1. Graphing Form of Sine and Cosine Functions

2. The length of one cycle of a graph.
Period
The length of one cycle of a graph.

3. Requirements for a Sine/Cosine Graph
x-intercept 2
Arrows (to show that there infinite cycles) 3 1 5 4
At least one Period
(in other words, at least 5 consecutive critical points accurately plotted)

4. The Amplitude and the Effect of “a”
Half of the distance between the maximum and minimum values of the range of a periodic function with a bounded range.
a < 0
a > 1
a = 1
0<a<1
Amplitude = 1 1
0.5 3
The amplitude is the absolute value of a! It is a positive distance.

5. You need at least 5 consecutive critical points.
Example: Sine
Transformation: Flip the parent graph and translate it 3Pi/2 units to the left.
Transformation:
New Equation:
y = 0
Period:
x = -3π/2
You need at least 5 consecutive critical points.

6. You need at least 5 consecutive critical points.
Example: Cosine
Transformation: Translate the parent graph Pi/2 units to the left and 1 unit down.
Transformation:
New Equation:
Period:
y = -1
x = -π/2
You need at least 5 consecutive critical points.

7. Sine v Cosine
Sine
Cosine
(Press the Graph)

8. Example: Sine or Cosine?
Transformation:
Amplitude - 2
Graph -
Translation -
3 units up and …
Period - 2π
Orientation -
New Equation:
Since the Sine and Cosine graphs are periodic and translations of each other, there are infinite equations that represent the same curve. Here are two examples.
y = 3

9. Example: Sine or Cosine?
Transformation:
Amplitude - 2
Graph -
Sine
Translation -
3 units up and 3π/4 to the left
3 units up and …
Period - 2π
Orientation -
Positive
New Equation:
y = 3
x = -3π/4

10. OR

11. Example: Sine or Cosine?
Transformation:
Amplitude - 2
Graph -
Cosine
Translation -
3 units up and π/4 to the left
3 units up and …
Period - 2π
Orientation -
Positive
New Equation:
y = 3
x = -π/4

12. Find the period for each graph and generalize the result.
Changing the Period
Find the period for each graph and generalize the result.
1 cycle in 2π
1/4 cycle in 2π
Period = 2π
Period = 8π
2 cycles in 2π
4 cycles in 2π
Period = π
Period = 0.5π

13. Determining the Period of Sine/Cosine Graph
If or , the period (the length of one cycle) is determined by:
Ex: What is the period of ?

14. Changing the Period w/o Affecting (h,k)
The key point (h,k) is a point on the sine graph. Also, multiplying x by a constant changes the period. Below are two different ways to write a transformation. In order for the equation to be useful, it must directly change the graph in a specific manner. Which equation changes the period and contains the point (-3,4)? or

15. Graphing Form for Sine k h

16. Graphing Form for Cosinek h

17. Notation: Trigonometric Functions
Correct way for the calculator!
is equivalent to

18. You need at least 5 consecutive critical points.
Example: Sine
Transformation: Change the amplitude to 0.5 and the period to π. Then translate it π/2 units to the right and 1 unit down.
Transformation:
Not in Graphing form
New Equation:
Period:
y = -1
x = π/2
You need at least 5 consecutive critical points.

19. You need at least 5 consecutive critical points.
Example: Cosine
Transformation: Change the period to 4π and translate the parent graph 1 unit up.
Transformation:
New Equation:
y = 1
Period:
x = 0
You need at least 5 consecutive critical points.

20. Example: Sine or Cosine?
Transformation:
Amplitude -
1.5
Graph -
Translation -
2 units down and …
Period -
π/2
Orientation -
New Equation:
Since the Sine and Cosine graphs are periodic and translations of each other, there are infinite equations that represent the same curve. Here are two examples.
y = -2
Period:

21. Example: Sine or Cosine?
Transformation:
Amplitude -
1.5
Graph -
Cosine
Translation -
2 units down and …
2 units down
Period -
π/2
Orientation -
Positive
x = 0
New Equation:
y = -2
Period:

22. OR

23. Example: Sine or Cosine?
Transformation:
Amplitude -
1.5
Graph -
Sine
Translation -
2 units down and …
2 units down and 5π/8 to the right
Period -
π/2
Orientation -
Negative
x = 5π/8
New Equation:
y = -2
Period: