
Graphs of Rational Functions
Section 4.3 Graphs of Rational Functions
Now, we will graph rational functions. This process is a bit different from the
one in the book ut I think
it is easier to do:
1. Factor, state domain, and THEN reduce.
2. Plot intercepts. For xintercepts, state whether graph touches (multiplicity
is even) or
crosses (multiplicity is odd) xaxis.
3. Draw vertical asymptotes (the zeros of the denominator).
4. Draw horizontal asymptotes:
If deg num < deg denom, y = 0 is a horizontal asymptote
If deg num = deg denom, y = L is a horizontal asymptote
If deg num = deg denom + 1, use long division to find oblique asymptote, y = mx
+ b
5. Plot where the graph crosses a horizontal or oblique asymptote, if it does.
That is, solve
the equation:
Function = asymptote
R(x) = y
6. If needed, plot a few extra points.
7. Connect the dots.
Ex: Graph
1. Factor, state domain, and THEN reduce.
Domain:
Cannot reduce.
2. Plot intercepts. For xintercepts, state whether graph
touches (multiplicity is even) or
crosses (multiplicity is odd) xaxis.
xintercept:
The factor x + 2 has multiplicity 1, which is odd, so
graph crosses here.
yintercept:
3. Draw vertical asymptotes (the zeros of the
denominator).
These occur when the denominator is 0. That is, at x = 3 and x = 3.
4. Draw horizontal asymptotes:
Since deg num < deg denom, HA is y = 0
5. Plot where the graph crosses a horizontal or oblique asymptote, if it does.
That is, solve
the equation:
Solve R(x) = y 


So, the graph crosses the horizontal
asymptote at x = –2. 
6. If needed, plot a few extra points.
R(4) = 0.3
R(4) = 0.9
7. Connect the dots.
