Notes on Graph Sketching:

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.

1. 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.

2. 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.

3. Use f' to find regions where f is increasing and decreasing.

4. Use f'' to find the regions where f is concave up and down.

5. 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.

6. If appropriate, locate the y intercept (by computing f(0)) and plot it.

7. If appropriate, compute the x intercepts (by solving for x in f(x) = 0) and plot them.

8. 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.

9. Add extra points as needed to help locate sections of the graph where you don't have any extrema or inflections.

10. 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).