The trapezoidal function
f
e
(
x
)
{f_e}(x)
is defined for fixed
e
∈
(
0
,
1
/
2
)
e \in (0,1/2)
by
f
e
(
x
)
=
(
1
/
e
)
x
{f_e}(x) = (1/e)x
for
x
∈
[
0
,
e
]
,
f
e
(
x
)
=
1
x \in [0,e],{f_e}(x) = 1
for
x
∈
(
e
,
1
−
e
)
x \in (e,1 - e)
, and
f
e
(
x
)
=
(
1
/
e
)
(
1
−
x
)
{f_e}(x) = (1/e)(1 - x)
for
x
∈
[
1
−
e
,
1
]
x \in [1 - e,1]
. For a given
e
e
and the associated one-parameter family of maps
{
λ
f
e
(
x
)
|
λ
∈
[
0
,
1
]
}
\{ \lambda {f_e}(x)|\lambda \in [0,1]\}
, we show that if
A
A
is an aperiodic kneading sequence, then there is a unique
λ
∈
[
0
,
1
]
\lambda \in [0,1]
so that the itinerary of
λ
\lambda
under the map
λ
f
e
\lambda {f_e}
is
A
A
. From this, we conclude that the "stable windows" are dense in
[
0
,
1
]
[0,1]
for the one-parameter family
λ
f
e
\lambda {f_e}
.