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4.5 Sinusoidal Graphs

4.5 Sinusoidal Graphs. Sketching and Writing Equations. Graphs of sine and cosine functions. Sinusoidal Graphs. Periodic Function. Functions whose graphs have a repeating pattern. Sinusoidal Graphs. Period. The horizontal length of each cycle in a periodic graph. Midline.

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4.5 Sinusoidal Graphs

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  1. 4.5Sinusoidal Graphs Sketching and Writing Equations

  2. Graphs of sine and cosine functions. Sinusoidal Graphs Periodic Function Functions whose graphs have a repeating pattern. Sinusoidal Graphs Period The horizontal length of each cycle in a periodic graph. Midline The horizontal line halfway between a sinusoid’s max/min values.

  3. The Graph of y = sin x y x

  4. Characteristics of the Sine Function • The domain is the set of all real numbers. • The range consists of all real numbers from -1 to 1, inclusive. • The sine function is an odd function (symmetric with respect to the origin). • The sine function has a period of 2π • Midline is the x-axis

  5. 3 2 1 -1 -2 -3 Period y = sinx pattern starts pattern starts pattern starts 1 period 1 period Period= 2

  6. The Graph of y = cos x y x

  7. Characteristics of the Cosine Function • The domain is the set of all real numbers. • The range consists of all real numbers from -1 to 1, inclusive. • The cosine function is an even function (symmetric with respect to the y-axis). • The cosine function has a period of 2π • Midline is the x-axis

  8. Period y = cosx 3 2 1 -1 -2 -3 pattern starts pattern starts pattern starts 1 period 1 period Period = 2

  9. Period 3 2 1 -1 -2 -3 pattern starts pattern starts 1 period Period= 

  10. 3 2 Period 3 2 1 -1 -2 -3 pattern starts pattern starts 1 period Period =

  11. Amplitude Amplitude Half the distance between the maximum and minimum of a sinusoidal graph.

  12. Amplitude 3 2 1 -1 -2 -3 max. Amplitude min. Amplitude = 1

  13. Amplitude max. 3 2 1 -1 -2 -3 Amplitude Amplitude = 2 min.

  14. Amplitude max. 3 2 1 -1 -2 -3 Amplitude min. Amplitude = 1

  15. Amplitude Amplitude = 2 3 2 1 -1 -2 -3 max. 3 Amplitude min.

  16. Amplitude = Period = Sinusoidal Graphs General Form of Equations

  17. Sinusoidal Graphs y = -4cos5x y = 3sin4x Amplitude = Amplitude = Period = Period =

  18. Writing Equations Answer these questions first: • Is it a “sin” or “cos” graph? • What is the amplitude? • What is the period? • Is the graph reflected? • Is it shifted up/down? • Is it shifted left/right?

  19. Writing Equations b = a = 4 Write an equation: cosine graph amplitude = 4 Period =  b = = 2

  20. Writing Equations b = a = 3 Write an equation: sine graph amplitude = 3 Period = 4 b =

  21. Writing Equations b = Write an equation: sine graph amplitude = 2 Period = 3 Reflected over x-axis a = ±2 b =

  22. Write an equation: cosine graph amplitude = ½ Period = Reflected over x-axis Writing Equations b = Invert and multiply! a = b =

  23. Writing Equations Write an Equation Sine graph Amplitude = 5 Period = Shifted up 2 a = b = Changing the midline causes vertical shifts

  24. Sinusoidal Graphs “sin” a =1 b =  Period = 3 2 1 -1 -2 -3 Amplitude = 1 y = sin2x

  25. Sinusoidal Graphs “sin” 2 Period = a =3 b = Amplitude = 3 3 2 1 -1 -2 -3 y = 3sinx

  26. Sinusoidal Graphs “cos” a =1 Period = b = 1 3 2 1 -1 -2 -3 Amplitude =

  27. Sinusoidal Graphs “cos” -2 a = b = Period = 3 3 2 1 -1 -2 -3 Amplitude = 2 Reflected over x-axis!

  28. Sinusoidal Graphs “cos” b = Period = 6 a = -3 Amplitude = 3 3 2 1 -1 -2 -3 Reflected over x-axis!

  29. Graphing Sinusoidal Functions 3 2 1 -1 -2 -3 y = 3sin4x

  30. Graphing Sinusoidal Functions 3 2 1 -1 -2 -3 y = 2sinx

  31. Graphing Sinusoidal Functions 3 2 1 -1 -2 -3 y = -3sin2x

  32. Graphing Sinusoidal Functions 3 2 1 -1 -2 -3

  33. Graphing Sinusoidal Functions 3 2 1 -1 -2 -3

  34. Graphing Sinusoidal Functions 3 2 1 -1 -2 -3

  35. Phase Shift • Horizontal translations of sine and cosine graphs. Right Left Phase Shift Phase Shift

  36. right Phase Shift Amp = 3 T = P.S. =

  37. left Phase Shift Amp = 2 T = P.S. =

  38. Phase Shift “Cosine” Amplitude = 5 Period =  Phase shift =

  39. Phase Shift “sine” Amplitude = 3 Period = 4 Phase shift =

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