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Graphs of Sine and Cosine
Graph y = sin
90º-90º 270º-270º
1
-1
2
-2
sin
0°
45°
90°
135°
180°
225°
270°
315°
360°
0
0
0
1
-1
0.707
0.707
-0.707
-0.707
180º 360º
2
2
2
2
2
2
2
2
y = sin x
90º-90º 270º-270º
1
-1
2
-2 Period: 360°
Period: the least amount of space (degrees or radians) the function takes to complete one cycle.
y = sin x
90º-90º 270º-270º
1
-1
2
-2
Amplitude = 1
Amplitude: half the distance between the maximum and minimum
In other words, how high does it go from its axis?
Graph y = cos
1
-1
2
-2
cos 0 1
1
-1
0
0
0.707
-0.707
-0.707
0.707
2
2
2
2
2
2
2
2
2
3
2
22
3
2
4
3
4
5
4
7
4
2
2
3
2
y = cos x
1
-1
2
-2
2--2
Period: 2
Period: the least amount of space (degrees or radians) the function takes to complete one cycle.
y = cos x
1
-1
2
-2
2--2
Amplitude = 1
How high does it go from its axis?
y= sin and y = cos are the mother functions.
Changing the equations changes the appearance of the graphs
We are going to talk about the AMPLITUDE, TRANSLATIONS, and PERIOD of relative equations
y = d + a sin (bx - c)
y = d + a cos (bx - c)a is the amplitude
set (bx-c)=0 to find the horizontal translation—THIS WILL BE YOUR STARTING POINT
d is the vertical translation (sinusoidal asymptote)
period = 360
b
Summary:
or 2b
Increments= Period 4
Mother Function relative function change?
y1 = sin x y2 = - sin x reflection over x-axis
y1 = sin x
y1 = sin x
y2 = 4 sin x
y2 = sin x1
2
amplitude = 4
amplitude = 1
2
generalization?
y = a sin x amplitude = a
Mother Function relative function change?
y1 = sin x y2 = sin (x - 45)
y2 = sin (x + 60)
horizontal translation, 45 degrees to the right.
horizontal translation, 60 degrees to the left.
y1 = sin x
generalization?
y2 = sin (2x + 60)y1 = sin xhorizontal translation, 30 degrees to the left.
y2 = sin (3x - 270)y1 = sin xhorizontal translation, 90 degrees to the right.
Set the parenthesis equal to zero to find the left right shift
Mother Function relative function change?
y1 = cos x y2 = 2 + cos x vertical translation, 2 units up.
y1 = cos x y2 = -3 + cos xvertical translation, 3 units down.
generalization?
y = d + cos x ‘d’ is the vertical translation
when d is positive, the graph moves up.
when d is negative, the graph moves down.
Mother Function relative function change?
y1 = sin x
y1 = sin x
y2 = sin 2x
y2 = sin x1
2
Period = 180or
Period = 7204or
generalization?
y = sin bx Period = 360
b
or 2b
THE PERIOD IS HOW LONG IT
TAKES THE GRAPH TO GO THROUGH ALL VALUES
Ex. #6
Analyze the graph of y x 2 3 2cos
amplitude =
vertical translation:
horizontal translation:
period = 360
2
180
Up 2
3
none
Ex. #4
Analyze the graph of y FHG
IKJ
1
2 3sin
amplitude =
vertical translation:
horizontal translation:
1
2
period = 2
1
2
3
to the right
none
Ex. #5
Analyze the graph of
amplitude =
vertical translation:
horizontal translation:
period = 2
2
1
none
3
2 to the left
)4 θ2cos(3 y
2 4 0
y = -2 + 3 cos (2x - 90°)
amplitude = 360
2
period =
vertical translation:
horizontal translation:
x y
45°
225°
90°
135°
180°
180 4 = 45table goes in increments of 45
1
-2
-5-2
1
1) horiz. tells you where to start
2) add the period to find out where to finish
3) divide period by 4 to find increments
4) plot points and graph45 + 180 = 225
Ex #6b Graph
45to the right
3
= 180°
down 2
high
low
high
mid
mid
y = 1 + 3 sin (2 + )
amplitude =period =
vertical translation:
horizontal translation:
x y
0
1
4
1
-2
1
1) horiz. tells you where to start
2) add the period to find out where to finish
3) divide period by 4 to find increments
4) plot points and graph
Ex #6c Graph
2
2
2
=
2
2
4
4
3
up 1
2 2
4table goes in increments of
4
mid
mid
mid
high
lowto the left
Write an equation of the cosine function whose
amplitude = , period = 270,
vertical translation: down 3, and horizontal trans: right 60,
1
2
amplitude: a = 1
2
y = d + a cos (bx - c)
horiz.:
period: 360b
= 270
b = 4
3y x F
HGIKJ3
1
2
4
380cos
a = 1
2
vert.: d = -3
Ex. #7b
60bx c c43
60 c 80
-180 -150 -120 -90 -60 -30 30 60 90 120 150 180
-4
-3
-2
-1
1
2
3
4
Write the equation of the sine graph
-320 -240 -160 -80 80 160 240 320
-4
-3
-2
-1
1
2
3
4
Write the equation of the cosine graph
