1. Graphs of Sine and Cosine

2. Label your graph paper... 2 1
90º
-270º
270º
360º
-90º -1
180º -2

3. In radians... 2 1  2 -1 -2

4. Characteristics of the Sine Function
1. The domain is the set of all real numbers.
2. The range consists of all real numbers from -1 to 1, inclusive.
3. The sine function is an odd function (symmetric with respect to the origin). 4. 5.

5. Characteristics of the Sine Function6.

6. Graph y = sin  sin  0° 0.707 45° 90° 1 135° 0.707 180° 225°2 0°
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°

7. y = sin x 2 1 -90º 90º -270º 270º -1 -2 Maximum Maximum intercept
Minimum
Minimum -2

8. 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°

9. 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?

11. Characteristics of the Cosine Function
1. The domain is the set of all real numbers.
2. The range consists of all real numbers from -1 to 1, inclusive.
3. The cosine function is an even function (symmetric with respect to the y-axis). 4. 5.

12. Characteristics of the Cosine Function6.

13. 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

14. y = cos x -2 -  2 2 1 -1 -2 Maximum Maximum intercept intercept
Minimum
Minimum -2

15. 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

16. 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

17. Try it on your calculator!
y = sin x
y = cos x
Try it on your calculator! 2 1 -1 -2

18. 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

19. 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

20. is the horizontal translation
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?
y = sin (bx - c)
to the right
is the horizontal translation
y = sin (bx – (- c))
to the left

21. ‘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.

22. 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

23. Summary: y = d + a sin (bx - c) y = d + a cos (bx - c)
a is the amplitude
period = or
is the horizontal translation
d is the vertical translation

24. Analyze the graph of amplitude = period = horizontal translation:
vertical translation:
none

25. Analyze the graph of 3 (to the left) amplitude = period =
horizontal translation:
vertical translation:
none

26. Analyze the graph of 3 amplitude = period = horizontal translation:
none
vertical translation:
Up 2

27. Graph and Analyze y = -2 + 3 cos (2x - 90°) 3 x y high 1 amplitude =
45° 3
amplitude =
90° -2
mid
period =
= 180°
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
2) add the period to find out where to finish
table goes in increments of 45
4) plot points and graph
= 225

29. You must know how to analyze the equation before you can graph it.
The most important thing to remember about graphing is determining the starting point and the stopping point on the t-table.

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