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Translating Trigonometric Functions: Rules and Examples

Learn how to transform trigonometric functions using shifts in variables and understand sine and cosine functions with examples and equations. Practice problems included.

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Translating Trigonometric Functions: Rules and Examples

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  1. Aim: How do we totally transform an trigonometric function by manipulating the variables? Do Now:

  2. Transforming Functions • If k and h are positive numbers and • f(x) is a function, then • f(x) + k shifts f(x) up k units • f(x) – k shifts f(x) down k units • f(x + h) shifts f(x) left h units • f(x – h) shifts f(x) right h units f(x) = (x + h)2 + k - parabolic f(x) = |x + h| + k - absolute value ex. f(x) = (x – 4)2 + 4 is the image of g(x) = x2 after a shift of 4 units to the right and four units up or a translation of T4,4.

  3. Transforming Sine & Cosine Functions parent function y = sin x y = cos x y = a sin b(x – h) + k y = a cos b(x – h) + k |a| = amplitute (vertical stretch or shrink) |b| = frequency h = phase shift, or horizontal shift k = vertical shift

  4. Phase Shift y = a sin b(x – h) + k a = 1 b = 1 k = 0

  5. Vertical Shift y = a cos b(x – h) + k a = 2 b = 1 k = 3 y = cos x y = cos x y = 2cos x y = 2cos x + 3

  6. The Whole Shebang! a = 2 b = 2 k = -3/2

  7. Model Problem Describe any phase and/or vertical shifts. y = 4 cos (x + 1) – 2 y = .5 sin 3(x - ) – /3 Write an equation for each transformation. y = sin x; /2 units to right and 3.5 units up

  8. Regents Prep Which function is a translation of y = sin  that is /3 units up and /2 units to the left? What is the period of the function

  9. Transforming Sine & Cosine Functions parent function y = sin x y = cos x y = a sin (bx – h) + k y = a cos (bx – h) + k |a| = amplitude (vertical stretch or shrink) |b| = frequency k = vertical shift

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