Here is a procedure for sketching the graph of a function. This works best on polynomials and rational functions (i.e., quotients of polynomials), but it can work for any function, if you are careful.
Hint: For rational functions, the points where the denominator is zero (but the numerator is not) will always be vertical asymptotes; you just have to decide which infinity to use. Check the one-sided limits carefully to see which side goes where.
Hint: For a rational function, the limits as x goes to infinity and negative infinity will agree, if they exist. Remember, a limit of infinity is a special way for a limit not to exist.
Hint: Use the f' information (first derivative test) to tell which is which.
Hint: The slope usually is zero at the extrema, so draw horizontal tangents there. Use f' to compute the slope at inflection points and at the intercepts, if you need them.
Hint: Connect the points by curves that correspond to the proper concavity and growth of f (see the table from our class notes).