4.5 graphs of sine and cosine functions graphing sine
TRANSCRIPT
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