3.3 Properties of Rational Functions
DAY 2
Objective ­ Find the vertical, horizontal and/or oblique
asymptotes of rational functions.
Title: Nov 11­7:38 PM (1 of 12)
Although, the asymptotes of a function are not part of the graph of the function, they provide information about how the graph looks.
F(x) = 5
x + 3
+ 4
horizontal asymptote
vertical asymptote
Title: Nov 11­7:39 PM (2 of 12)
A vertical asymptote, describes a certain behavior of the graph when it is close to some number. The graph of the function will never intersect a vertical asymptote.
R(x) = 1
+ 1
(x ­ 2)2
vertical asymptote
Title: Nov 11­7:42 PM (3 of 12)
Find the vertical asymptote: (set the denominator equal to zero)
x
1.
R(x) = 2.
F(x) = 3.
G(x) = x2 ­ 16
x = ­4 and x = 4
x + 1
x ­ 5
x2 ­ 9
x2 + 4x ­ 21
Title: Nov 11­9:23 PM (4 of 12)
x = 5
x = ­7 (not x = 3)
A horizontal asymptote, describes a certain end behavior. The graph of a function may intersect a horizontal asymptote.
R(x) = 1
+ 1
(x ­ 2)2
Title: Nov 11­7:41 PM (5 of 12)
An oblique asymptote is neither horizontal nor vertical. An oblique asymptote, describes the end behavior of the graph. The graph of a function may intersect an oblique asymptote.
x3
F(x) = x2 + 3x
Title: Nov 11­7:42 PM (6 of 12)
Rules for finding Horizontal & Oblique Asymptotes:
Consider rational function: R(x) =
p(x)
anxn + an­1xn­1 + ... + a 1x + a0
q(x)
bmxm + bm­1
xm­1
+ ... + b1x + b0
= in which the degree of the numerator is n and the degree of the denominator is m.
1.
If n < m, then R is a proper fraction and will have the horizontal asymptote y = 0.
x ­ 12
Example: R(x) = 4x2 + x + 1
Title: Nov 11­9:31 PM (7 of 12)
x 1
~
=
~
4x2 4x
R(x) can never equal 0
2.
If n > m, then R is improper and long division is used.
an
(a) If n = m, the quotient obtained will be a number
, and the bm
an
line y = is a horizontal asymptote.
bm
Example: F(x) = 3x + 2
HA: y = 3
x ­ 5
(b) If n = m + 1, the quotient obtained is of the form ax + b (a polynomial of degree 1), and the line y = ax + b is an oblique asymptote.
Example: G(x) = 3x4 ­ x2
Use long division to find OA: y = 3x + 3
x3 ­ x2 + 1
(c) If n > m + 1, the quotient obtained is a polynomial of degree 2 or higher and R has neither a horizontal nor an oblique asymptote.
Example: H(x) = Title: Nov 11­10:03 PM (8 of 12)
2x5 ­ x2
x2 + 1
No horizontal or oblique asymptote.
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Title: Nov 11­10:06 PM (9 of 12)
Find the horizontal asymptote:
1.
F(x) = 2.
G(x) = x
3x2 + 1
4x + 3
2x ­ 5
Title: Nov 11­9:22 PM (10 of 12)
y = 0
y = 2
Find the oblique asymptote:
1.
F(x) = 3x4 + 5x3 + 4
x3 + 3x
Title: Nov 11­9:23 PM (11 of 12)
y = 3x + 5
Homework: page 197 (41 ­ 52) all
Title: Snowman scene (12 of 12)