Factor the following completely:

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Presentation transcript:

Factor the following completely: 3x2-8x+4 11x2-99 16x3+128 4. x3+2x2-4x-8 2x2-x-15 10x3-80 (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)

9.3 Graphing General Rational Functions

Objectives/Assignment Graph general rational functions Use the graph of a rational function to solve problems Assignment: 13-37 odd

Last class, 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 rational function when x is not to the 1st power Find the x-intercepts. (Set numerator =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. 4. x y -2 -.4 -1 -.5 0 0 1 .5 2 .4 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 (No real solns.)

Domain: all real numbers 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.

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

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 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

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