The tangent function
f(x) = a*tan(bx+c)+d
and its properties such as graph, period, phase shift and asymptotes are explored interactively by changing the parameters a, b, c and d using an applet. See figure below for main panel of the applet showing the graph of tangent function in blue and the vertical asymptotes in red.
You may also want to consider another tutorial on the trigonometric unit circle .
Once you finish the present tutorial, you may want to go through a self test on trigonometric graphs .
Interactive Tutorial
How do the 4 coefficients a,b,c and d affect the graph of f(x)?

use the scrollbar to set a=1,b=1,c=0 and d=0. Write down f(x) and take note of the period phase shift and positions of asymptotes (in red)of f(x)? Now change a , how does it affect the graph?

set a=1,c=0,d=0 and change b. Find the period from the graph and compare it to
pi/b. How does b affect the graph of f(x)?How does it affect the asymptotes?

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.

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.

repeat 3 and 4 above for b=2,3 and 4.

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

What parameters affect the positions of the asymptotes? Explain algebraically.
More on
Graphing Tangent Functions. A step by step tutorial on graphing and sketching tangent functions. The graph, domain, range and vertical asymptotes of these functions and other properties are examined.
Derivative of tan(x).
Trigonometric Functions.

