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Section 4.5

Section 4.5. Graphs of Sine & Cosine Functions MA.PC.4.5 2000 Define and graph trigonometric functions ( i.e.,sine , cosine, tangent, cosecant, secant, cotangent). MA.PC.4.6 2000 Find domain, range, intercepts, periods, amplitudes, and asymptotes of trigonometric functions.

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Section 4.5

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  1. Section 4.5 • Graphs of Sine & Cosine Functions • MA.PC.4.5 2000 Define and graph trigonometric functions (i.e.,sine, cosine, tangent, cosecant, secant, cotangent). • MA.PC.4.6 2000 Find domain, range, intercepts, periods, amplitudes, and asymptotes of trigonometric functions. • MA.PC.4.7 2000 Draw and analyze graphs of translations of trigonometric functions, including period, amplitude, and phase shift.

  2. Graph of y = sin x 1 π/2 3π/2 2π π -1

  3. Domain: all real numbers Range: -1<y<1

  4. Periodic Function • sin(t+2)=sin t • cos(t+2)=cos t • The sine and cosine functions are periodic functions and have period 2

  5. Graph of y = cos x 1 π/2 3π/2 2π π -1

  6. Domain: all real numbers Range: -1<y<1

  7. Amplitudes and Periods • Given the graph of y=AsinBx • Amplitude = |A| = distance up or down from center • Period = 2/B = time taken to repeat

  8. Graphing y=½ sin x

  9. Graphing y=-2 sin x

  10. Graphing y=3 sin 2x

  11. Example • Determine the amplitude and period of the following and graph:

  12. Example • Determine the amplitude and period of the following and graph:

  13. The graph of y = A sin(Bx + C) • The graph of y = A sin(Bx + C) is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the cycle is shifted from x=0 to x=-C/B. This is called the phase shift. • Amplitude = |A| • Period = 2π/B

  14. The Effect of Horizontally Shifting the Graph

  15. Example • Determine the amplitude, period and phase shift of

  16. Example • Determine the amplitude, period and phase shift of

  17. Cosine with a Different Amplitude and Period

  18. Sine and Cosine Function • y = A sin (Bx + C) + D or y = Acos (Bx + C) + D • Amplitude = |A| • How high above center it travels • Period = 2πB • How long it takes to complete a cycle (start over) • Phase Shift = - C where B > 0B • If C > 0 the shift is to the left. If C < 0 the shift is to the right. • Vertical Shift = D • If D > 0, the shift is upward. If D < 0, the shift is downward. • The midline (horizontal axis) is y = D.

  19. Homework • Page 533: • 20-56 every four • Do NOT graph. Just identify the values.

  20. Section 4.5 • Graphs of Sine & Cosine Functions • MA.PC.4.5 2000 Define and graph trigonometric functions (i.e.,sine, cosine, tangent, cosecant, secant, cotangent). • MA.PC.4.6 2000 Find domain, range, intercepts, periods, amplitudes, and asymptotes of trigonometric functions. • MA.PC.4.7 2000 Draw and analyze graphs of translations of trigonometric functions, including period, amplitude, and phase shift.

  21. Graphing Sine and Cosine Functions • Step 1Determine the vertical shift and graph the midline with a dashed line. • Step 2Determine the amplitude. Use dashed lines to indicate the maximum and minimum values of the function. • Step 3Determine the period of the function and graph the appropriate sine or cosine curve. • Step 4Determine the phase shift and translate the graph accordingly.

  22. Graphing • State the amplitude, period, phase shift, and vertical shift for y = 4 cos (x/2 + π) – 6. Then graph the function. 1: Vertical shift = midline = D = (graph dotted midline) 2: Amplitude = |A| = (dotted max and min lines) 3: Period = 2π/B = (dotted function) 4: Phase shift = -C/B = (graph translation) 4π

  23. Graphing • State the amplitude, period, phase shift, and vertical shift for y = 2 cos (x/4 + π) – 1. Then graph the function.

  24. Modeling Periodic Behavior

  25. Example • The tides are very high in some areas of Canada. The depth of the water at a boat dock varies from a high tide of 20 feet to a low tide of 6 feet. On a certain day, high tide is at 2am and low tide is at 8am. Use a sine function to model the waters depth. 20 10 2 6 10 14 18 22 26 The number of hours after midnight.

  26. Simple Harmonic Motion

  27. Simple harmonic motion

  28. Frequency = 1 / Period

  29. (a) (b) (c) (d)

  30. (a) (b) (c) (d)

  31. (a) (b) (c) (d)

  32. Homework • Page 533: • 58-64 even, 85

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