Functions that have values occurring at regular intervals are known as periodic functions. Many everyday processes can be referred to as periodic for example, the temperature in a day rise all the way till noon, begins to fall as night approach, and repeats the same pattern every single day.
Amplitude of a periodic wave is the average difference between the maximum and minimum values, i.e., A = (max-min/2). The period of the wave is the time it takes to complete one oscillation/cycle.
To important periodic functions to be studied are the sine and cosine functions. Their graphs are equally important. It is to be noted that the graph of the cosine function is the graph of the sign function, shifted pi/2 to the left.
The graph for sin(x) looks as follows:
As can be seen it has an amplitude of 1 and a period of 2(pi).
The graph for cos(x) looks as follows:
As can be seen, it has an amplitude of 1 and a period of 2(pi) as well.
For an expression of a sine or cosine curve where
y = Asin(Bx+C)+D or y = Acos(Bx+C)+D
A affects the amplitude of the curve. Infact the absolute value of A is the amplitude of the curve.
B affects the period, i.e., period = 2(pi)/absolute value of B.
C affects the horizontal shift of the curve, and
D affects the vertical shift.
The part of graphing such sine and cosine curves that I found most challenging was when determining the amount by which and the direction in which curves moved when experiencing a horizontal shift. The positive and negative signs of C were slightly confusing.
Again, as mentioned before, periodic functions are extremely related to Economics because of business cylces are usually recurring phenomenon, i.e., periodic functions of recessions and booms. Hence the ways in which the operate is directly related to the shifts experienced by a regular sine or cosine function.
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