Section
2.6: Rational Functions and Asymptotes
A rational number is a number that can be written in a ratio.
A rational function is a function that can be written as:
In this, N and D are both polynomials.
The domain of f is all numbers that do not make the denominator equal to 0.
Example 1.
Find the domain of
Solution: Factor the denominator. (x+3)(x-3)
Solution: Factor the denominator. (x+3)(x-3)
Solve to find values of x that make the
denominator = 0, x=3,-3
Domain: all real numbers except -3 and 3,
Horizontal
and Vertical Asymptotes
Definitions of Asymptotes
1. A graph has a vertical asymptote if D(x)=0. This
means that all values that are not in the domain are vertical asymptotes.
2. A horizontal line is a horizontal
asymptote of
the graph if the graph approaches the line as x approaches positive or negative
infinity.
The graph will never cross a
vertical asymptote, but it may cross
a horizontal asymptote
Rules for
Asymptotes of Rational Functions
Let f be a rational
function:
1. The graph of f has a vertical asymptote at x = a if D(a) = 0. A vertical asymptote occurs at the value(s)
of x that make the denominator equal to 0, which makes the function undefined.
2. The graph of f has one horizontal asymptote or no horizontal asymptote, depending
on the degree of N and D.
a.
If the degree of the
numerator (n) is less than the degree of the
denominator (m),
then
the horizontal asymptote is y=0.
b.
If the degree of the numerator
(n) is equal to the degree of the denominator
(m), then the
horizontal asymptote is quotient of the leading coefficients.
c.
If the degree of the numerator (n)
is greater than the degree of the denominator (m), then there is
no horizontal asymptote.
3. The graph of f has a slant asymptote if the degree of the
numerator (n) is 1 greater the degree of the denominator (m).
Example 2.
Find any horizontal and vertical
asymptotes of the following.
Solution: The degree of the numerator is 3, the degree
of the denominator is 1. Since the
degree of the numerator is greater than the degree of the denominator, there is
no horizontal asymptote. The vertical asymptote is -8 because -8 makes the
denominator equal to 0.
Solution: The degree of the numerator is 1 and the
degree of the denominator is 1. Since
the degree of the numerator is equal to than the degree of the denominator, the
horizontal asymptote is the quotient of the leading coefficient of the
numerator and the denominator, so it is:
Set the denominator
equal to zero and solve for x.
4x-6=0, so the vertical asymptote is:
Solution: The degree of the numerator is 1 and the
degree of the denominator is 2. Since
the degree of the numerator is less than the degree of the denominator, the horizontal
asymptote is y=0
Set the denominator equal to zero and
solve for x.
X2-1=0
X=1,-1, so the
vertical asymptotes are x=1 and x=-1
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