1. Graphing Sinusoidal Functions y=sin x

2. Recall from the unit circle that: –Using the special triangles and quadrantal angles, we can complete a table.

3.  y Quadrant I Quadrant 2 y.5.707.866 1  y y.707.5 Table of Values

4.  y Quadrant III Quadrant IV Table of Values  y

5. Parent Function y=sin x

6. Domain Recall that we can rotate around the circle in either direction an infinite number of times. Thus, the domain is (- ,  )

7. Range Recall that –1  sin   1. Thus the range of this function is [-1, 1 ]

8. Period One complete cycle occurs between 0 and 2 . The period is 2 .

9. How many periods are shown?

10. Between 0 and 2 , there is one minimum point at (, -1). Critical Points Between 0 and 2 , there is one maximum point at (, 1). Between 0 and 2 , there are three zeros at (0,0), ( ,0) and (2 ,0).

11. Parent Function Key Points * Notice that the key points of the graph separate the graph into 4 parts.

12. y= a sin b(x-c)+d a = amplitude, the distance from the center to the maximum or minimum. If |a| > 1, vertical stretch If 0<|a|<1, vertical shrink If a is negative, the graph reflects about the x-axis.

13. y = 3 sin x What changed?

14. y= sin x

15. y= -2 sin x

16. y=a sin b (x-c)+d b= horizontal stretch or shrink Period = If |b| > 1, horizontal shrink If 0 < |b|< 1, horizontal stretch If b < 0, the graph reflects about the y-axis.

17. Tick Marks Recall that the key points separate the graph into 4 parts. If we alter the period, we need to alter the x-scale. This can be done by dividing the new period by 4.

18. y = sin 2x What is the period of this function? If we wanted to graph only one period, what would the tick marks need to be?

19. y = sin x

20. y = a sin b(x- c ) + d c = horizontal shift If c is negative, the graph shifts left c units. (x+c)=(x-(-c)) If c is positive, the graph shifts right c units. (x-c)=(x-(+c)) In trigonometric functions, these horizontal shifts are called phase shifts.

21. y = sin(x- ) What changed? Which way did the graph shift? By how many units?

22. y = sin (x + )

23. y=a sin b(x-c) + d d= vertical shift If d is positive, graph shifts up d units. If d is negative, graph shifts down d units. In trigonometric functions, these vertical shifts are called the vertical displacement.

24. y = sin x +2 What changed? Which way did the graph shift? By how many units?

25. y = sin x - 3

26. y = 3 sin(2(x-  )) - 2 Can you list all the transformations?

27. y=-2sin(2x-  ) +1