Up: Notes for Math 13
Notes on Graph Sketching:
Even though computers and calculators can draw nice graphs of functions,
it is important to know how to understand a graph yourself by finding the
important features of a graph. This will let you use your computer
intelligently to make sure you do not overlook anything important (like
relative extrema that are off the edge of the screen, or details that are
too small for the resolution you have selected).
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
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.
- 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
- Use f'' to find the regions where f is concave up and
- 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
where f(x) = 0) and plot them.
- 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).
Up: Notes for Math 13
Created: Oct 14 1997 ---
Last modified: Mar 8, 1999 9:55:56 PM