2.9. Vertical Asymptotes http://www.ck12.org
There are two discontinuities: one is a hole and one is a vertical asymptote. The hole occurs at (1, 5). The vertical
asymptote occurs atx=^23.
Notice that holes are identified as points while vertical asymptotes are identified as lines of the formx=awhere
ais some constant.
Example C
Draw the vertical asymptotes for the following function.
f(x) =(x− 4 )(x−^12 )(x+ 3 )
Solution:
Note that you may not know the characteristics of what the function does inside these vertical lines. You will soon
learn how to use sign tests as well as techniques you’ve already learned to fill in the four sections that this function
is divided into.
Concept Problem Revisited
Holes occur when factors from the numerator and the denominator cancel. When a factor in the denominator does
not cancel, it produces a vertical asymptote. Both holes and vertical asymptotes restrict the domain of a rational
function.
Vocabulary
Avertical asymptoteis a dashed vertical line that indicates that as a function approaches, it shoots off to positive or
negative infinity without ever actually touching the line.
Arational functionis a function with at least one rational expression.
Arational expressionis a ratio of two polynomial expressions.
Guided Practice
- Write a function that fits the following criteria:
- Vertical asymptotes at 0 and 3
- Zeroes at 2 and 5
- Hole at (4, 2)