A tutorial on exploring the cosecant function given by
f(x) = a * csc( b x + c ) + d
and its period, phase shift, asymptotes domain and range is presented. An interactive applet is used where parameters a, b, c and d are changed and their effects explored.
Interactive Tutorial
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. Now change a , how does it affect the graph? Does it affect the range? If yes how?

3  Set a = 1,c = 0,d = 0 and change b. For each value of b find 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?

4  Set a = 1,b = 1,d = 0 and change c starting from zero going slowly to positive large values. Take note of the shift, is it left or right, and compare it to  c / b.

5  Set a = 1, b = 1, d = 0 and change c starting from zero going slowly to negative smaller values. Take note of the shift, is it left or right, and compare it to  c / b.

6  Repeat 4 and 5 above for b = 2, 3 and 4.

7  Set a,b and c to non zero values and change d. What is the direction of the shift of the graph?

8  Which of the parameters affect the positions of the asymptotes (in red)? Explain analytically.

8  Which of the parameters affect the domain of the cosecant function? Explain analytically.

9  Which of the parameters affect the range of the cosecant function? Explain analytically.
More references on the trigonometric functions.
Trigonometric Functions.

