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 sine.htm - an interactive applet from AnalyzeMathhttp:// sine.htm es/colleges/blackpoolsixthform/cameron/Tr ansform%20of%20Trig1.xls - another interactive applethttp://ferl.becta.org.uk/content_files/resourc es/colleges/blackpoolsixthform/cameron/Tr ansform%20of%20Trig1.xls

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, p , Q1- 12