1. 8.3 Graph General Rational Functions

2. Last class, we graphed rational functions where x was to the first power only. What if x is not to the first power?
Such as:

3. Steps to graph rational function when x is not to the 1st power
Find the x-intercepts. (Set numerator =0 and solve)
Find vertical asymptote(s). (set denom=0 and solve)
Find horizontal asymptote. 3 cases:
If degree of top < degree of bottom, y=0
If degrees are =,
If degree of top > degree of bottom, no horiz. asymp, but there will be a slant asymptote.
4. Make a T-chart: choose x-values on either side & between all vertical asymptotes.
Graph asymptotes, pts., and connect with curves.
Check solutions on calculator.

4. Ex: Graph. State domain & range.
4. x y
x-intercepts: x=0
vert. asymp.: x2+1=0
x2= -1
No vert asymp
horiz. asymp:
1<2
(deg. of top < deg. of bottom)
y=0
(No real solns.)

5. Domain: all real numbers
Range:

6. Ex: Graph, then state the domain and range.
x-intercepts:
3x2=0
x2=0
x=0
Vert asymp:
x2-4=0
x2=4
x=2 & x=-2
Horiz asymp:
(degrees are =)
y=3/1 or y=3
x y
On right of x=2 asymp.
Between the 2 asymp.
On left of x=-2 asymp.

7. Domain: all real #’s except -2 & 2
Range: all real #’s except 0<y<3

8. Ex: Graph, then state the domain & range.
x-intercepts:
x2-3x-4=0
(x-4)(x+1)=0
x-4=0 x+1=0
x=4 x=-1
Vert asymp:
x-2=0
x=2
Horiz asymp: 2>1
(deg. of top > deg. of bottom)
no horizontal asymptotes, but there is a slant!
x y
Left of x=2 asymp.
Right of x=2 asymp.

10. Slant asymptotes
Do synthetic division (if possible); if not, do long division!
The resulting polynomial (ignoring the remainder) is the equation of the slant asymptote.
In our example:
Ignore the remainder, use what is left for the equation of the slant asymptote: y=x-1

11. Domain: all real #’s except 2
Range: all real #’s