13-7 Translating Sine and Cosine Functions

Objectives

To graph translations of trigonometric functions

To write equations of translations

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Recall that for any function f, you can graph f ( x − h ) f open x minus h close  by translating the graph of f by | h | absolute value of h ,  units horizontally. You can graph f ( x ) + k f open x close plus k  by translating the graph of f by | k | absolute value of k ,  units vertically.

Essential Understanding You can translate periodic functions in the same way that you translate other functions.

Each horizontal translation of certain periodic functions is a phase shift.

g ( x ) : horizontal translation of f ( x ) g open x close colon . horizontaltranslationof . f open x close
g ( x ) = f ( x − h ) g open x close equals f open x minus h close

h ( x ) : vertical translation of f ( x ) h open x close colon . verticaltranslationof . f open x close
h ( x ) = f ( x ) + k h open x close equals f open x close plus k

When g ( x ) = f ( x − h ) , g open x close equals f open x minus h close comma  the value of h is the amount of the shift left or right. If h > 0 , h greater than 0 comma  the shift is to the right. If h < 0 , h less than 0 comma  the shift is to the left.

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