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Factor the following completely:

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. By: L. Keali’i Alicea.

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Factor the following completely:

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  1. 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)

  2. 9.3 Graphing General Rational Functions By: L. Keali’i Alicea

  3. In the past, we graphed rational functions where x was to the first power only. What if x is not to the first power? Such as:

  4. 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.

  5. 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.)

  6. Domain: all real numbers Range:

  7. 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.

  8. Domain: all real #’s except -2 & 2 Range: all real #’s except 0<y<3

  9. 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.

  10. 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

  11. Domain: all real #’s except 2 Range: all real #’s

  12. Assignment

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