8.2 - 8.3: Graphing Rational Functions
Below are a few examples of rational functions:
,
,
Rational functions are undefined for certain values of x.
is undefined at _____________________
is undefined at _____________________
is undefined at _____________________
Because of this, graphs of rational functions have breaks in
continuity that appear as asymptotes or as points of
discontinuity (holes).
Vocabulary
asymptote - a line that the graph of a function approaches but
never (or rarely) crosses. (The graph will never cross an
asymptote as
.)
point of discontinuity - a hole in the graph.
asymptotes
hole
1
Graphing Steps for Rational Functions
1. Simplify.
2. Find vertical asymptote(s) and/or holes.
3. Find horizontal or oblique asymptote.
4. Find the x-intercept(s) and the y-intercept.
5. Find and graph points in each section
separated by the vertical asymptote(s).
2
How to find the horizontal or oblique asymptote:
*There are 3 scenarios
If...
numerator degree = denominator degree
then...
the horizontal asymptote is
If...
numerator degree < denominator degree
then...
the horizontal asymptote is
If...
numerator degree > denominator degree
then...
there is NO horizontal asymptote. There is
an oblique asymptote IF the difference in
degree is by 1 AND it hasn't simplified to a
linear function. We will explore this through
the examples to follow.
*Note: There will never be a horizontal asymptote
AND an oblique asymptote on the same graph. It
will be one or the other or neither.
3
Do all 5 graphing steps for all of the examples.
Example 1: Graph
Vertical asymptote(s) ______________
Horizontal asymptote _____________
Oblique asymptote ________________
Hole(s) __________________________
x-intercept(s) _____________________
y-intercept _______________________
Domain _________________________
4
Example 2: Graph
Vertical asymptote(s) ______________
Horizontal asymptote _____________
Oblique asymptote ________________
Hole(s) __________________________
x-intercept(s) _____________________
y-intercept _______________________
Domain _________________________
5
Example 3: Graph
Vertical asymptote(s) ______________
Horizontal asymptote _____________
Oblique asymptote ________________
Hole(s) __________________________
x-intercept(s) _____________________
y-intercept _______________________
Domain _________________________
6
Example 4: Graph
Vertical asymptote(s) ______________
Horizontal asymptote _____________
Oblique asymptote ________________
Hole(s) __________________________
x-intercept(s) _____________________
y-intercept _______________________
Domain _________________________
7