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Translating Sine and Cosine Functions Section 13.7

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Objectives… 1. Given a translation equation, determine both the “parent” function and the shifts needed in order to graph the translation 2. Graph a translation equation 3. Given both the “parent” function and the shifts, write the translation equation

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A little review about translations… a “translation” is an operation that shifts a graph horizontally, vertically, or both (diagonally) ONLY changes the location of the graph (NOT the size or shape) “translations” start with “parent” functions (things are added to the “parent” function to cause the movements of the graph to occur)

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Vertical, Horizontal, and Diagonal Translations if the “parent” functions are y = a sin bθ and y = a cos bθ, then the “translation” functions are y = a sin b(θ – h) + k and y = a cos b(θ – h) + k h = horizontal shift (“phase shift”) k = vertical shift

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Vertical, Horizontal, and Diagonal Translations if h > 0, then the graph is shifted to the right if h < 0, then the graph is shifted to the left if k > 0, then the graph is shifted up if k < 0, then the graph is shifted down

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Remember the following… horizontal translations are found within the parent function (parentheses) vertical translations are found at the end of the parent function example: y = sin x + 2 and y = sin (x + 2) are different!

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Examples… Given the parent functions y = sin x and y = cos x, graph the following translations: A) y = sin x + 2 B) y = cos(x – pi)

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More Examples… Given the parent function y = -3 sin 2x and y = 2 cos 2x, graph the following translation functions: A) y = -3 sin 2(x – (pi/3)) – 3/2 B) y = 2 cos 2(x + 1) – 3

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And Some More Fun… Given the following information, write the translation equation: A) y = cos θ, (pi/2) units up B) y = 2 sin x, (pi/4) units to the right C) y = sin 3θ, pi units down D) y = -cos x, 3 units to the left E) y = -3 cos 4x, 2.5 units to the left, 4 units up

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Homework! pgs. 746-747, #’s 1-6, 11-40 (skip #37) ** for the “graphing” problems, just state the parent function and the shift(s)

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