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)

Graphing General Rational Graphing General Rational FunctionsFunctions

Yesterday, we graphed rational Yesterday, 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=0

b. 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