Factor the following completely:
1. 3x2-8x+4
(3x-2)(x-2)
2. 11x2-99
4. x3+2x2-4x-8
(x-2)(x+2)2
5. 2x2-x-15
11(x+3)(x-3)
(2x+5)(x-3)
3. 16x3+128
6. 10x3-80
16(x+2)(x2-2x+4)
10(x-2)(x2+2x+4)
8.3
Graphing General Rational
Functions
Algebra II
Friday, we graphed rational
functions where x was to the first
power only.
What if x is not to the first power?
Such as:
x
f ( x)  2
x 1
Steps to graph when x is not to the 1st power
1. Find the x-intercepts. (Set numer. =0 and solve)
2. Find vertical asymptote(s). (set denom=0 and solve)
3. Find horizontal asymptote. 3 cases:
a. If degree of top < degree of bottom, y=0
lead. coeff. of top
y

b. If degrees are =,
lead. coeff. of bottom
c. 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.
5. Graph asymptotes, pts., and connect with curves.
Ex 1: Graph. State domain & range.
x
y 2
x 1
1. x-intercepts: x=0
2. vert. asymp.: x2+1=0
x2= -1
No vert asymp x  1
(No real solns.)
4. x
y
-2
-.4
-1
-.5
0
0
1
.5
3. horiz. asymp:
2
1<2
(deg. of top < deg. of bottom)
y=0
.4
Domain: all real numbers
Range:  1  y  1
2
2
Steps to graph when x is not to the 1st power
1. Find the x-intercepts. (Set numer. =0 and solve)
2. Find vertical asymptote(s). (set denom=0 and solve)
3. Find horizontal asymptote. 3 cases:
a. If degree of top < degree of bottom, y=0
lead. coeff. of top
y

b. If degrees are =,
lead. coeff. of bottom
c. 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.
5. Graph asymptotes, pts., and connect with curves.
Ex 2: Graph, then state the domain and
range.
3x 2
y
1. x-intercepts:
4. x
3x2=0
4
x2=0
x=0
3
2. Vert asymp:
1
x2-4=0
0
x2=4
-1
x=2 & x=-2
-3
3. Horiz asymp:
-4
(degrees are =)
y=3/1 or y=3
x2  4
y
4
5.4
On right of x=2
asymp.
-1
0
Between the 2
asymp.
-1
5.4
4
On left of x=-2
asymp.
•
Domain: all real #’s except -2 & 2
Range: all real #’s except 0<y<3
Steps to graph when x is not to the 1st power
1. Find the x-intercepts. (Set numer. =0 and solve)
2. Find vertical asymptote(s). (set denom=0 and solve)
3. Find horizontal asymptote. 3 cases:
a. If degree of top < degree of bottom, y=0
lead. coeff. of top
y

b. If degrees are =,
lead. coeff. of bottom
c. 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.
5. Graph asymptotes, pts., and connect with curves.
Ex 3: Graph, then state the domain &
range.
x 2  3x  4
y
x2
1. x-intercepts:
x2-3x-4=0
4. x
y
(x-4)(x+1)=0
-1
0
x-4=0 x+1=0
Left of x=2
x=4 x=-1
0
2
asymp.
2. Vert asymp:
1
6
x-2=0
3
-4
Right of
x=2
x=2
asymp.
4
0
3. Horiz asymp: 2>1
(deg. of top > deg. of bottom)
no horizontal asymptotes, but there is a slant!
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.
Ignore the remainder,
In our example:
use what is left for the
2 1 -3 -4
equation of the slant
2 -2
asymptote: y=x-1
1 -1 -6
•
Domain: all real #’s except 2
Range: all real #’s
Assignment