# State the amplitude and period for each, and relative to the basic function the phase shift and vertical translation. (a.) y=-2 cos (x minus `pi` /4 ) + 1 (b.) y=2cos 2`pi` x + 1

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### 1 Answer

The general formula for the graph of basic cosine function is:

`Y=Acos(Bx-C)+D`

where A=amplitude of the function

B=stretch/shrink on the x-axis. B has the following relationship with the period:

B=`(2pi)/P` where, P is the period. So, `P=(2pi)/B`

C/B=the phase shift of the graph (the shift left (if C/B is neg.) or right (if C/B is pos.))

D=the vertical shift of the graph.

(a) `y=-2 cos (x-pi/4) + 1`

Here, amplitude =A= -2

B=1

Hence, period, `P=(2pi)/B=(2pi)/1=2pi`

Phase shift=C/B=`(pi/4)/1=pi/4` since, it is positive the shift is to the right.

Vertical translation=D=1. There is a vertical shift of 1 up.

(b)`y=2cos 2pix + 1`

Here, amplitude =A=2. The graph is stretched vertically by a factor of 2.

B=`2pi`

Hence, period, `P=(2pi)/B=(2pi)/(2pi)=1 `

Phase shift=C/B=`0/(2pi)=0.`

Vertical translation=D=1. There is a vertical shift of 1 up.

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