Precalculus
Name ____________________
2.6 Notes: Rational Functions and Asymptotes
Rational Functions:
f (x ) =
N (x )
D (x )
where N (x ) and D (x ) are polynomials functions and D (x ) ≠ 0 .
N(x) represents the __________________and D(x) represents the __________________ .
Why can’t D (x ) = 0 ?
Domain of rational functions includes _______________________________________________ .
Asymptotes are lines that a graph of a function _____________________ . Asymptotes are a type of
discontinuity because there is a “break” in the graph of a function.
Let f be the rational function f (x ) =
N (x )
D (x )
where N (x ) and D (x ) have no common factors.
Vertical Asymptotes
The graph of f has vertical asymptotes at ______________________________________________.
Horizontal Asymptotes:
The graph of f has at most one horizontal asymptote, representing the function’s _________________.
Horizontal asymptotes may be determined by ____________________________________________
or by evaluating a function at a very large input value.
Ø If the degree of N (x ) is less than the degree of D (x ) , the graph of f has a HA at _________.
Ø If the degree of N (x ) is equal to the degree of D (x ) , the graph of f has a HA at __________,
where a and b are the leading coefficients of N (x ) and D (x ) , respectively.
Ø If the degree of N (x ) is greater than the degree of D (x ) , the graph of f has ____________.
Examples: Find all horizontal and vertical asymptotes.
1.
f (x ) =
2x 3 + 1
x −4
2.
g (x ) =
2
x 2 − 5x + 6
3.
f (x ) =
6x 2
x 2 − 25
Hole(s) of a Rational Function
Let f be the rational function f (x ) =
N (x )
D (x )
where N (x ) and D (x ) have at least one common factor.
The graph will display a hole at the zero of that common factor, rather than a vertical asymptote.
To find the coordinates of the hole, substitute the x-coordinate (the zero of the common factor) into
the simplified form of the rational function. Plot the hole as an “open dot” on the graph of the rational
function.
Examples: Identify and horizontal and vertical asymptotes and identify any holes. Verify your
results with a graphing calculator.
4.
f (x ) =
x 2 + 6x + 9
x 2 + 4x + 3
Vertical asymptote(s):
Horizontal asymptote:
Holes:
Domain:
5.
f(x) =
2
(x + 2)2
Vertical asymptote(s):
Horizontal asymptote:
Holes:
Domain:
6.
f(x) =
2x2
x2 − 9
Vertical asymptote(s):
Horizontal asymptote:
Holes:
Domain:
7.
x2 −x −2
f (x ) =
x −2
Vertical asymptote(s):
Horizontal asymptote:
Holes:
Domain: