1. Vertical Asymptote
x=4
Hole Horizontal Asymptote
At x = -2
y=1

2. Graphing rational functions
1. Factor numerator and denominator
2. Discontinuities occur where denominator equals
zero
• Holes are indicated by factors of denominator
that cancel out
• Vertical asymptotes are indicated by factors of
denominator that do not cancel

3. 3. Horizontal Asymptotes (predict end behavior) – look at
highest degree term of numerator and denominator
• Degree of top > degree of bottom: no HA
• Degree of top < degree of bottom: HA is y = 0
• Degrees equal: HA is y =

4. Determine the equations of any asymptotes and the
values of x for any holes in the graph of
Undefined for x = –2 and –3
Hole at x = -2
Vertical asymptote is the line x = -3
Horizontal asymptote:
the line y = 1

5. Graph
VA: x = -3
HA: y = 1
Hole at x = -2 (-2, -4)
-2
= -4
-2

6. Graph
x
-6 -8/-3 or 8/3
-5 -7/-2 or 7/2
-4 -6/-1 or 6
-1 -3/2
0 -2/3
1 -1/4

7. Graph
VA: x = –1
No holes
HA: y = 1

8. Graph
x f (x)
–4 -4/-3 = 4/3
–3 -3/-2 = 3/2
–2 -2/-1 = 2
0 0/1 = 0
1 1/2
2 2/3
3 3/4

9. Graph
Undefined for x = 2
Hole at x = 2 (2, 4)
2
x
-1 1
0 2
1 3