http://www.ck12.org Chapter 2. Polynomials and Rational Functions
2.12 Sign Test for Rational Function Graphs
Here you will learn to predict the nature of a rational function near the asymptotes.
The asymptotes of a rational function provide a very rigid structure in which the function must live. Once the
asymptotes are known you must use the sign testing procedure to see if the function becomes increasingly positive
or increasingly negative near the asymptotes. A driving question then becomes how close does near need to be in
order for the sign test to work?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/60822http://www.youtube.com/watch?v=OEQnQNvJtG0 James Sousa: Graphing Rational Functions
Guidance
Consider mentally substituting the number 2.99999 into the following rational expression.
f(x) =(x−(^1 x)(+x 2 +)(^3 x)(−x 4 −)(^5 x)(−x 3 +)^10 )Without doing any of the arithmetic, simply note the sign of each term:
f(x) =(+)(+)·(+)·(−·()−·()−·(+))The only term where the value is close to zero is(x− 3 )but careful subtraction still indicates a negative sign. The
product of all of these signs is negative. This is strong evidence that this function approaches negative infinity as
xapproaches 3 from the left.
Next consider mentally substituting 3.00001 and going through the same process.
f(x) =(+)(+)·(+)·(−·()−·(+))·(+)The product of all of these signs is positive which means that from the right this function approaches positive infinity
instead.