1. Graphs Transformation of Sine and Cosine
Consider the form
y = A sin (Bx – C) + D
and
y = A cos (Bx – C) + D
where A, B, C, and D are all constants. These constants have the effect of translating, reflecting, stretching, and shrinking the basic graphs.

2. Vertical Shift
Let’s observe the effect of the constant D.

3. Vertical Shift

4. The Constant D The constant D in y = A sin (Bx – C) + D and
y = A cos (Bx – C) + D
translates the graphs up D units if D > 0 or down |D| units if D < 0.

5. The Amplitude The amplitude of the graphs of
Let’s observe the effect of the constant A.

6. The Amplitude

7. The Constant |A| is the amplitude of the graph
If |A| > 1, then there will be a vertical stretching.
If |A| < 1, then there will be a vertical shrinking.
If A < 0, the graph is also reflected across the x-axis.

8. The Constant B
Let’s observe the effect of the constant B.

9. The Constant B

10. The Constant B

11. The Constant B

12. Copyright © 2009 Pearson Education, Inc.
The Constant B
If |B| < 1, then there will be a horizontal stretching.
If |B| > 1, then there will be a horizontal shrinking.
If B < 0, the graph is also reflected across the y-axis.
Copyright © 2009 Pearson Education, Inc.

13. Period The period of the graphs of y = A sin (Bx – C) + D and
y = A cos (Bx – C) + D is

14. Period The period of the graphs of y = A csc (Bx – C) + D and
y = A sec (Bx – C) + D is

15. Period The period of the graphs of y = A tan (Bx – C) + D and
y = A cot (Bx – C) + D is

16. The Constant C
Let’s observe the effect of the constant C.

17. The Constant C

18. The Constant C

19. The Constant C

20. The Constant C If B = 1, then
if |C| < 0, then there will be a horizontal translation of |C| units to the right, and
if |C| > 0, then there will be a horizontal translation of |C| units to the left.

21. Combined Transformations
It is helpful to rewrite
y = A sin (Bx – C) + D
and
y = A cos (Bx – C) + D as
and

22. Phase Shift
The phase shift of the graphs
and
is the quantity

23. Phase Shift
If C/B > 0, the graph is translated to the right |C/B| units.
If C/B < 0, the graph is translated to the right |C/B| units.

24. Transformations of Sine and Cosine Functions
To graph
and
follow the steps listed below in the order in which they are listed.

25. Transformations of Sine and Cosine Functions
1. Stretch or shrink the graph horizontally according to B.
|B| < Stretch horizontally
|B| > Shrink horizontally
B < Reflect across the y-axis
The period is

26. Transformations of Sine and Cosine Functions
2. Stretch or shrink the graph vertically according to A.
|A| < Shrink vertically
|A| > Stretch vertically
A < Reflect across the x-axis
The amplitude is A.

27. Transformations of Sine and Cosine Functions
3. Translate the graph horizontally according to C/B.
The phase shift is

28. Transformations of Sine and Cosine Functions
4. Translate the graph vertically according to D.
D < |D| units down
D > D units up

29. Example Sketch the graph of
Find the amplitude, the period, and the phase shift.
Solution:

30. Example Solution continued
To create the final graph, we begin with the basic sine curve, y = sin x.
Then we sketch graphs of each of the following equations in sequence.

31. Example
Solution continued

32. Example
Solution continued

33. Example
Solution continued

34. Example
Solution continued

35. Example
Solution continued