4.5 – g raphs of s ine and c osine f unctions. i n this section, you will learn to : sketch the...
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4.5 – GRAPHS OF SINE AND COSINE FUNCTIONS
IN THIS SECTION, YOU WILL LEARN TO:
Sketch the graphs of sine and cosine functions
Use the period and amplitude to sketch the sine and cosine graphs
Sketch the translation of the sine and cosine graphs
THE GRAPH OF THE SINE CURVE:
THE GRAPH OF THE SINE CURVE:
a) The sine curve is symmetric with respect
to the origin. , ,x y x y
b) The period of sine is 2 , the domain is ,
and the range is 1,1 .
THE GRAPH OF THE COSINE CURVE:
a) The cosine curve is symmetric with respect
to the . , ,y axis x y x y
b) The period of cosine is 2 , the domain is
, and the range is 1,1 .
a) Definition of the Amplitude of the Sine and Cosine Functions:
The amplitude of sin and
cos is half the distance
between the maximum and minimum values
of the function and is given by .
y a bx c d
y a bx c d
a
Graphical effects of constants , , and in
sin and cos
functions :
a b c d
y a bx c d y a bx c d
AMPLITUDE:Amplitude of the sine and cosine function
sin and cos :y x y x 1
1 1 12
1 1a
AMPLITUDE:Amplitude of the sine and cosine function
sin and cos :y x y x 1
1 1 12
1 1a
AMPLITUDE OF SINE & COSINE FUNCTIONS:
a) Amplitude of 2sin :y x 2 2
1b) Amplitude of sin :
2y x
1 1
2 2
REFLECTION ACROSS THE X-AXIS:
If a is positive, then there is no reflection about the x-axis.
If a is negative, then there is a reflection about the x-axis.
Blue : cosy x
Green : cosy x
90 180 270 360-90-180-270-360
1
2
-1
-2
x
y
REFLECTION ACROSS THE Y-AXIS:
If b is positive, then there is no reflection about the y-axis.
If b is negative, then there is a reflection about the y-axis.
Blue: siny x
Green: siny x
90 180 270 360-90-180-270-360
1
2
-1
-2
x
y
PERIOD OF SINE AND COSINE FUNCTIONS:
The period of sine and cosine functions
sin and cos
2is .
y a bx c d y a bx c d
b
PERIOD OF SINE AND COSINE FUNCTIONS:
2a) Period of sin : 2
1y x
1) Period of
2sin : 4
122
c y x
) Period 2
of sin 2 :2
y xb
PERIOD OF SINE AND COSINE FUNCTIONS:
2a) Period of cos : 2
1y x
1) Period of
2cos : 4
122
c y x
) Period 2
of cos 2 :2
y xb
PERIOD OF SINE AND COSINE FUNCTIONS:
The period of sine and cosine functions
sin and cos
2is .
y a bx c d y a bx c d
b
a) Period of 4sin 3 1:4
y x
2 2
3b
1 1b) Period of sin 3:
3 4y x
2 2
81/ 4b
LEFT AND RIGHT ENDPOINTS:
The endpoints of a one cycle interval of
sine and cosine functions can be found by
solving the equation 0 and 2 .bx c bx c
* Find the left and right endpoints for the function :
4sin 3 14
y x
LEFT AND RIGHT ENDPOINTS:
4sin 3 1:4
y x
0 and 2bx c bx c
3 0 and 3 24 4
x x
3and
12 4x x
3 and 3 24 4
x x
LEFT AND RIGHT ENDPOINTS:
This means that the left endpoint where this
sine graph begins is and the right endpoint 12
3where the graph ends is .
4
VERIFY THE ENDPOINTS BY USING THE PERIOD OF THE FUNCTION: You can verify the distance between
the two endpoints with the period of the sine/cosine functions. The distance between the endpoints must be equal to the period of the function.
2Period
3
Distance between endpoints:
3 9 8 2
4 12 12 12 12 3
HORIZONTAL TRANSLATION OR PHASE SHIFT:
The constant determines the horizontal
translation of the graph.
c
a) If is , then the shift is toward
th
posi
e
tive
r .ight
c
b) If is , then the shift is toward
th
negative
lee ft.
c
HORIZONTAL TRANSLATION:
Because is , the horizontal translation 2
is to the left.2
c
HORIZONTAL TRANSLATION:
Because is , the horizontal translation 2
is to the right.2
c
VERTICAL TRANSLATION OR PHASE SHIFT:
The constant determines the vertical
translation of the graph.
d
a) If is , then the shift ispositive upw .ardd
b) If is , then the shift is negative downwa rd.d
VERTICAL TRANSLATION:
Because is 2, the vertical translation
is 2 units upward.
d
VERTICAL TRANSLATION:
Because is 3 , the vertical translation
is 3 units downward.
d
x
yy = sinx
y = sinx -3
siny x
sin 3y x
FIND THE AMPLITUDE, PERIOD, REFLECTIONS, HORIZONTAL SHIFT, VERTICAL SHIFT, ENDPOINTS, DOMAIN, RANGE AND SKETCH THE GRAPH.
Example #1: 3cos 22
y x
a) Amplitude:
b) Period:
c) Vertical Translation:
d) Reflection:
Example #1: 3cos 22
y x
3 3a
2 2
2b
none
none
e) Endpoints:
Verify distance with the period:
f) Horizontal Translation:
Example #1: 3cos 22
y x
2 0 and 2 22 2
x x
3
4 4
32 2
2 2x x
3
4 4x x
to the left4
Graph of 3cos 22
y x
Graph of 3cos 22
y x
Domain : , Range : 3,3
FIND THE AMPLITUDE, PERIOD, REFLECTIONS, HORIZONTAL SHIFT, VERTICAL SHIFT, ENDPOINTS, DOMAIN, RANGE AND SKETCH THE GRAPH.
Example #2: 2sin 2 34
y x
a) Amplitude:
b) Period:
c) Vertical Translation:
d) Reflection:
Example #2: 2sin 2 34
y x
2 2a
2 2
2b
3 units downward
about the -axisx
e) Endpoints:
Verify distance with the period:
f) Horizontal Translation:
Example #2: 2sin 2 34
y x
2 0 and 2 24 4
x x
9
8 8
92 2
4 4x x
9
8 8x x
to the right8
Graph of 2sin 2 34
y x
Graph of 2sin 2 34
y x
Domain : , Range : 5, 1
FIND THE AMPLITUDE, PERIOD, REFLECTIONS, HORIZONTAL SHIFT, VERTICAL SHIFT, ENDPOINTS, DOMAIN, RANGE AND SKETCH THE GRAPH.
Example #3: cosy x
a) Amplitude:
b) Period:
c) Vertical Translation:
d) Reflection:
Example #3: cosy x
1 1a
2 22
b
none
none
e) Endpoints:
Verify distance with the period:
f) Horizontal Translation:
Example #3: cosy x
0 and 2x x
2 0 2
0 2x x
none
Graph of cosy x
Graph of cosy x
x
y
21
1
1
0.5 1.5
Domain : , Range : 1,1
1Example #4: sin 1
2y x
FIND THE AMPLITUDE, PERIOD, REFLECTIONS, HORIZONTAL SHIFT, VERTICAL SHIFT, ENDPOINTS, DOMAIN, RANGE AND SKETCH THE GRAPH.
a) Amplitude:
b) Period:
c) Vertical Translation:
d) Reflection:
1Example #4: sin 1
2y x
1 1
2 2a
2 22
1b
1 units upward
about the -axisy
e) Endpoints:
Verify distance with the period:
f) Horizontal Translation:
1Example #4: sin 1
2y x
0 and 2x x
2
x x
to the left
1Graph of sin 1
2y x
1Graph of sin 1
2y x
Domain : , Range : 0.5,1.5
90 180-90-180
0.5
1
1.5
2
2.5
3
-0.5
-1
x
y