9 3 Graphing General Rational Functions p 547

9. 3 Graphing General Rational Functions p. 547

Steps to graphing rational functions 1. 2. 3. 4. 5. 6. Find the y-intercept. Find the x-intercepts. Find vertical asymptote(s). Find horizontal asymptote. Find any holes in the function. Make a T-chart: choose x-values on either side & between all vertical asymptotes. 7. Graph asymptotes, pts. , and connect with curves. 8. Check your graph with the calculator.

How to find the intercepts: § y-intercept: – Set the y-value equal to zero and solve § x-intercept: – Set the x-value equal to zero and solve

How to find the vertical asymptotes: § A vertical asymptote is vertical line that the graph can not pass through. Therefore, it is the value of x that the graph can not equal. The vertical asymptote is the restriction of the denominator! § Set the denominator equal to zero and solve

How to find the Horizontal Asymptotes: 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.

How to find 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.

How to find the points of discontinuity (holes): § When simplifying the function, if you cancel a polynomial from the numerator and denominator, then you have a hole! § Set the cancelled factor equal to zero and solve.

Steps to graphing rational functions 1. 2. 3. 4. 5. 6. Find the y-intercept. Find the x-intercepts. Find vertical asymptote(s). Find horizontal asymptote. Find any holes in the function. Make a T-chart: choose x-values on either side & between all vertical asymptotes. 7. Graph asymptotes, pts. , and connect with curves. 8. Check your graph with the calculator.

Ex: Graph. State domain & range. 2. x-intercepts: x=0 3. vert. asymp. : x 2+1=0 x 2= -1 5. Function doesn’t simplify so NO HOLES! No vert asymp (No real solns. ) 4. horiz. asymp: 1<2 (deg. top < deg. bottom) y=0 6. x y -2 -. 4 -1 -. 5 0 0 1 . 5 2 . 4

Domain: all real numbers Range:

Ex: Graph 2. 3. 4. x-intercepts: 3 x 2=0 x=0 Vert asymp: x 2 -4=0 x 2=4 x=2 & x=-2 Horiz asymp: (degrees are =) y=3/1 or y=3 then state the domain and range. 6. x y 4 4 3 5. 4 1 -1 0 0 -1 -1 -3 5. 4 -4 4 5. Nothing cancels so NO HOLES! 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. 1. y-intercept: -2 5. Nothing cancels 2. x-intercepts: so no holes. 2 x -3 x-4=0 6. x y (x-4)(x+1)=0 x-4=0 x+1=0 -1 0 Left of x=2 asymp. x=4 x=-1 0 2 2. Vert asymp: 1 6 x-2=0 Right of x=2 3 -4 x=2 asymp. 3. Horiz asymp: 2>1 0 (deg. of top > deg. of bottom) 4 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

Assignment Workbook page 61 #1 -9 Find each piece of the function.