Hprec7.3
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Hprec7.3 Presentation Transcript

  • 1. 7.3: Periodic Graphs and Amplitude © 2008 Roy L. Gover(www.mrgover.com) Learning Goals: •Understand period and amplitude of a Trigonometric function. •Introduce real world applications of periodic graphs.
  • 2. Important Idea Period and amplitude are important in describing the nature of many real world phenomenon such as light waves, sound waves and radio waves.
  • 3. Larger Period, Smaller Frequency Important Idea Radio Visible X-Ray
  • 4. Smaller Period Greater Frequency Important Idea Radio Visible X-Ray
  • 5. Important Idea Radio Waves
  • 6. Important Idea Medicine
  • 7. Important Idea Ocean Tides
  • 8. Definition A complete waveform of a periodic function with no repeats is a cycle.
  • 9. Important Idea A cycle may be measured in different ways but for a given function, the period will be the same . Period
  • 10. Definition The Period is the time required for the function to repeat. t ( )f t Go to Sketchpad
  • 11. Definition Frequency (in cycles per second) is the reciprocal of the period.
  • 12. Start Movie
  • 13. Try This •What are your observations about the movement of the bridge? •What do you think is the connection between this bridge and today’s lesson?
  • 14. Important Idea The x-axis is now the t-axis (time). The y-axis is now the f (t) axis. t ( )f t
  • 15. Important Idea For the trigonometric functions: ( ) sinf t bt= ( ) cosg t bt= or The constant b changes the period.
  • 16. Definition The trigonometric functions: ( ) sinf t bt= ( ) cosg t bt= have periods (repeat) every 2 b π radians.
  • 17. Example What is the period of: ( ) sinf t t= 0 2π Period= 2 2 1 π π=
  • 18. Example What is the period of: ( ) sin(2 )g t t= 0 π 2π 2 2 π π=Period=
  • 19. Example What is the period of: ( ) sin(.5 )h t t= Period= 2 4 .5 π π= 0 4π 8π
  • 20. Important Idea ( ) sin( )f t x= ( ) sin(2 )g t t= ( ) sin(.5 )h t t= 0 2π 4π As b gets small, period gets large.
  • 21. Try This What is the period (how often does the function repeat)? : ( ) cosp t t= 2 2 1 π π= 0 2π 4π
  • 22. Try This What is the period (how often does the function repeat)? : ( ) cos1.2p t t= 2 1.67 1.2 π π= 0 1.67π
  • 23. Try This Which is the graph of ( ) sin(4 )f t t= 0 red 2 π π
  • 24. Try This What is the equation of the function graphed in blue? ( ) sin(2 )f t t= 0 2 π π
  • 25. Important Idea The trigonometric function: ( ) tanf t bt= has a period (repeats) every b π radians. t ( )f t b π b π
  • 26. Example Determine how often the function repeats (period): ( ) tan(2 )k t t= ( ) tan 3 t j t   =  ÷  
  • 27. Try This Determine how often the function repeats (period): ( ) tan 6 t p t   =  ÷   1 6 b = 6π; period=
  • 28. Try This Using your calculator, graph: 1 tany t= 2 tan 6 t y   =  ÷   Compare and contrast.
  • 29. Definition Amplitude is where is the distance from the middle to the top and a a+ a− is the distance from the middle to the bottom. a a− Geom Sketchpad
  • 30. Try This Compar e and Contrast ( ) 5sinf t t= ( ) 2sing t t=
  • 31. Try This What is the amplitude of ( ) 6tanp t t= Hint: this is a trick question. ?
  • 32. Important Idea ( ) sinf t a bt= ( ) cosg t a bt= The constant a controls the amplitude (vertical stretch). The constant b controls the horizontal stretch. Amplitude is . Period is . a 2 b π
  • 33. Try This Find the amplitude and period for ( ) sinf t t= Amplitude=1; Period = 2π
  • 34. Try This Find the amplitude and period for ( ) 3cos2f t t= Amplitude=3; Period = π
  • 35. Try This Find the amplitude and period for ( ) 3cos2f t t= − Amplitude=3; Period = π
  • 36. Try This Find the amplitude and period for ( ) 3tan(.5 )f t t= − Amplitude: None; Period=2π
  • 37. Example For what value(s) of t between is sin 2 1tπ = 0 1t≤ ≤
  • 38. Example For what value(s) of t between [0, ] is cos2 0t = 2π
  • 39. Example For what value(s) of t between [0, ] is cos3 0t = 2π
  • 40. Try This For what value(s) of t between [0, ] is sin3 0t = π 0, 3 t t π = =
  • 41. Try This For what value(s) of t between [0,1] is sin3 0tπ = 1 2 0, , ,1 3 3 t =
  • 42. Lesson Close Period and amplitude will be important when we study modeling and simple harmonic motion in section 8.4.