2.12. Sign Test for Rational Function Graphs http://www.ck12.org
Example A
Identify the vertical asymptotes and use the sign testing procedure to roughly sketch the nature of the function near
the vertical asymptotes.
f(x) =(x+ 2 ) (^21) ·(x− 1 )
Solution: The vertical asymptotes occur atx=−2 andx=1. The points to use the sign testing procedure with are
-2.001, -1.9999, 0.9999, 1.00001. The number of decimals does matter so long as the number is sufficiently close
to the asymptote. Note that any real number squared is positive.
f(− 2. 001 ) =(+)(+)·(−)=−
f(− 1. 9999 ) =(+)(+)·(−)=−
f( 0. 9999 ) =(+)(+)·(−)=−
f( 1. 0001 ) =(+)(+)·(+)= +
Later when you sketch everything you will use your knowledge of zeroes and intercepts. For now, focus on just the
portions of the graph near the asymptotes. Note that the graph below is NOT complete.
Example B
Identify the vertical asymptotes and use the sign testing procedure to roughly sketch the nature of the function near
the vertical asymptotes.
f(x) =(x+^1 )(x−^4 )
(^2) (x− 1 )(x+ 3 ) 3
100 (x− 1 )^2 (x+ 2 )
Solution: Note thatx=−2 is clearly an asymptote. It may be initially unclear whetherx=1 is an asymptote or a
hole. Just like holes have priority over zeroes, asymptotes have priority over holes. The four values to use the sign
testing procedure are -2.001, -1.9999, 0.9999, 1.00001.