Chapter 6 Section 6.4 Translations of the Graphs of Sine and Cosine Functions

Period and Phase Shift The period of a trigonometric function is the distance along the x -axis it takes for the graph to go through a complete cycle. The phase shift is where it begins one of it cycles. The period and the phase shift are determined by the values for b and c in the equations: y = sin( bx+c )and y = cos( bx+c )

To find the interval on which a function of the form y = sin( bx + c ) or y = cos( bx + c ) will go through one of it cycles take the inside and set it equal to 0 and 2  and solve. Solving for x when ( bx + c ) is set equal to 0 will give you the starting value of a cycle (on the x - axis). Solving for x when ( bx + c ) is set equal to 2  will give you the ending value of a cycle (on the x - axis). Subtracting the values for the two solutions will give the period. beginning ending Phase Shift Period Graph between  /6 and -  /6 Example: y = sin(6 x +  )

Displacement The entire graph of y = sin( x ) or y = cos( x ) can be shifted up or down by adding a number onto the end. This amount the graph gets moved up or down is called the displacement. It is determined by the number d in the following equations: y = sin( x ) + d and y = cos( x ) + d If d is positive the graph is shifted up by d and d is negative the graph is shifted down by d. y = sin( x ) + 2 y = cos( x ) - 1

Putting it Together The graphing information for the equations: y = a sin( bx + c ) + d and y = a cos( bx + c ) + d can be broken down into the following parts: a : amplitude controls how far above and below the center line a cycle is b and c : determine the period and phase shift which are where a cycle starts and how long its is in the x direction d : the displacement determine how far up or down the centerline of the graph is moved. y = 2 cos(3 x +  ) – 1 amplitude = 2 displacement = -1

Find the equation of the graph pictured to the left.