Factor the following completely:Factor the following completely:
1. 3x2-8x+4
2. 11x2-99
3. 16x3+128
4. x3+2x2-4x-8
5. 2x2-x-15
6. 10x3-80
(3x-2)(x-2)(3x-2)(x-2)
11(x+3)(x-3)11(x+3)(x-3)
16(x+2)(x16(x+2)(x22-2x+4)-2x+4)
(x-2)(x+2)(x-2)(x+2)22
(2x+5)(x-3)(2x+5)(x-3)
10(x-2)(x10(x-2)(x22+2x+4)+2x+4)
9.3 9.3 Graphing General Rational Graphing General Rational
FunctionsFunctions
By: L. Keali’i AliceaBy: L. Keali’i Alicea
In the past, we graphed rational In the past, we graphed rational functions where x was to the first functions where x was to the first
power only. power only. What if x is not to the first power?What if x is not to the first power?
Such as: Such as:
1)(
2 x
xxf
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.a. If degree of top < degree of bottom, y=0If degree of top < degree of bottom, y=0b. If degrees are =, 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.
6.Check solutions on calculator.
bottom of coeff. lead.
topof coeff. lead.y
Ex: Graph. State domain & range.Ex: Graph. State domain & range.
1. x-intercepts: x=02.2. vert. asymp.: xvert. asymp.: x22+1=0+1=0 x2= -1No vert asymp
3. horiz. asymp: 1<2
(deg. of top < deg. of bottom)y=0
12 x
xy
1x
4. x y
-2 -.4
-1 -.5
0 0
1 .5
2 .4
(No real solns.)
Domain: all real numbers
Range:
2
1
2
1
y
Ex: Graph, then state the domain and range.
1. x-intercepts:
3x2=0
x2=0
x=0x=0
2. Vert asymp:
x2-4=0
x2=4
x=2 & x=-2
3. Horiz asymp:
(degrees are =)
y=3/1 or y=3
4
32
2
x
xy
4. x y
4 4
3 5.4
1 -1
0 0
-1 -1
-3 5.4
-4 4On left of x=-2
asymp.
Between the 2 asymp.
On right of x=2 asymp.
Domain: all real #’s except -2 & 2
Range: all real #’s except 0<y<3
Ex: Graph, then state the domain & range.
1. x-intercepts:
x2-3x-4=0
(x-4)(x+1)=0
x-4=0 x+1=0x-4=0 x+1=0
x=4 x=-1
2. Vert asymp:
x-2=0
x=2
3. Horiz asymp: 2>1
(deg. of top > deg. of bottom)
no horizontal asymptotes, but there is a slant!
2
432
x
xxy
4. x y
-1 0
0 2
1 6
3 -4
4 0
Left of x=2 Left of x=2 asymp.asymp.
Right of Right of x=2 x=2
asymp.asymp.
Slant asymptotesSlant asymptotes• Do synthetic division (if possible); if not, do long Do synthetic division (if possible); if not, do long
division!division!
• The resulting polynomial (ignoring the The resulting polynomial (ignoring the remainder) is the equation of the slant remainder) is the equation of the slant asymptote.asymptote.
In our example:In our example:
2 1 -3 -42 1 -3 -4
1 -1 -61 -1 -6
2 -22 -2
Ignore the remainder, Ignore the remainder, use what is left for the use what is left for the equation of the slant equation of the slant asymptote: y=x-1asymptote: y=x-1
Domain: all real #’s except 2
Range: all real #’s
Assignment