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# Hprec7.3

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### Hprec7.3

1. 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. 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. 3. Larger Period, Smaller Frequency Important Idea Radio Visible X-Ray
4. 4. Smaller Period Greater Frequency Important Idea Radio Visible X-Ray
5. 5. Important Idea Radio Waves
6. 6. Important Idea Medicine
7. 7. Important Idea Ocean Tides
8. 8. Definition A complete waveform of a periodic function with no repeats is a cycle.
9. 9. Important Idea A cycle may be measured in different ways but for a given function, the period will be the same . Period
10. 10. Definition The Period is the time required for the function to repeat. t ( )f t Go to Sketchpad
11. 11. Definition Frequency (in cycles per second) is the reciprocal of the period.
12. 12. Start Movie
13. 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. 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. 15. Important Idea For the trigonometric functions: ( ) sinf t bt= ( ) cosg t bt= or The constant b changes the period.
16. 16. Definition The trigonometric functions: ( ) sinf t bt= ( ) cosg t bt= have periods (repeat) every 2 b π radians.
17. 17. Example What is the period of: ( ) sinf t t= 0 2π Period= 2 2 1 π π=
18. 18. Example What is the period of: ( ) sin(2 )g t t= 0 π 2π 2 2 π π=Period=
19. 19. Example What is the period of: ( ) sin(.5 )h t t= Period= 2 4 .5 π π= 0 4π 8π
20. 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. 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. 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. 23. Try This Which is the graph of ( ) sin(4 )f t t= 0 red 2 π π
24. 24. Try This What is the equation of the function graphed in blue? ( ) sin(2 )f t t= 0 2 π π
25. 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. 26. Example Determine how often the function repeats (period): ( ) tan(2 )k t t= ( ) tan 3 t j t   =  ÷  
27. 27. Try This Determine how often the function repeats (period): ( ) tan 6 t p t   =  ÷   1 6 b = 6π; period=
28. 28. Try This Using your calculator, graph: 1 tany t= 2 tan 6 t y   =  ÷   Compare and contrast.
29. 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. 30. Try This Compar e and Contrast ( ) 5sinf t t= ( ) 2sing t t=
31. 31. Try This What is the amplitude of ( ) 6tanp t t= Hint: this is a trick question. ?
32. 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. 33. Try This Find the amplitude and period for ( ) sinf t t= Amplitude=1; Period = 2π
34. 34. Try This Find the amplitude and period for ( ) 3cos2f t t= Amplitude=3; Period = π
35. 35. Try This Find the amplitude and period for ( ) 3cos2f t t= − Amplitude=3; Period = π
36. 36. Try This Find the amplitude and period for ( ) 3tan(.5 )f t t= − Amplitude: None; Period=2π
37. 37. Example For what value(s) of t between is sin 2 1tπ = 0 1t≤ ≤
38. 38. Example For what value(s) of t between [0, ] is cos2 0t = 2π
39. 39. Example For what value(s) of t between [0, ] is cos3 0t = 2π
40. 40. Try This For what value(s) of t between [0, ] is sin3 0t = π 0, 3 t t π = =
41. 41. Try This For what value(s) of t between [0,1] is sin3 0tπ = 1 2 0, , ,1 3 3 t =
42. 42. Lesson Close Period and amplitude will be important when we study modeling and simple harmonic motion in section 8.4.