Hello,

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

2. Asympotes:

Thus you have 3 asymptotes: x = -3 or x = 3 or y = 1

3. Y-intercept:

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 relativemaximumat 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.