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TF.03.3a - Transforming Sinusoidal Functions

TF.03.3a - Transforming Sinusoidal Functions. MCR3U - Santowski. (A) Review  y = sin(x). Recall the appearance and features of y = sin(x) The amplitude is 1 unit The period is 2  rad. The equilibrium axis is at y = 0

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TF.03.3a - Transforming Sinusoidal Functions

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  1. TF.03.3a - Transforming Sinusoidal Functions MCR3U - Santowski

  2. (A) Review  y = sin(x) • Recall the appearance and features of y = sin(x) • The amplitude is 1 unit • The period is 2 rad. • The equilibrium axis is at y = 0 • One cycle begins at (0,0), on the equilibrium axis and rises up to its maximum • The five keys points on the sin function are (0,0), (/2,1), (,0), (3/2,-1) and (2,0)

  3. (A) Review  y = cos(x) • Recall the appearance and features of y = cos(x) • The amplitude is 1 unit • The period is 2 rad. • The equilibrium axis is at y = 0 • One cycle begins at (0,1), at the maximum and decreases to the equilibrium axis and the minimum • The five keys points on the cos function are (0,1), (/2,0), (,-1), (3/2,0) and (2,1)

  4. (A) Review  y = tan(x) • Recall the appearance and features of y = tan(x) • There is no amplitude as the curve rises along the asymptotes • The period is  rad. • The equilibrium axis is at y = 0 • One cycle begins at x = -/2 where we have an asymptote, rises to the x-intercept and then rise along the asymptote at x = /2 • The five keys points on the tan function are (-/2,undef), (-/4,-1), (0,0), (/4,1) and (/2,undef)

  5. (B) Review - Transformations • Recall our work with transforming functions and the various notations that communicate the different types of transformations. • If y = f(x) is our “standard, base” function, then: • f(x) + a is a vertical translation up • f(x) – a is a vertical translation down • f(x-a) is a horizontal translation to the right • f(x+a) is horizontal translation to the left • af(x) is a vertical dilation by a factor of a • f(ax) is a horizontal dilation by a factor of 1/a • -f(x) is a reflection in the x axis • f(-x) is a reflection in the y-axis

  6. (C) Transformations - Investigation • Open up WINPLOT and a WORD document  copy all graphs into your document and include descriptions and analysis in your document • In WINPLOT, set the domain to [–2,2] and when analyzing a graph, state the location of the 5 keys points • Your analysis will describe the amplitude, period, location of the equilibrium axis, and where one cycle starts

  7. (D) Transforming y = sin(x) • Graph y = sin(x) as our reference curve • (i) Graph y = sin(x) + 2 and y = sin(x) – 1 and analyze  what features change and what don’t? • (ii) Graph y = 3sin(x) and y = ¼sin(x) and analyze  what features change and what don’t? • (iii) Graph y = sin(2x) and y = sin(½x) and analyze  what features change and what don’t? • (iv) Graph y = sin(x+/4) and y = sin(x-/3) and analyze  what changes and what doesn’t? • We could repeat the same analysis with either y = cos(x) or y = tan(x)

  8. (E) Combining Transformations • We continue our investigation by graphing some other functions in which we have combined our transformations • (i) Graph and analyze y = 2sin(x - /4) + 1  identify transformations and state how the key features have changed • (ii) Graph and analyze y = -½ cos[2(x + /60]  identify transformations and state how the key features have changed • (iii) Graph and analyze y = tan( ½ x + /4) – 3  identify transformations and state how the key features have changed

  9. (F) Transformations  Generalizations • If we are given the the general formula f(x) = a sin [k(x + c)] + d, then we have the following features in our transformed sinusoidal curve: • (i) amplitude = a • (ii) period = 2/k • (iii) equilibrium axis  y = d • (iv) phase shift  c units to the left or right, depending on whether c>0 or c<0

  10. (G) Internet Links • http://www.analyzemath.com/trigonometry/sine.htm - an interactive applet from AnalyzeMath • http://ferl.becta.org.uk/content_files/resources/colleges/blackpoolsixthform/cameron/Transform%20of%20Trig1.xls - another interactive applet

  11. (H) Examples Ex 1 – Given f(x) = 4sin(2x-/2) + 1, determine the period, amplitude, equilibrium axis and phase shift Ex 2 – If a cosine curve has a period of  rad, an amplitude of 4 units, and the equilibrium axis is at y = -3, write the equation of the curve.

  12. (I) Homework • From the Nelson textbook, p456-457, Q1-12

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