# How do you graph #f(x)=tanx# and include two full periods?

##### 1 Answer

graph{tanx [-19.25, 20.75, -9.44, 10.56]}

#### Explanation:

Lets first analyse the function for the period **[-π/2,π/2]**

tan(-x)=-tanx thus **Odd Function**

Some specific points

**tan(0)=0, tan(π/2)=infinity, tan(-π/2)= - infinity**

Lets see the graph of tanx lies above or below y=x

graph{x [-20.12, 19.88, -9.32, 10.68]}

Consider the function tanx - x

Differentiating it w.r.t to x

since

at x=0 ,tan x -x =0

since its an increasing function

for **x>0** => tanx -x >0 thus **tanx >x**

similarly for **x<0** **tanx < x**

Discussing the **Slopes and Concavity**

**Slope**

d(tanx)/dx =

**At x=0 Slope is π/4**

**At x= π/2 and -π/2 it is infinity** thus parallel to y axis

**Concavity**

Taking second order derivative

2secx* secx*tanx

which is **positive for x>0**

thus slope is increasing **Concave up**

graph{tanx [-1.67, 3.33, -0.11, 2.39]}

Similarly its **concave down for x<0**

graph{tanx [-2.805, 2.195, -2.445, 0.055]}

for the period [- π/2,π/2 ]

Graph is

**Periodicity**

**Period of tanx is π** Just repeat the graph

The required graph is

graph{tanx [-4.93, 15.07, -4.71, 5.29]}