# Rational Expressions, Vertical Asymptotes, and Holes

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OBJECTIVE:
Find Asymptotes and Holes
RELEVANCE:
Learn how to evaluate data from real-world applications that fit into a quadratic model.
1. Warm Up
2. Rational Expressions,
Vertical Asymptotes,
and Holes
3. Objective
Asymptotes and
4. Relevance
Learn how to evaluate
data from real world
applications that fit
model.
5. Rational Expression
 It is the quotient of two polynomials.
 A rational function is a function defined by a
rational expression.
x 2 3x 2  2 x  5
y y 3
x 5 x  4 x2  5x  7
Not Rational:
4x x
y y 2
x2 x 5
6. 1
Find the domain: f ( x) 
x
Graph it:
7. 1
Find the domain: f ( x) 
x
Graph it:
8. 1
Find the domain: f ( x) 
x2
Graph it:
9. 1
Find the domain: f ( x) 
x2
Graph it:
10. Vertical Asymptote
 Ifx – a is a factor of the denominator of a
rational function but not a factor of the
numerator, then x = a is a vertical asymptote
of the graph of the function.
11. x 3
Find the domain: f ( x)  2
x  x  12
Graph it using the
graphing calculator.
What do you see?
12. x 3
Find the domain: f ( x)  2
x  x  12
Graph it using the
graphing calculator.
What do you see?
13. Hole (in the graph)
 Ifx – b is a factor of both the numerator and
denominator of a rational function, then there
is a hole in the graph of the function where
x = b, unless x = b is a vertical asymptote.
 The exact point of the hole can be found by
plugging b into the function after it has been
simplified.
14. Find the domain and identify
vertical asymptotes & holes.
x 1
f ( x)  2
x  2x  3
15. Find the domain and identify
vertical asymptotes & holes.
x 1
f ( x)  2
x  2x  3
16. Find the domain and identify
vertical asymptotes & holes.
x
f ( x)  2
x 4
17. Find the domain and identify
vertical asymptotes & holes.
x
f ( x)  2
x 4
18. Find the domain and identify
vertical asymptotes & holes.
x 5
f ( x)  2
2x  x  3
19. Find the domain and identify
vertical asymptotes & holes.
x 5
f ( x)  2
2x  x  3
20. Find the domain and identify
vertical asymptotes & holes.
3  2x  x2
f ( x)  2
x x 2
21. Find the domain and identify
vertical asymptotes & holes.
3  2x  x2
f ( x)  2
x x 2
22. Horizontal Asymptotes
& Graphing
23. Horizontal Asymptotes
 Degree of numerator = Degree of denominator
coefficient of numerator
Horizontal Asymptote: y
coefficient of denominator
 Degree of numerator < Degree of denominator
Horizontal Asymptote: y 0
 Degree of numerator > Degree of denominator
Horizontal Asymptote: none
24. Find all asymptotes & holes & then graph:
x 2
f ( x) 
2x  3
25. Find all asymptotes & holes & then graph:
x 2
f ( x) 
2x  3
26. Find all asymptotes & holes & then graph:
x 1
f ( x)  2
x  2x  3
27. Find all asymptotes & holes & then graph:
x 1
f ( x)  2
x  2x  3
28. Find all asymptotes & holes & then graph:
x 2  3x  4
f ( x) 
x 4
29. Find all asymptotes & holes & then graph:
x 2  3x  4
f ( x) 
x 4
30. Find all asymptotes & holes & then graph:
3x 2  7 x  6
f ( x) 
x2  9
31. Find all asymptotes & holes & then graph:
3x 2  7 x  6
f ( x) 
x2  9
32. Find all asymptotes & holes & then graph:
2
 x  3
f ( x) 
x2
33. Find all asymptotes & holes & then graph:
2
 x  3
f ( x) 
x2
34. Find all asymptotes & holes & then graph:
3x 2
f ( x)  2
2x  5
35. Find all asymptotes & holes & then graph:
3x 2
f ( x)  2
2x  5
36. Find all asymptotes & holes & then graph:
x
f ( x)  2
x  2x  3
37. Find all asymptotes & holes & then graph:
x
f ( x)  2
x  2x  3
38. Slant Asymptotes (Oblique)
 Find a slant asymptote when the H.A.
DNE.
 Occurs when the degree of the top is
exactly 1 more than the degree of the
bottom.
 Tofind the slant asymptote, we divide them
and the answer is the asymptote.
39. Slant Asymptote
If the degree of the numerator is exactly one more than the degree of
the denominator, then the graph of the function has a slant (or oblique)
asymptote.
40. Find the oblique (slant)
asymptote: x2  4
f ( x) 
x 1
x 1
2
x  1 x  0x  4
The line x + 1 is an oblique asymptote
41. Find the slant asymptote: x2  9
f ( x) 
x 1
x 1
2
x 1 x  0 x  9
The line x - 1 is an oblique asymptote
42. 3
Find the slant asymptote: f  x   x  4 x
2
x  3x  2
x 3
2 3 2
x  3x  2 x  0 x  4 x
The line x +3 is an oblique asymptote
43. Slant Asymptote
Sketch the graph:
44. Slant Asymptote
Sketch the graph:
45. CW: Polynomial Review
HW: Textbook p.148 (17 – 23) Odd and
(37 – 41) Odd
Textbook p. 157 (18, 22, 24, 32, 52, 54)