1. Lesson 2.9

3. ©Carolyn C. Wheater, 2000 3  Sine  The most fundamental sine wave, y=sin(x), has the graph shown.  It fluctuates from 0 to a max of 1, min of –1, with a period of 2 .

4. ©Carolyn C. Wheater, 2000 4  The graph of is determined by four numbers, a, b, h, and k.  The amplitude, a, tells the height of each peak and the depth of each trough.  The frequency, b, tells the number of full wave patterns that are completed in a space of 2 .  The period of the function is  The two remaining numbers, h and k, tell the translation of the wave from the origin.

5. ©Carolyn C. Wheater, 2000 5  Which of the following equations best describes the graph shown?  (A) y = 3sin(2x) - 1  (B) y = 2sin(4x)  (C) y = 2sin(2x) - 1  (D) y = 4sin(2x) - 1  (E) y = 3sin(4x)           

6. ©Carolyn C. Wheater, 2000 6  Find the baseline between the high and low points.  Graph is translated -1 vertically.  Find height of each peak.  Amplitude is 3  Count number of waves in 2   Frequency is 2            y = 3sin(2x) - 1

7. ©Carolyn C. Wheater, 2000 7  Cosine  The graph of y=cos(x) resembles the graph of y=sin(x) but is shifted, or translated, units to the left.  It also fluctuates from A max of 1 to a min of –1, with a period of 2 .

8. ©Carolyn C. Wheater, 2000 8 AmplitudeaHeight of each peak FrequencybNumber of full wave patterns Period 2  /bSpace required to complete wave Translation h, kHorizontal and vertical shift  The values of a, b, h, and k change the shape and location of the wave as for the sine.

9. ©Carolyn C. Wheater, 2000 9  Which of the following equations best describes the graph?  (A) y = 3cos(5x) + 4  (B) y = 3cos(4x) + 5  (C) y = 4cos(3x) + 5  (D) y = 5cos(3x) +4  (E) y = 5sin(4x) +3     

10. ©Carolyn C. Wheater, 2000 10  Find the baseline  Vertical translation + 4  Find the height of peak  Amplitude = 5  Number of waves in 2   Frequency =3      y = 5cos(3x) + 4

11. ©Carolyn C. Wheater, 2000 11  Tangent  The tangent function has a discontinuous graph, repeating in a period of . The graph is discontinuous at  Cotangent  Like the tangent, cotangent is discontinuous. Discontinuities of the cotangent are units left of those for tangent.

12. ©Carolyn C. Wheater, 2000 12  Secant and Cosecant  The secant and cosecant functions are the reciprocals of the cosine and sine functions respectively.  Imagine each graph is balancing on the peaks and troughs of its reciprocal function.