1. Section 5.6 - Rational Functions

2. Rational Functions
A rational function is a function of the form 𝑝(𝑥) 𝑞(𝑥) where 𝑝(𝑥) and 𝑞(𝑥) are polynomial functions and 𝑞(𝑥)≠0.
The domain of the rational function is the set of all inputs for which 𝑞(𝑥)≠0.

3. Graph Features Vertical asymptotes Holes Horizontal Asymptotes
Slant Asymptotes

4. Vertical Asymptotes and Holes
The vertical line 𝑥=𝑎 is a vertical asymptote for the graph of 𝑓 if any of the following is true:
𝑓(𝑥)→∞ as 𝑥→ 𝑎 − , or 𝑓 𝑥 →−∞ as 𝑥→ 𝑎 − , or
𝑓(𝑥)→∞ as 𝑥→ 𝑎 + , or 𝑓 𝑥 →−∞ as 𝑥→ 𝑎 +
To find Vertical Asymptotes and Holes:
Find the domain of the function
Factor the numerator and denominator.
Factors in the denominator that do NOT cancel give vertical asymptotes.
Factors in the denominator that do cancel give holes.

5. Vertical Asymptotes
Determine the vertical asymptotes for the graph of the function.
Vertical Asymptote: 𝒙=−𝟏𝟎

6. Vertical Asymptotes Hole at 𝒙=𝟑
Determine vertical asymptotes , if they exist.
Hole at 𝒙=𝟑

7. Vertical Asymptotes 𝒙=𝟎, 𝒙=−𝟓, 𝒙=𝟐
Determine the vertical asymptotes for the graph of the function.
Vertical Asymptote:
𝒙=𝟎, 𝒙=−𝟓, 𝒙=𝟐

8. Horizontal Asymptotes
The horizontal line 𝑦=𝑏 is a horizontal asymptote for the graph of 𝑓 if either or both of the following are true:
𝑓(𝑥)→𝑏 as 𝑥→∞, or 𝑓 𝑥 →𝑏 as 𝑥→−∞
Determining a Horizontal Asymptote:
Degree on top < the degree on bottom: 𝒚=𝟎
Degree on top = the degree on bottom: 𝒚= 𝒍𝒆𝒂𝒅𝒊𝒏𝒈 𝒄𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒍𝒆𝒂𝒅𝒊𝒏𝒈 𝒄𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕
Degree on top > the degree on bottom: No horizontal asymptote

9. Horizontal Asymptotes
Determine the horizontal asymptote for the graph of the function.
Since the degree on top is less than the degree on bottom, we have a horizontal asymptote at 𝒚=𝟎

10. Horizontal Asymptotes
Determine the horizontal asymptote for the graph of the function.
Since the degree on top = the degree on bottom, we have a horizontal asymptote at 𝑦= 1 1 or 𝒚=𝟏
***Note that the software does not indicate the hole in the graph at 𝒙=𝟎

11. Horizontal Asymptotes
Determine the horizontal asymptote for the graph of the function.
Since the degree on top = the degree on bottom, we have a horizontal asymptote at 𝑦= 8 2 or 𝒚=𝟒

12. Slant Asymptotes
Sometimes a line that is not horizontal is as asymptote. Such a line is called an oblique asymptote or slant asymptote.
To find a Slant Asymptote (In the reduced function, if the degree on top is exactly one more than the degree on bottom) :
Perform polynomial division.
The quotient, without the remainder will give the slant asymptote.
Horizontal asymptotes and slant asymptotes can NOT occur together!

13. Slant Asymptotes
Determine the oblique asymptote for the graph of the function.
Since the degree on top is exactly one more than the degree on bottom, we have a slant asymptote.
Slant Asymptote: 𝒚=𝒙 −𝟏

14. Graphs of Rational Functions
Vertical Asymptotes: These occur at x-values that are not in the domain of the function and not in the domain of the reduced function.
Holes: These occur at x-values that are not in the domain of the function, but are in the domain of the reduced function.
Horizontal Asymptotes: 𝑦=0; 𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟<𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑦= 𝑎 𝑏 ; 𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑛𝑢𝑚𝑏𝑒𝑟𝑎𝑡𝑜𝑟=𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑁𝑜𝑛𝑒; 𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟>𝐷𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟
Oblique or Slant Asymptote: These occur when the degree of the numerator is equal to the degree of the denominator plus 1 (in the reduced function).
The graph will NEVER cross a vertical asymptote. It may cross other types!

15. Graph Can Cross Horizontal and Slant Asymptotes
8. 𝑓 𝑥 = (𝑥−3) 2 𝑥 2 −5𝑥
𝑓 𝑥 = (𝑥−3)(𝑥−3) 𝑥(𝑥−5)
Domain: 𝑥≠0, 𝑥 ≠5
Vertical Asymptotes: 𝑥=0, 𝑥=5
Holes: None
Horizontal Asymptote: 𝑦=1
Slant Asymptote: None
If you would like to find where the graph crosses the horizontal asymptote, you can set the function equal to 1 and solve!

16. Graph Rational Function
To find the x-intercept, let y = 0.
Domain:
Vert. Asym:
Holes:
Hor. Asym:
Slant Asym:
To find the y-intercept, let x = 0.

17. Graph Rational Functionx
g(x) -2 2
Domain:
Vert. Asym:
Slant Asym: