Chapter 2
Polynomial and Rational Functions
2.6 Rational Functions and
Asymptotes
Objectives:
Find the domains of rational functions.
Find the horizontal and vertical asymptotes of
graphs of rational functions.
Use rational functions to model and solve real-life
problems.
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Rational Functions
A rational function can be written in the form
N ( x)
f ( x) 
D( x)
where N(x) and D(x) are polynomials.
A rational function is not defined at values of x for
which D(x) = 0.
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Reciprocal Function
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Asymptotes
An asymptote is a boundary line that the graph of a
function approaches, but never touches or crosses.
The line x = a is a vertical asymptote of the graph of
f if, as x approaches a from either the left or the right,
f (x) approaches ∞ or –∞.
The line y = b is a horizontal asymptote of the
graph of f if, as x approaches ∞ or –∞, f (x)
approaches b.
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Examples
The following graphs show horizontal and vertical
asymptotes of two rational functions.
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Finding Asymptotes
Let f be a rational function:
Vertical Asymptotes:
Occur when the denominator equals zero.
Simplify the function if possible.
Set D(x) = 0 and solve for x.
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Horizontal Asymptotes
The graph of f has at most one horizontal asymptote
determined by comparing the degrees of N(x) and D(x).
Let n be the degree of the numerator and m be the
degree of the denominator.
Let an be the leading coefficient of the numerator and
bn be the leading coefficient of the denominator.
If n < m
If n = m
If n > m
HA: y = 0
HA: y = an/bn
No HA
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Examples
Find all HA and VA of each rational function.
1.
2x
f ( x)  2
3x  1
2x2
2. g ( x)  2
x 1
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Example
Find all HA and VA of the rational function.
x2  x  2
f ( x)  2
x  x6
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For a person with sensitive skin, the amount of time T,
in hours, the person can be exposed to the sun with a
minimal burning can be modeled by
0.37 s  23.8
T
, 0  s  120
s
where s is the Sunsor Scale reading (based on the level
of intensity of UVB rays).
a. Find the amount of time a person with sensitive skin
can be exposed to the sun with minimal burning
when s = 10, s = 25, and s = 100.
b. If the model were valid for all s > 0, what would be
the horizontal asymptote of this function, and what
would it represent?
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Homework 2.6
Worksheet 2.6
# 7 – 12 (matching), 15, 17, 19, 35, 39
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