If $f$ and its derivatives, $f'$, $f''$, $f'''$, and so forth, exist in an interval containing a real number $a$, we may approximate $f(x)$ for $x$ near $a$ using a Taylor polynomial. A degree-one Taylor polynomial is a tangent line. The higher the degree, the more accurate the Taylor approximation near $a$.
This program graphs Taylor polynomials up to order eight at an arbitrary point, marked by a red dot. Type a function of $x$, and press TAB to graph. The left and right arrow keys move $a$. The up and down arrow keys increase or decrease the order of approximation by one. The less-than and greater-than keys shrink or expand the domain.
Currently graphing the Taylor polynomial: