Clues to find the graphs of functions

Question

How do I find the graph for functions $f_n(x)=x^n$ where $n\in \mathbb{N} \bigcup \{0 \}$?

Thanks for all the help!

Answer 1

This can be characterized in a few points:

$0^n=0$

$1^n=1$

even and odd powers give even and odd functions

$0<x<1$ gets closer to zero with higher powers

$x>1$ gets larger with higher powers

Just look at positive $x$ first: All the above points suggest fixing $0$ and $1$ for every $n$, and then drawing functions whose rise from 0 to 1 gets sharper and sharper. So $x^2$ gives you a parabola, $x^4$ gives you a "sharper/more square parabola", etc. The higher odd powers are just sharper versions of $x^3$ as well.

Technically I should add more statements such as"each is monotonically increasing on $x\geq 0$" and "each has positive slope for $x\geq 0$", but I think the above gets the gist of it. In general you can tell a lot about a function by investigating its fixed points, symmetries, and how it increases/decreases and where.