1. 9.3 Graphing General Rational Functions

2. 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.

3. 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

4. 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

5. 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.

6. 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.

7. 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.

8. 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.

9. 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
(No real solns.)

10. Domain: all real numbers
Range:

11. Ex: Graph then state the domain and range.
6. x y
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!

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

13. 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
Left of x=2 asymp.
Right of x=2 asymp.

15. 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:
Ignore the remainder, use what is left for the equation of the slant asymptote: y=x-1

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

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