Graphs of Even and Odd Functions

Graphs of even and odd functions have following properties:

  1. If function is even then its graph is symmetric about y-axis.
  2. If function is odd then its graph is symmetric about origin.

Example 1. Draw graph of the function `y=|x|`.graph of the function y=|x|

Since `f(-x)=|-x|=|x|=f(x)` then function is even. This means that its graph is symmetric about y-axis.

If `x>=0` then `|x|=x`, i.e. when `x>=0` we have that `y=x`. Graph of the function y=x when `x>=0` is bisector of first coordinate angle. Now draw it symmetrically about y-axis and you will obtain graph of the function `y=|x|`.

Example 2. Draw graph of the function `y=x|x|`.graph of the function y=x|x|

Since `f(-x)=(-x)|-x|=-x|x|=-f(x)` then function is odd. This means that its graph is symmetric about origin.

If `x>=0` then `|x|=x`, i.e. when `x>=0` we have that `y=x*x=x^2`. Graph of the function `y=x^2` when `x>=0` is part of parabola. Now draw it symmetrically about origin and you will obtain graph of the function `y=|x|`.