# Graph of y = tan x

y = tan x is periodic function. The period of y = tan x is π. Therefore, we will draw the graph of y = tan x in the interval [-π, 2π].

For this, we need to take the different values of x at intervals of 10°. Then by using the table of natural tangent we will get the corresponding values of tan x. Take the values of tan x correct to two place of decimal. The values of tan x for the different values of x in the interval [-π, 2π] are given in the following table.

We draw two mutually perpendicular straight lines XOX’ and YOY’. XOX’ is called the x-axis which is a horizontal line. YOY’ is called the y-axis which is a vertical line. Point O is called the origin.

Now represent angle (x) along x-axis and y (or tan x) along y-axis.

Along the x-axis: Take 1 small square = 10°.

Along the y-axis: Take 10 small squares = 1 unity.

Now plot the above tabulated values of x and y on the co-ordinate graph paper. Then join the points by free hand. The continuous curve obtained by free hand joining is the required graph of y = tan x.

Properties of y = tan x:

(i) The tangent-graph is not a continuous curve, but consists of infinite separate branches parallel to one another, the points of discontinuities are at x = (2n + 1)$$\frac{π}{2}$$ where n = 0 or any integer.

(ii) As x passes through any point of discontinuity from the left to the right, the value of tan x suddenly changes from (+∞ ) to (-∞).

(iii) Each branch of the curve approaches continuously the two lines parallel to y- axis at two points of discontinuity of the graph. Such lines are called asymptotes to the curve.

(iv) Since the function y = tan x is periodic of period π hence each branch is simply a repetition of the branch from -$$\frac{π}{2}$$ to $$\frac{π}{2}$$.