Fill in blanks to write the particular equation of this
transformed cosine graph

f ( x ) = -3*cos ( x ) -2                    

Step-by-step explanation:

Solution:-

The standard generalized cosine function is given in the form:

                             f ( x ) = a* cos ( w*x - k ) + b

Where,

                   a: The magnitude of the waveform

                   w: the frequency of the waveform

                   k: The phase difference of the waveform

                   b: The vertical offset of central axis from the origin

To determine the amplitude ( a ) of the waveform. We will first determine the central axis of the waveform. This can be determined by averaging the maximum and minimum values attained. So from graph:

                   Maximum: 1

                   Minimum: -5

The average would be:

                  Central axis ( y ) = [ 1 - 5 ] / 2

                                              = -4 / 2

                                         y   = -2

The amplitude ( a ) is the difference between either the maximum value and the central axis or minimum value and the central axis. Hence,

                   a = Maximum -  Central value

                   a = 1 - (-2)

                   a = 3  

The waveform is inverted for all values of ( x ). That means the direction of amplitude is governed to the mirror image about x-axis. Hence, a = -3 not +3.

The offset of central axis from the x - axis ( y = 0 ) is denoted by the value of ( b ).

                   b = ( y = -2 ) - ( y = 0 )

                   b = -2   ... Answer

The frequency of the waveform ( w ) is given as the number of cycles completed by the waveform. The peak-peak distance over the domain of [ 0, 2π ]. We see from the graph is that two consecutive peaks are 2π distance apart. This means the number of cycles in the domain  [ 0, 2π ] are w = 1.

The phase difference ( k ) is determined by the amount of "lag" or "lead" in the waveform. This can be determined from the x-distance between x point value of peak and the origin value ( x = 0 ). The peak and the origin coincides with one another. Hence, there is no lag of lead in the waveform. Hence, k = 0.

The waveform can be written as:

                      f ( x ) = -3*cos ( x ) -2                    

step-by-step explanation:

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