1. 9.3 Graphing General Rational Functions
Obj: to graph general rational functions
We’ve only graphed rational functions where x was to the first power. What if x is not to the first power?
Do Now: read (do not copy)
Such as:

2. Steps to graph when x is not to the 1st power
x-intercepts: set numerator=0 and solve
2. vertical asymptote(s): set denominator=0 and solve
3. horizontal asymptote: 3 cases:
If degree of top < bottom, y=0
If degrees are =,
If degree of top > bottom, there is no H.A., but there will be a slant asymptote.
4. table: choose x-values on either side & between all vertical asymptotes.

3. Ex 1: Graph . State domain & range.
4. x y
x-int x=0
V.A. x2+1=0
x2= -1
No V.A
H.A.
degree of top < bottom
y=0
(No real solns.)

4. Domain: all real #s
Range:
x-int
(0, 0)
H.A. y=0

5. Ex 2: x-int 3x2=0 x2=0 x y x=0 V.A. x2-4=0 x2=4 x=2 and x=-2 H.A.
degrees are =
y=3
x y
On right of V.A.
Between the V.A.s
On left of V.A.

6. HW: Domain: all real #’s except -2 & 2
V.A. x=-2
V.A. x=2
H.A. y=3
HW:
Domain: all real #’s except -2 & 2
Range: all real #’s except 0<y≤3

7. Ex 3: x-int x2-3x-4=0 (x-4)(x+1)=0 x-4=0 x+1=0 x=4 x=-1 -1 0 V.A. 0 2
Continued…
x-int
x2-3x-4=0
(x-4)(x+1)=0
x-4=0 x+1=0
x=4 x=-1
V.A.
x-2=0
x=2
H.A.
deg. of top > bottom
no H.A., but there is a slant!
x y
Left of V.A.
Right of V.A.

8. Slant asymptotes Use synthetic division
(ignoring the remainder) is the equation of the slant asymptote.
In our example: 2
y=x-1

9. Domain:
all real #’s except 2
Range:
all real #’s
y=x-1
V.A. x=2