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3x 2 -8x+4 11x 2 -99 16x 3 +128. 4. x 3 +2x 2 -4x-8 2x 2 -x-15 10x 3 -80. Factor the following completely:. (3x-2)(x-2). (x-2)(x+2) 2. (2x+5)(x-3). 11(x+3)(x-3). 16(x+2)(x 2 -2x+4). 10(x-2)(x 2 +2x+4). 9.3 Graphing General Rational Functions. p. 547.

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factor the following completely
3x2-8x+4

11x2-99

16x3+128

4. x3+2x2-4x-8

2x2-x-15

10x3-80

Factor the following completely:

(3x-2)(x-2)

(x-2)(x+2)2

(2x+5)(x-3)

11(x+3)(x-3)

16(x+2)(x2-2x+4)

10(x-2)(x2+2x+4)

slide3
Yesterday, we graphed rational functions where x was to the first power only. What if x is not to the first power?

Such as:

steps to graph when x is not to the 1 st power
Steps to graph when x is not to the 1st power
  • Find the x-intercepts. (Set numer. =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.
ex graph state domain range
Ex: Graph. State domain & range.
  • 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

4. x y

-2 -.4

-1 -.5

0 0

1 .5

2 .4

(No real solns.)

ex graph then state the domain and range
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
  • 4 4
  • 3 5.4
  • 1 -1
  • 0 0
  • -1 -1
  • -3 5.4
  • -4 4

On right of x=2 asymp.

Between the 2 asymp.

On left of x=-2 asymp.

slide8

Domain: all real #’s except -2 & 2

Range: all real #’s except 0<y<3

ex graph then state the domain range
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
  • -1 0
  • 0 2
  • 1 6
  • 3 -4
  • 4 0

Left of x=2 asymp.

Right of x=2 asymp.

slant asymptotes
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:

2 1 -3 -4

1 -1 -6

Ignore the remainder, use what is left for the equation of the slant asymptote: y=x-1

2 -2

slide12

Domain: all real #’s except 2

Range: all real #’s