1. Graphs of Sine and Cosine

2. Graph y = sin  sin  0° 0.707 45° 90° 1 135° 0.707 180° 225°0° 2
0.707
45° 1
90° 1
-270º
-90º
90º
180º
270º
360º
135°
0.707 -1
180° -2
225°
-0.707 -1
270°
-0.707
315°
360°

3. y = sin x
Period: the least amount of space (degrees or radians) the function takes to complete one cycle. 2 1
-90º
90º
-270º
270º -1 -2
Period: 360°

4. In other words, how high does it go from its axis?
y = sin x
Amplitude: half the distance between the maximum and minimum 2
Amplitude = 1 1
-90º
90º
-270º
270º -1 -2
In other words, how high does it go from its axis?

5. Graph y = cos cos   1 0.707 -0.707 -1 -0.707 0.707 1 2 1 -1 -21 2
0.707 1
-0.707  2 -1  -1
-0.707 -2
0.707 2 1

6. y = cos x -2 -  2 Period: 2
Period: the least amount of space (degrees or radians) the function takes to complete one cycle. 2 1
-2 -  2 -1 -2
Period: 2

7. How high does it go from its axis?
y = cos x
How high does it go from its axis? 2
Amplitude = 1 1
-2 -  2 -1 -2

8. 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

9. Summary: y = d + a sin (bx - c) y = d + a cos (bx - c)
a is the amplitude
period = or
set (bx-c)=0 to find the horizontal translation—THIS WILL BE YOUR STARTING POINT
d is the vertical translation (sinusoidal asymptote)
Increments= Period 4

10. Mother Function
relative function
change?
y1 = sin x
reflection over x-axis
y2 = - sin x
y1 = sin x
y2 = 4 sin x
amplitude = 4
amplitude =
y2 = sin x
y1 = sin x
generalization?
y = a sin x
amplitude = a

11. Set the parenthesis equal to zero to find the left right shift
Mother Function
relative function
change?
y1 = sin x
y2 = sin (x - 45)
horizontal translation, 45 degrees to the right.
horizontal translation, 60 degrees to the left.
y1 = sin x
y2 = sin (x + 60)
horizontal translation, 30 degrees to the left.
y2 = sin (2x + 60)
y1 = sin x
horizontal translation, 90 degrees to the right.
y1 = sin x
y2 = sin (3x - 270)
generalization?
Set the parenthesis equal to zero to find the left right shift

12. ‘d’ is the vertical translation
Mother Function
relative function
change?
y1 = cos x
y2 = 2 + cos x
vertical translation, units up.
vertical translation, units down.
y1 = cos x
y2 = -3 + cos x
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.

13. Mother Function
relative function
change?
y1 = sin x
y2 = sin 2x
Period = 180 or
Period = 720 or
y1 = sin x
y2 = sin x
generalization?
Period =
y = sin bx or
THE PERIOD IS HOW LONG IT
TAKES THE GRAPH TO GO THROUGH ALL VALUES

14. Ex. #6 Analyze the graph of 3 amplitude = period =
horizontal translation:
none
vertical translation:
Up 2

15. Ex. #4 Analyze the graph of amplitude = period =
horizontal translation:
vertical translation:
none

16. Ex. #5 Analyze the graph of 3 to the left amplitude = period =
horizontal translation:
vertical translation:
none

17. y = -2 + 3 cos (2x - 90°) Ex #6b Graph 3 x y high 1 amplitude = -2 mid
45° 3
amplitude =
90° -2
mid
= 180°
period =
135°
low -5
horizontal translation:
180° -2
mid
to the right
225° 1
high
vertical translation:
down 2
1) horiz. tells you where to start
3) divide period by 4 to find increments
= 45
table goes in increments of 45
2) add the period to find out where to finish
4) plot points and graph
= 225

19. y = 1 + 3 sin (2 + ) =  Ex #6c Graph 3 x y mid 1 amplitude = high 4
period = 1
mid
horizontal translation:
to the left -2
low
vertical translation:
up 1
mid 1
On Calculator, go to table setup and change independent to ask.
3) divide period by 4 to find increments
1) horiz. tells you where to start
table goes in increments of
2) add the period to find out where to finish
4) plot points and graph

21. Ex. #7b y = d + a cos (bx - c) horiz.: a = vert.: d = -3 = 270 b =
Write an equation of the cosine function whose
amplitude = , period = 270,
vertical translation: down 3, and horizontal trans: right 60,
y = d + a cos (bx - c)
amplitude: a =
horiz.:
a =
period:
= 270
vert.: d = -3
b =

22. Write the equation of the sine graph

23. Write the equation of the cosine graph