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5/7/13 Obj : SWBAT apply properties of periodic functions Bell Ringer :

5/7/13 Obj : SWBAT apply properties of periodic functions Bell Ringer : Construct a sinusoid with amplitude 2, period 3 π , point 0,0 HW Requests: Pg 395 #72-75, 79, 80 WS Amplitude, Period, Phase Shift In class: 61-68 Homework : Study for Quiz, Bring your Unit Circle

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5/7/13 Obj : SWBAT apply properties of periodic functions Bell Ringer :

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  1. 5/7/13 Obj: SWBAT apply properties of periodic functions Bell Ringer: Construct a sinusoid with amplitude 2, period 3π, point 0,0 HW Requests: Pg395 #72-75, 79, 80 WS Amplitude, Period, Phase Shift In class: 61-68 Homework: Study for Quiz, Bring your Unit Circle Read Section 5.1 Project Due Wed. 5/8 Each group staple all projects together Education is Power! Dignity without compromise!

  2. To find the phase or horizontal shift of a sinusoid where a, b, c, and d are constants and neither a nor b is 0 Let c = -2 the shift is to the right or left Let c = +2 the shift is to the right or left Engineers and physicist change the nomenclature +c becomes -h What does this change mean? where a, b, c, and d are constants and neither a nor b is 0 Let h = -2 the shift is to the right or left Let h = +2 the shift is to the right or left Find the relationship between h and c Solve for h: (bx+c) = b(x-h) Go to phase shift pdf http://www.analyzemath.com/trigonometry/sine.htm

  3. To find the phase or horizontal shift of a sinusoid where a, b, c, and d are constants and neither a nor b is 0 where a, b, c, and d are constants and neither a nor b is 0 For #2, factor b out of the argument, the resulting h is the phase shift For #1, the phase shift is -c/b Note: the phase shift can be positive or negative Go to phase shift pdf http://www.analyzemath.com/trigonometry/sine.htm

  4. Horizontal Shift and Phase Shift (use Regent) Go to phase shift pdf

  5. Determining the Period and Amplitude of y = a sin bx Given the function y = 3sin 4x, determine the period and the amplitude. . The period of the function is . Therefore, the period is The amplitude of the function is | a |. Therefore, the amplitude is 3. y = 3sin 4x 4.3.10

  6. Graphing a Periodic Function Graph y = sin x. 1 Domain: Period: 2p all real numbers Amplitude: 1 Range: y-intercept: 0 x-intercepts: 0, ±p, ±2p, ... -1 ≤ y ≤ 1 4.3.3

  7. Graphing a Periodic Function Graph y = cos x. 1 Period: 2p Domain: all real numbers Range: -1 ≤ y ≤ 1 Amplitude: 1 y-intercept: 1 x-intercepts: , ... 4.3.4

  8. Graphing a Periodic Function Graph y = tan x. Period:p Asymptotes: Domain: Range: all real numbers 4.3.5

  9. Determining the Period and Amplitude of y = a sin bx Sketch the graph of y = 2sin 2x. The period is p. The amplitude is 2. 4.3.11

  10. Determining the Period and Amplitude of y = a sin bx Sketch the graph of y = 3sin 3x. The period is . The amplitude is 3. 4.3.12

  11. Writing the Equation of the Periodic Function Period Amplitude p = 2 b = 2 Therefore, the equation as a function of sine is y = 2sin 2x. 4.3.13

  12. Writing the Equation of the Periodic Function Period Amplitude 4 p b = 0.5 = 3 Therefore, the equation as a function of cosine is y = 3cos 0.5x. 4.3.14

  13. Summary of Transformations • a = vertical stretch or shrink amplitude • b = horizontal stretch or shrink period/frequency • c = horizontal shift (phase shift) phase • h = horizontal shift (phase shift) phase • d = vertical translation/shift • k = vertical translation/shift Exit Ticket pg 439 #61-64

  14. Horizontal Shift and Phase Shift (use Regent)

  15. Audacity Sinusoid- Periodic Functions A function is a sinusoid if it can be written in the form where a, b, c, and d are constants and neither a nor b is 0 Domain: Range: Continuity: Increasing/Decreasing Symmetry: Bounded: Max./Min. Horizontal Asymptotes Vertical Asymptotes End Behavior

  16. Sinusoid – a function that can be written in the form below. Sine and Cosine are sinusoids. The applet linked below can help demonstrate how changes in these parameters affect the sinusoidal graph: http://www.analyzemath.com/trigonometry/sine.htm

  17. For each sinusoid answer the following questions.What is the midline? X = What is the amplitude? A =What is the period? T = (radians and degrees)What is the phase? ϴ =

  18. Definition: A function y = f(t) is periodic if there is a positive number c such that f(t+c) = f(t) for all values of t in the domain of f. The smallest number c is called the period of the function. - a function whose value is repeated at constant intervals

  19. Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants and neither a nor b is 0 Why is the cosine function a sinusoid? http://curvebank.calstatela.edu/unit/unit.htm

  20. Read page 388 – last paragraph Vertical Stretch and Shrink baseline On your calculator ½ cos (x) -4 sin(x) What are the amplitudes? What is the amplitude of the graph? 2

  21. Vertical Stretch and Shrink Amplitude of a graph Abs(max value – min value) 2 For graphing a sinusoid: To find the baseline or middle line on a graph y = max value – min value 2 Use amplitude to graph. baseline

  22. Vertical Stretch and Shrink Amplitude of a graph Abs(max value – min value) 2 For graphing a sinusoid: To find the baseline or middle line on a graph y = max value – amplitude baseline

  23. Horizontal Stretch and Shrink b = number complete cycles in 2π rad. On your calculator T = sin(2x) T = sin) T = sin(5x) T = What are the periods (T)? Horizontal Stretch/Shrink y = f(cx) stretch if c< 1 factor = 1/c shrink if c > 1 factor = 1/c

  24. See if you can write the equation for the Ferris Wheel

  25. We can use these values to modify the basic cosine or sine function in order to model our Ferris wheel situation.

  26. Audacity Sinusoid- Periodic Functions A function is a sinusoid if it can be written in the form where a, b, c, and d are constants and neither a nor b is 0

  27. Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants and neither a nor b is 0 Why is the cosine function a sinusoid? http://curvebank.calstatela.edu/unit/unit.htm

  28. Read page 388 – last paragraph Vertical Stretch and Shrink On your calculator ½ cos (x) -4 sin(x)

  29. Horizontal Stretch and Shrink On your calculator sin2(x) sin) sin3(x) Horizontal Stretch/Shrink y = f(bx) stretch if |b| < 1 shrink if |b |> 1 Both cases factor = 1/|b|

  30. The frequency is the reciprocal of the period. f = .

  31. Periodic Functions Functions that repeat themselves over a particular interval of their domain are periodic functions. The interval is called the period of the function. In the interval there is one complete cycle of the function. To graph a periodic function such as sin x, use the exact values of the angles of 300, 450, and 600. In particular, keep in mind the quadrantal angles of the unit circle. http://curvebank.calstatela.edu/unit/unit.htm http://www.analyzemath.com/trigonometry/sine.htm (0, 1) The points on the unit circle are in the form (cosine, sine). (-1, 0) (1, 0) (0, -1) 4.3.2

  32. Determining the Amplitude of y = a sin x Graph y = 2sin x and y = 0.5sin x. y = 2sin x y = sin x y = sin x y = 0.5sin x 4.3.6

  33. Comparing the Graphs of y = a sin x y = sin x y = 2sin x y = 0.5sin x 2p 2p 2p Period Amplitude Domain Range 1 2 0.5 all real numbers all real numbers all real numbers -1 ≤ y ≤ 1 -2 ≤ y ≤ 2 -0.5 ≤ y ≤ 0.5 The amplitude of the graph of y = a sin x is | a |. When a > 1, there is a vertical stretch by a factor of a. When 0 < a < 1, there is a vertical shrink by a factor of a. 4.3.7

  34. Determining the Period for y = sin bx, b > 0 Graph y = sin 2x y = sin x y = sin x y = sin 2x y = sin x 4.3.8

  35. Comparing the Graphs of y = sin bx y = sin x y = sin 2 x y = sin 0.5 x 2p p 4p Period Amplitude Domain Range 1 1 1 all real numbers all real numbers all real numbers -1 ≤ y ≤ 1 -1 ≤ y ≤ 1 -1 ≤ y ≤ 1 The period for y = sin bx is When b > 1, there is a horizontal shrink. When 0 < b < 1, there is a horizontal stretch. 4.3.9

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