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Transformations of Sine and Cosine Functions MHF4UI Tuesday November 13th, 2012 Relating Trig Functions to Angles in Standard Position Last class we learned about the general shape and characteristics of the six Trig Functions For each angle that that was inputted into a trig function, the resultant output was the trig ratio for that particular angle. We can construct the graphs of each Trig Function with the help of our calculators OR We are able to draw the angle in standard position to determine what its corresponding Trig Ratios will be For example, let’s choose the angle 2 The General Form of Sinusoidal Functions A sinusoidal function is a function can be produced by shifting, stretching or compressing the sine function. The general form of a Sinusoidal Function can be written as: = a • sin − + OR = a • cos − + Where, a affects the Amplitude of the function affects the Period of the function affects the Phase Shift (or horizontal shift) of the function affects the vertical shift of the function The General Form of Sinusoidal Functions (Continued) = a • sin − + OR = a • cos − + The absolute value of a, or , is the amplitude of the graph, if a is negative, it represents a reflection in the x-axis. , will stretch or compress the Period of the Function. The period of the function will be 2 . , will result in a Phase Shift to the right units for positive . Conversely the Phase Shift will be units to the left for a negative . , will result in a Vertical Shift up units for positive . Conversely the Vertical Shift will be units down for a negative . The Function = sin The Sine function that we discussed yesterday is in general form with : =1 =0 =0 =1 The period was 2 1 = 2 It is important to realize that the characteristics of the Sine Function will only apply if we do not apply any transformations. When we apply any transformations we are potentially altering the amplitude, period, Max/Min and intercepts of the function. Note that if we apply a Phase shift of − to the Sine function the 2 graph will look identical to the Cosine Function. Applying Transformations Example 1 Given the following equation, state the amplitude, period, phase shift and vertical shift. = 3 sin 2 + −2 2 Applying Transformations Example 2 Given the following equation, state the amplitude, period, phase shift and vertical shift. = −2 sin 4 − 2 + 2 Mapping Notation for Sinusoidal Functions Recall that for Logarithmic Functions we used mapping notation to map key points to their coordinates for the transformed function. The formula for mapping notation of Sinusoidal Functions is identical to the mapping notation of Logarithmic Functions: 1 (x, ) ( + , + ) What we must determine now is what key points we will choose map to our transformed function. = sin The Key Points for = sin As we just pointed out, the key points for the sine function are: , = (0,0) , = ( , 1) 2 , = (, 0) 3 , = ( , −1) 2 , = 2, 0 You MUST memorize the key points! = cos The Key Points for = cos As we just pointed out, the key points for the cosine function are: , = (0,1) , = ( , 0) 2 , = (, −1) 3 , = ( , 0) 2 , = 2, 1 You MUST memorize the key points! Graphing Sine and Cosine Functions When sketching a sine or cosine function, you must: Label all 5 key points (After you have applied the mapping notation) Show the general shape of the graph. Graphing Sine and Cosine Functions Example 1 Find the key points and sketch the function = 2 sin 2 − 1 = 2, = 2, = 0, = −1 1 (x, ) ( , 2 − 1) 2 (0,0) ( 2 , 1) (, 0) 3 ( 2 , −1) 2, 0 1 2 0 , 2 0 − 1 = (0, −1) 1 2 2 1 , 2 1 − 1 = ( 4 , 1) (2 , 2(0) − 1) = ( 2 , −1) 1 3 3 (2 ( 2 ), 2(−1) − 1) = ( 4 , −3) 1 2 ( 2, 2(0) − 1) = (, −1) Graphing Sine and Cosine Functions Example 1 Find the key points and sketch the function = 2 sin 2 − 1 Graphing Sine and Cosine Functions Example 2 2 Find the key points and sketch the function = −5 sin 2( − ) + 2 = −5, = 2, = , = 2 2 1 (x, ) ( + , −5 + 2) 2 2 (0,0) ( 2 , 1) (, 0) 3 ( 2 , −1) 2, 0 1 2 0 + 2 , −5 0 + 2 = ( 2 , 2) 1 2 2 3 + 2 , −5 1 + 2 = ( 4 , −3) 1 (2 + 2 , −5(0) + 2) = (, 2) 1 3 5 (2 ( 2 ) + 2 , −5(−1) + 2) = ( 4 , 7) 1 2 2 3 , 2) 2 ( 2 + , −5(0) + 2) = ( Graphing Sine and Cosine Functions Example 2 2 Find the key points and sketch the function = −5 sin 2( − ) + 2 Graphing Sine and Cosine Functions Example 3 1 Find the key points and sketch the function = − cos ( + ) 2 1 = −1, = , = −, = 0 2 (x, ) (2 − , −) (0,1) 2 0 − , − 1 2 − , − 0 = (−, −1) ( 2 , 0) 2 = (0,0) (, −1) (2 − , −(−1)) = (, 1) 3 (2( 2 ) − , −(0)) = (2, 0) 2, 1 (2(2) − , −(1)) = (3, −1) ( 2 , 0) 3 Graphing Sine and Cosine Functions Example 3 Find the key points and sketch the function = 1 − cos ( 2 + ) Homework Questions: Complete the Transformations Worksheet Handout