1. 4.5 – G RAPHS OF S INE AND C OSINE F UNCTIONS

2. I N THIS SECTION, YOU WILL LEARN TO :  Sketch the graphs of sine and cosine functions  Use the period and amplitude to sketch the sine and cosine graphs  Sketch the translation of the sine and cosine graphs

3. T HE GRAPH OF THE SINE CURVE :

5. T HE GRAPH OF THE COSINE CURVE :

6. a) Definition of the Amplitude of the Sine and Cosine Functions:

7. A MPLITUDE :

9. A MPLITUDE OF S INE & C OSINE F UNCTIONS :

10. R EFLECTION ACROSS THE X - AXIS :  If a is positive, then there is no reflection about the x-axis.  If a is negative, then there is a reflection about the x-axis.

11. R EFLECTION ACROSS THE Y - AXIS :  If b is positive, then there is no reflection about the y-axis.  If b is negative, then there is a reflection about the y-axis.

12. P ERIOD OF S INE AND C OSINE F UNCTIONS :

16. L EFT AND R IGHT E NDPOINTS :

19. V ERIFY THE ENDPOINTS BY USING THE PERIOD OF THE FUNCTION :  You can verify the distance between the two endpoints with the period of the sine/cosine functions. The distance between the endpoints must be equal to the period of the function.

20. H ORIZONTAL T RANSLATION OR P HASE S HIFT :

21. H ORIZONTAL T RANSLATION :

23. V ERTICAL T RANSLATION OR P HASE S HIFT :

24. V ERTICAL T RANSLATION :

26. FIND THE AMPLITUDE, PERIOD, REFLECTIONS, HORIZONTAL SHIFT, VERTICAL SHIFT, ENDPOINTS, DOMAIN, RANGE AND SKETCH THE GRAPH.

27. a) Amplitude: b) Period: c) Vertical Translation: d) Reflection:

28. e) Endpoints: Verify distance with the period: f) Horizontal Translation:

31. FIND THE AMPLITUDE, PERIOD, REFLECTIONS, HORIZONTAL SHIFT, VERTICAL SHIFT, ENDPOINTS, DOMAIN, RANGE AND SKETCH THE GRAPH.

32. a) Amplitude: b) Period: c) Vertical Translation: d) Reflection:

33. e) Endpoints: Verify distance with the period: f) Horizontal Translation:

36. FIND THE AMPLITUDE, PERIOD, REFLECTIONS, HORIZONTAL SHIFT, VERTICAL SHIFT, ENDPOINTS, DOMAIN, RANGE AND SKETCH THE GRAPH.

37. a) Amplitude: b) Period: c) Vertical Translation: d) Reflection:

38. e) Endpoints: Verify distance with the period: f) Horizontal Translation:

41. FIND THE AMPLITUDE, PERIOD, REFLECTIONS, HORIZONTAL SHIFT, VERTICAL SHIFT, ENDPOINTS, DOMAIN, RANGE AND SKETCH THE GRAPH.

42. a) Amplitude: b) Period: c) Vertical Translation: d) Reflection:

43. e) Endpoints: Verify distance with the period: f) Horizontal Translation: