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.

Review: Some of the properties of the graph of f(x) = tan(x) are as follows:
Example 1: Graph
Over one period. Solution to Example 1: tan x is undefined for values of x equal to π/2 and π/2. However we need to understand the behavior of the graph of tan x as x approaches π/2 and π/2. Let us look at the values of tan x for x close to π/2 such that x is smaller then π/2.
We now look at the values of tan x for x close to π/2 such that x is larger then π/2.
tan x has an asymptotic behavior close to π/2 and π/2. Using the values of tan x above plus the following values: tan 0 = 0, tan (π/4) = 1 and tan (π/4) = 1, we start by plotting the points (0,0) , (π/4,1) and (π/4,1) and the vertical asymptotes. We then draw a smooth curve passing by the points calculated. Close to the vertical asymptotes, the graph either goes upward indefinitely (close to x = π/2 vertical asymptote) and downward indefinitely (close to x = π/2 vertical asymptote). Step 1: Make a table of values over one period.
Step 2: Plot the points and the vertical asymptotes. Step 3: Draw a curve passing through all points and rising or falling vertically along the vertical asymptotes.
Example 2: Graph function f given by
Over one period. Solution to Example 2:
We now use the relationship between x and t, t = 2 x  π / 4, to find the values of x corresponding to the values of t used in the above table. Solve t = 2 x  π/4 for x. x = t / 2 + π / 8 A row showing the x values may be added to the above table: These values of x have been found using x = t / 2 + π / 8 found above and the values of t in the table.
We now have the values of the function 2 tan t and the corresponding x values. We have enough information to graph the given function.
Example 3: Graph function f defined by
Over one period. Solution to Example 3: Let t = x + π/2. We first make a table using t over one period.
Solve t = x + π/2 for x. x = t  π / 2 A row showing the x values is added to the above table.
We now have the values of the function  tan t and the corresponding x values. More references and links on graphing. Graphing Functions Tangent Function. The tangent function f(x) = a tan(b x + 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 Home Page 