Translating Sine and Cosine Functions Section 13.7

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

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)

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

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

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!

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)

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

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

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