Answer:
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
