sketching the graph of functions Good morning!
How to draw the graph of the following function and discuss the increasing and decreasing intervals of y. 
$y = $x$  + \sqrt[]{x^2}$
Also, I would like to know free online graphing calculator
Thanks in advance.
 A: Hint: Recall that $\sqrt{x^2} = |x|$, so
$$f(x) = x + |x| = \left\{\begin{array}{lr} 2x &: \text{ if } x \ge 0 \\ 0 &: \text{ if } x < 0 \end{array}\right.$$

For an online graphic calculator, I would suggest Wolfram Alpha.
A: You can try Calcpad, it is free for both online and desktop: 
http://calcpad.net/Calculator
Just type the following:
f(x) = x + sqr(x^2)
$Plot{f(x)  @ x = -1 : 1}
This is the graph you will get
Plot 1
You can also define a function with three segments as follows:
f(x) = -2*(x > 0) + 0.5*(x ≡ 0) - x^3 * (x < 0)
The conditions in the brackets return 1 for "true" and 0 for "false" which makes possible to create an expression like the above one. It will evaluate correctly: 
f(-2) = 8
f(0) = 0.5
f(2) = -2
However, it will not plot correctly as one function. You have to use the "multiple" plot command. It has the following syntax:
$Plot{x1 | y1 & x2 | y2 & x3 | y3 @ x = a : b}
For your example, type the following...
$Plot{x*(x > 0) | -2 & 0 | 0.5 & x*(x < 0) | -x^3*(x < 0) @ x = -1 : 1}
...and you will get this plot:
Plot 2
