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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

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objectives
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 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
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 translations1
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
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
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
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
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
Homework!
  • pgs. 746-747, #’s 1-6, 11-40 (skip #37)

** for the “graphing” problems, just state the parent function and the shift(s)