MHF 4U1
Eric Zhang
Problem Solving Assignment: Rational Functions
The function I have created is:
h
(
x
)
=
45
(
x
+
4
)
2
49
(
x
+
6
)
(
where x ≠
−
9
)
A graph is as follows:
y
=
45
49
(
x
+
2
)
x
=−
6
h
(
x
)

MHF 4U1
Eric Zhang
Required Characteristics:
a)
has a linear oblique asymptote
Linear oblique asymptotes exist when the order of
x
in the numerator is 1 greater than the
order of
x
in the denominator. It is conclusive that
h
(
x
)
has a linear oblique asymptote, as
the order of
x
in the denominator is 1 while the numerator’s is 2. The asymptote can be
determined through polynomial division:
h
(
x
)
=
45
(
x
+
4
)
2
49
(
x
+
6
)
=
180
49
(
x
+
6
)
+
45
49
(
x
+
2
)
, where the
linear oblique asymptote is the quotient:
y
=
45
49
(
x
+
2
)
.
Thus, the function has a linear oblique asymptote.
b)
has a hole in quadrant III
Since a condition for the function is
x≠
−
9
, there is a point of discontinuity (a ‘hole’) at
x
=−
9
. The limit as
x→
−
9
is
lim
x →
−
9
h
(
x
)
=
−
375
49
and so the hole is at
(−
9,
−
375
49
)
.

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- Fall '15
- Rational function, linear oblique asymptote, Eric Zhang