Factor the following completely: 1.3x 2 -8x x x x 3 +2x 2 -4x-8 5.2x 2 -x x (3x-2)(x-2) 11(x+3)(x-3) 16(x+2)(x 2 -2x+4) (x-2)(x+2) 2 (2x+5)(x-3) 10(x-2)(x 2 +2x+4)

9.3 Graphing General Rational Functions p. 547 What information can you get from the equation of a rational graph? What is local minimum?

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 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.If 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.

Ex: Graph. State domain & range. 1.x-intercepts: x=0 2.vert. asymp.: x 2 +1=0 x 2 = -1 No vert asymp 3.horiz. asymp: 1<2 (deg. of top < deg. of bottom) y=0 4. x y (No real solns.)

Domain: all real numbers Range:

Ex: Graph, then state the domain and range. 1.x-intercepts: 3x 2 =0 x 2 =0 x=0 2.Vert asymp: x 2 -4=0 x 2 =4 x=2 & x=-2 3.Horiz asymp: (degrees are =) y= 3 / 1 or y=3 4. x y On 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: x 2 -3x-4=0 (x-4)(x+1)=0 x-4=0 x+1=0 x-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! 4. x y Left of x=2 asymp. Right of x=2 asymp.

Slant asymptotes Do synthetic division (if possible); if not, do long division!Do synthetic division (if possible); if not, do long division! The resulting polynomial (ignoring the remainder) is the equation of the slant asymptote.The resulting polynomial (ignoring the remainder) is the equation of the slant asymptote. In our example: Ignore the remainder, use what is left for the equation of the slant asymptote: y=x-1

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

What information can you get from the equation of a rational graph?What information can you get from the equation of a rational graph? X-intercept from setting the numerator=0 Vertical asymptotes from setting the denominator = 0 If m<n, the horizontal asymptote is y = 0 If m=n, the horizontal asymptote is y =coefficient of numerator/coefficient of denominator If m>n, there is no horizontal asymptote What is local minimum?What is local minimum? Tells where the most efficient solution is in application problems.

Assignment p odd,35