9.3 Graphing General Rational Functions

Steps to graphing rational functions 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. Graph asymptotes, pts., and connect with curves. 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: 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.

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 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. Graph asymptotes, pts., and connect with curves. Check your graph with the calculator.

Ex: Graph. State domain & range. 5. Function doesn’t simplify so NO HOLES! 2. x-intercepts: x=0 3. vert. asymp.: x2+1=0 x2= -1 No vert asymp 4. horiz. asymp: 1<2 (deg. top < deg. bottom) y=0 6. x y -2 -.4 -1 -.5 0 0 1 .5 2 .4 (No real solns.)

Domain: all real numbers Range:

Ex: Graph 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 2. x-intercepts: 3x2=0 x2=0 x=0 3. Vert asymp: x2-4=0 x2=4 x=2 & x=-2 4. Horiz asymp: (degrees are =) y=3/1 or y=3 On right of x=2 asymp. Between the 2 asymp. On left of x=-2 asymp. 5. Nothing cancels so NO HOLES!

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

Ex: Graph, then state the domain & range. y-intercept: -2 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! 5. Nothing cancels so no holes. 6. 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

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