1 / 12

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

janine
Download Presentation

TF.03.3a - Transforming Sinusoidal Functions

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. 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

More Related