5
5
−5
−5
x
y
MATH 107 – Handout 3.6
Rational Functions: .0)(,
)(
)(
)( ≠
=xq
functionpolynomial
functionpolynomial
xq
xp
xf
Asymptotes are lines that our graphs approach.
Although these lines are not part of the graph of a rational function,
they do serve as guidelines for the graph our function.
Vertical Asymptotes: (Form: x = #)
If
∞
→
)
(
x
f
or
−∞
→
)
(
x
f
as
a
x
→
, then you have the vertical asymptote: x = a.
To find the vertical asymptote(s), you set the denominator equal to zero and solve .
Horizontal Asymptotes: (Form: y = #)
If
b
x
f
→
)
(
as
∞
→
x
or
−∞
→
x
, then you have the horizontal asymptote: y = b.
To find the horizontal asymptote, you first examine the degrees of the numerator and denominator.
If the degrees are equal, then the horizontal asymptote is )(
)(
xqoftcoefficienleading
xpoftcoefficienleading
y=.
If the degree is higher in the denominator, then the horizontal asymptote is y = 0.
If the degree is higher in the numerator, then there is no horizontal asymptote.
Slant Asymptotes: (Form: y = mx + b)
If the degree of the numerator is exactly one degree higher than the degree of the denominator, then the function has a slant asymptote.
To find the slant asymptote, use long division.
The slant asymptote is y = quotient (do not include the remainder).
y-intercept: (0, #): Set x = 0 and solve.
x-intercept(s): (#, 0): Set y = 0 and solve.
(Notice: For rational functions, x-intercept values are whatever turns the numerator into zero.)
Helpful Points: Choose x-values surrounding the x-values of the vertical asymptotes (and the zeros) to evaluate on a t-table.
1. f(x) =
4
122
+
−
x
x
(a) Vertical Asymptote(s): _________________
(b) Horizontal Asymptote: _________________
(c) y – intercept: _____________________
(d) x – intercept(s): ___________________
Other helpful points:
x y
Domain: _______________________
Range: _________________________
Parent Function: f(x) =
x
1
