I'm not sure what you mean by "graphing a curve with the derivative correctly". So I'm going to show you what you can do as an examination with the given example:
1. Domain: The denominator must be unequal zero. The denominator is zero at x = -3 or x = 3. Thus the domain is
Thus you have 3 asymptotes: x = -3 or x = 3 or y = 1
4. Range: and
That means the range is
5. Zeros . A fraction is zero if the numerator equals zero and the denominator unequals zero. The term will never become zero.
6. Relative extrema. Calculate first the first derivative of f:
. Don't forget to cancel the factor (x²-9)
f'(x) = 0 ==> x = 0
that means you have a relative maximum at x = 0
7. Point of inflection. Condition: . (see my remark to #5) thus there is no point of inflection.
8. I've attached a sketch of the graph including the asymptotes.