Graphing General Rational Functions Yesterday we graphed rational

Graphing General Rational Functions

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 (No real solns. ) 4. x y -2 -. 4 -1 -. 5 0 0 1 . 5 3. horiz. asymp: 2 1<2 (deg. of top < deg. of bottom) y=0 . 4

Domain: all real numbers Range:

Ex: Graph, then state the domain and range. 1. x-intercepts: 4. x 3 x 2=0 4 x 2=0 x=0 3 2. Vert asymp: 1 x 2 -4=0 0 x 2=4 -1 x=2 & x=-2 -3 3. Horiz asymp: -4 (degrees are =) y=3/1 or y=3 y 4 5. 4 On right of x=2 asymp. -1 0 Between the 2 asymp. -1 5. 4 4 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. 1. x-intercepts: x 2 -3 x-4=0 4. x y (x-4)(x+1)=0 -1 0 x-4=0 x+1=0 Left of x=2 x=4 x=-1 0 2 asymp. 2. Vert asymp: 1 6 x-2=0 3 -4 Right of x=2 asymp. 4 0 3. Horiz asymp: 2>1 (deg. of top > deg. of bottom) no horizontal asymptotes, but there is a slant!

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. Ignore the remainder, In our example: use what is left for the 2 1 -3 -4 equation of the slant 2 -2 asymptote: y=x-1 1 -1 -6

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