

- J

Amplitude, Period and Phase Shift of y = J4 COS(BX — C) Tlie graph of _v = A cos{Bx •••••• C) is obtained by horizontally siiifting the graph

of }' = .4 cos Bx so that the starting point of the c\clc h shifted from x = 0 to

X = 4 - I f 1^ > 0' the shift is to the right.

If 77 < 0. the shift is to the left. The D

number -— is called the phase shift. B

amplitude = \A

period =

Example: Graphing y = A cos(Bx - C) j» Determine the amplitude, period, and phase shift of y = ^cos (^2x + Then graph one function.

period of th

Amplitude: /\ 1 1

c Phase shift: -

Transformations:

."IT

Q> '(p

fe> b t?

\ A: y

IT V 1 0 \ 1 "K It I

\ 0

Is: r \ V

i ^ \

3k

21 o\ 0

6>. $IT\ 27l\ 1 \

1

y = fix) + a Shifted up a units

y = f(x) - a Shifted down a units

y = nx + b) Shifted left b units

y = n x - b) Shifted right b units

y = -fix) Reflection over x-axis

y = fi-x) Reflection over y-axis

y = cfix) c > 1 Vertical stretch by a factor of c

y = cfix) 0 < c < 1 Vertical shrink by a factor of c

y = f{cx) c > 1 Horizontal shrink by a factor of c

y = J\cx) 0 < c < 1 Horizontal stretch by a factor of c

Vertical Shifts of Sinusoidal Graphs

• Equations are of the form y = A s in(Bx - C) -I- D or y = .4 cos(Bx - C) + D

m D is the midline, as it is in the middle of the sinusoidal wave

Example: Graphing y = A s in( f ix - C) + D and y = A cos(BAr - C) + D

Find the amplitude, period, phase shift (if any), and midline of the following function. Then graph period of the function y = 2 cos x - I-1 .

M i d l i n e " 1

(5)

" 0 IT

1

TT

2

\ 0

- \

0 \

Z 1 X N

0

2.

Z

'IT - 1

0

2TT 1

Example: Graphing y = A sin(Bx -C) + Dandy = A cosiBx - C) + D Find the amplitude, period, phase shift (if any), and midline of the following function. Then graph one period of the function y = sin(2x + TT) + 1.

@

z 0 I

I

(S) x - i r

@_ X V X \

0 - T T 0 - f t 2. 0

\ ^ -I \ -tt

0 0 0 0 0

0 2. T T

- \

0

4

1

-1 0

H

0

"5-

2 ! 0 1