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.

- Find the vertical asymptotes and draw the dashed lines for them.
For each side, determine if
*f*is going to positive or negative infinity.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.

- Find the horizontal asymptotes and draw dashed lines for these.
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. - Use
*f*' to find regions where*f*is increasing and decreasing. - Use
*f*'' to find the regions where*f*is concave up and down. - Locate and plot the extrema and the inflection points.
Hint: Use the second derivative test to distinguish maxima from minima. You can also use the

*f*' information to tell which is which. - If appropriate, locate the
*y*intercept (by computing*f*(0)) and plot it. - If appropriate, compute the
*x*intercepts (by solving for*x*in ) and plot them.*f*(*x*) = 0 - Compute the slope at important points and draw small tangent lines as
guides at these points.
Hint: The slope usualy 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. - Add extra points as needed to help locate sections of the graph where
you don't have any extrema or inflections.
- Sketch the curve.
Hint: Connect the points by curves that correspond to the proper concavity and growth of

*f*(see the table from our class notes).

Comments to: dpvc@union.edu

Created: Oct 14 1997 --- Last modified: Oct 15, 1997 2:03:03 PM