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Material Type: Notes; Class: PRECALCULUS ALGEBRA; Subject: MATHEMATICS - CALCULUS AND PRECALCULUS; University: Florida State University; Term: Fall 2007;
Typology: Study notes
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( )
qx
p x f x = , where p(x) and q(x) are two polynomials.
So if the q(x) does not have a zero, then the domain is all real numbers.
Exercise 1
Exercise 2
3. Recall Inverse Variation:
x
f x
Note that the lines never cross the axes -- they get closer and closer to x = 0 and y =
0, but x and y never equal zero. (See the supplementary material for more details.)
The function blows up, shooting vertically up or down, near the zeros of the denominator,
just as inverse variation function does. For example, if ( )
q x
px f x = and r is a zero of
q(x), then the graph of the function would approach (upward or downward) the vertical line
x=r, but never touches it. Such vertical line is called vertical asymptote.
More formally, the line of x=r is a vertical asymptote of the graph of f(x) if f ( x )→∞
as x → r.
Remark: A rational function could have many vertical asymptotes. One zero of q(x)
correspond to one vertical asymptote.
Exercise 5
Let ( )
q x
px f x = , the degree of p(x) is m, the degree of q(x) is n. Then when m ≤ n , the
graph of f(x) will have a horizontal asymptote.
z If m<n (i.e. f(x) is proper), then y=0 is the horizontal asymptote.
z If n=m, then
n
m
b
a y = is the horizontal asymptote, where a (^) m is the leading
coefficient of p(x) (including the sign), b (^) n is the leading coefficient of q(x) (including
the sign).
z Otherwise, NO horizontal asymptotes.
Remark: A rational function has at most one horizontal asymptote.
Exercise 6
Let ( )
q x
px f x = , the degree of p(x) is m, the degree of q(x) is n. Then when m = n + 1 ,
then there is an oblique asymptote, which is the QUOTIENT computed by long division.
Otherwise, NO oblique asymptote.
Exercise 7
Extra Credit
general rational function near the zeros of its denominator.
z When m<n, y=0 is the horizontal asymptote;
z When m=n,
n
m
b
a y = is the horizontal asymptote.
z When m=n+1, the quotient of
( )
q x
p x is the oblique asymptote. (Hint: Put ( )
q x
p x into
the form ( )
q x
rx quotient x qx
p x = +. What’s the degree of quotient(x)? What the
ends behavior of ( )
q x
r x ?)
z Otherwise, no horizontal asymptote nor oblique asymptote.
