1. Section 8.4 – Graphing Rational Functions
EQ: How do I graph a rational function using the vertical and horizontal asymptotes?

2. A Rational Function
An equation of the form𝑓 𝑥 = 𝑎(𝑥) 𝑏(𝑥) , where a(x) and b(x) are polynomial functions and b(x) ≠ 0

3. Vertical and Horizontal Asymptotes
Vertical Asymptotes
Whenever b(x) = 0
Horizontal Asymptotes (at most one)
If the degree of 𝑎 𝑥 is greater than the degree of 𝑏 𝑥 , there is no horizontal asymptote
If the degree of 𝑎 𝑥 is less than the degree of 𝑏 𝑥 , the horizontal asymptote is the line y = 0
If the degree of 𝑎 𝑥 equals the degree of 𝑏 𝑥 , the horizontal asymptote is the line 𝑦= 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑎(𝑥) 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑏(𝑥)

4. Graph by finding the following
Vertical Asymptote(s), if any: Set the denominator = 0 and solve
Horizontal Asymptote(s), if any. Compare the degree of the numerator (m) to the degree of the denominator (n)
m < n: HA is y = 0
m = n: HA is (LC of numerator/LC of denominator)
m > n: HA does not exist
y – intercept: Let x = 0 and solve
x – intercept(s): Set the numerator = 0 and solve for x
Choose other x – values, 2 on each side of an asymptote

5. Example 1 What are the asymptotes of the function 𝑓 𝑥 = 𝑥−3 𝑥−2 ?
Vertical Asymptote: Set the denominator = 0 and solve.
VA: x = 2
Horizontal Asymptote: compare the degree of the numberator (m) to the degree of the denominator (n).
HA: 1𝑥 1 1𝑥 1 = 1 1 →𝑦=1

6. Example 2 What are the asymptotes of the function 𝑓 𝑥 = 2𝑥−7 𝑥+4 ?
Vertical Asymptote: Set the denominator = 0 and solve.
VA: x = -4
Horizontal Asymptote: compare the degree of the numberator (m) to the degree of the denominator (n).
HA: 2𝑥 1 1𝑥 1 = 2 1 →𝑦=2

7. Example 3
Graph the function, then state the domain and range and any asymptote equations. a) 𝑓 𝑥 = 𝑥 𝑥 2 −9
VA: Set the denominator = 0 and solve.
𝑥 2 −9=(𝑥−3)(𝑥+3)
𝑥=−3, 3
HA: 𝑥 𝑥 2 →𝑚<𝑛→𝑦=0
*Use calculator to find other points, then connect the points
Domain: all real, 𝑥≠−3, 3
Range: all real, 𝑦≠0

8. Example 3
Graph the function, then state the domain and range and any asymptote equations. b) 𝑓 𝑥 = 𝑥 2 𝑥 2 −1
VA: Set the denominator = 0 and solve.
𝑥 2 −1=(𝑥−1)(𝑥+1)
𝑥=−1, 1
HA: 𝑥 2 𝑥 2 →𝑚=𝑛→𝑦=1
*Use calculator to find other points, then connect the points
Domain: all real, 𝑥≠−1, 1
Range: all real, 𝑦≠1

9. Point Discontinuity
A point where a graph is undefined, looks like a hole in the graph

10. Point of discontinuity is at x = 3
Example 4
What is the point of discontinuity of the following functions? a) 𝑥 2 −9 𝑥−3
(𝑥+3)(𝑥−3) 𝑥−3
Point of discontinuity is at x = 3

11. Point of discontinuity is at x = -1
Example 4
What is the point of discontinuity of the following functions? b) 2 𝑥 2 +2 𝑥+1
2(𝑥+1)(𝑥−1) 𝑥+1
Point of discontinuity is at x = -1

12. Point of discontinuity is at x = -5
Example 4
What is the point of discontinuity of the following functions? c) 𝑥 2 +4𝑥−5 𝑥+5
(𝑥+5)(𝑥−1) 𝑥+5
Point of discontinuity is at x = -5

13. Point of discontinuity is at x = 2
Example 5
Graph 𝑓 𝑥 = 𝑥 2 −4 𝑥−2
Hole at x = 2
(𝑥+2)(𝑥−2) 𝑥−2
Point of discontinuity is at x = 2
𝑓 𝑥 =𝑥+2

14. Point of discontinuity is at x = -4
Example 6
Graph 𝑓 𝑥 = 𝑥 2 −16 𝑥+4
(𝑥+4)(𝑥−4) 𝑥+4
Point of discontinuity is at x = -4
𝑓 𝑥 =𝑥−4
Hole at x = -4