http://www.ck12.org Chapter 2. Polynomials and Rational Functions
2.13 Graphs of Rational Functions by Hand
Here you will use your knowledge of zeroes, intercepts, holes and asymptotes to sketch rational functions by hand.
Sketching rational functions by hand is a mental workout because it combines so many different specific skills to
produce a single coherent image. It will require you to closely examine the equation of the function in a variety
of different ways in order to find clues as to the shape of the overall function. Since computers can graph these
complicated functions much more accurately than people can, why is sketching by hand important?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/60903http://www.youtube.com/watch?v=vMVYaFptvkk James Sousa: Ex. Match Equations of Rational Functions to
Graphs
Guidance
While there is no strict procedure for graphing rational functions by hand there is a flow of clues to look for in the
function. In general, it will make sense to identify different pieces of information in this order and record them on
a sketch.
STEPS FOR GRAPHING BY HAND
- Examine the denominator of the rational function to determine the domain of the function. Distinguish
between holes which are factors that can be canceled and vertical asymptotes that cannot. Plot the vertical
asymptotes. - Identify the end behavior of the function by comparing the degrees of the numerator and denominator and
determine if there exists a horizontal or oblique asymptote. Plot the horizontal or oblique asymptotes. - Identify the holes of the function and plot them.
- Identify the zeroes and intercepts of the function and plot them.
- Use the sign test to determine the behavior of the function near the vertical asymptotes.
- Connect everything as best you can.
Example A
Completely plot the following rational function.
f(x) =^4 x(^3) − 2 x (^2) + 3 x− 1
8 (x− 1 )^2 (x+ 2 )