We want to find an equation of the general form
f
(
x
)
=
a
b
x
+
c
+
d
f(x)=ab
x+c
+d. We use the description provided to find a, b, c, and d.
We are given the parent function
f
(
x
)
=
e
x
f(x)=e
x
, so b = e.
The function is stretched by a factor of 2, so a = 2.
The function is reflected about the y-axis. We replace x with –x to get:
e
−
x
e
−x
.
The graph is shifted vertically 4 units, so d = 4.
Substituting in the general form we get,
⎧
⎨
⎩
f
(
x
)
=
a
b
x
+
c
+
d
=
2
e
−
x
+
0
+
4
=
2
e
−
x
+
4
{
f
(
x
)
=a
b
x
+
c
+d =2
e
−
x
+
0
+4 =2
e
−
x
+4
The domain is
(
−
∞
,
∞
)
(−∞,∞); the range is
(
4
,
∞
)
(4,∞); the horizontal asymptote is
y
=
4
y=4.
