The cosecant function given by
f ( x ) = a * cot ( b x + c ) + d
and its period, phase shift, asymptotes domain and range are explored interactively. An applet is used, where parameters a, b, c and d are changed to investigate their effects on the graph of f.
1 - Click on the button above "click here to start" and maximize the window obtained.
2 - Use the scroll bar to set a = 1, b = 1, c = 0 and d = 0. Take note of the period, phase shift and positions of the asymptotes (in red) of the graph of f? Now change a , how does it affect the graph? Does it affect its range? If yes, how?
3 - Set a = 1,c = 0,d = 0 and change b. Approximate the period from the graph and compare it to 2pi/|b|. How does b affect the graph of f(x)? How does it affect the asymptotes of the graph of f?
4 - Set a = 1,b = 1,d = 0 and change c starting from zero increasing slowly to positive large values. Take note of the shift, is it to the left or to the right? Compare its measure to - c / b.
5 - Set a = 1,b = 1,d = 0 and change c starting from zero deceasing slowly to negative smaller values. Take note of the shift, is it a left or a right shift? Compare its measure to - c / b.
6 - Set a,b and c to non zero values and change d. What is the direction of the shift of the graph?
7 - Which parameters affect the positions of the asymptotes(in red)? Explain analytically.
8 - Which parameters affect the domain of the cosecant function? Explain analytically.
9 - Which parameters affect the range of the cosecant function? Explain analytically.
More references on the trigonometric functions.