Definition và Graph of the Cotangent Function
Angle ( heta ) with initial side on the positive x axis (in standard position) and terminal side OM is shown below.
Properties of cot x
1) cot x has a period equal lớn ( pi ).2) ( cot(x) ) has vertical asymptotes at all values of ( x = npi ) , ( n ) being any integer.3) The tên miền of ( cot(x) ) is the mix of all real numbers except ( x = npi ) , ( n ) being any integer.4) The graph of ( cot(x) ) is symmetric with respect to lớn the origin of the system of coordinates.5) The range of ( cot(x) ) is given by: ( (-infty , +infty) )6) ( cot(x) ) is odd and its graph is symmetric with respect lớn the origin of the system of axes.7) ( cot(x) ) is decreasing on intervals.
Interactive Tutorial on the General Cotangent Function
The general cotangent function given by
( f ( x ) = a cot ( b x + c ) + d )and its period, phase shift, asymptotes domain và range are explored interactively. An ứng dụng is used, where parameters a, b, c and d are changed lớn investigate their effects on the graph of f.
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|>1 - mix a = 1, b = 1, c = 0 và d = 0. Take lưu ý of the period, phase shift & positions of the asymptotes (vertical lines) of the graph of f? Now change a , how does it affect the graph? Does it affect its range? If yes, how?2 - mix a = 1, c = 0, d = 0 and change b. Approximate the period from the graph và compare it khổng lồ ( dfrac2pi b ). How does b affect the graph of f(x)? How does it affect the asymptotes of the graph of f?3 - phối a = 1, b = 1, d = 0 & change c starting from zero increasing slowly to lớn positive large values. Take note of the shift, is it khổng lồ the left or to lớn the right? Compare its measure to ( - c / b ).4 - mix a = 1, b = 1, d = 0 và change c starting from zero deceasing slowly to negative smaller values. Take chú ý of the shift, is it a left or a right shift? Compare its measure to lớn ( - c / b ).5 - set a,b & c khổng lồ non zero values and change d. What is the direction of the shift of the graph?6 - Which parameters affect the positions of the vertical asymptotes? Explain analytically.7 - Which parameters affect the domain name of the cotangent function? Explain analytically.8 - Which parameters affect the range of the cotagent function? Explain analytically.|
Cotangent function cot x
Cotangent Function cot x