1 / 39

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.

fay
Télécharger la présentation

4.5 Sinusoidal Graphs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 =

More Related