Answer:
f(x)=4log(x + 2) - 3
3Step-by-step explanation:
The parent function is the simplest form of the type of function given.
g
(
x
)
=
log
(
x
)
The transformation being described is from
g
(
x
)
=
log
(
x
)
to
f
(
x
)
=
4
log
(
x
+
2
)
−
3
.
g
(
x
)
=
log
(
x
)
→
f
(
x
)
=
4
log
(
x
+
2
)
−
3
The transformation from the first equation to the second one can be found by finding
a
,
c
and
d
for
f
(
x
)
=
4
log
(
x
+
2
)
−
3
.
f
(
x
)
=
a
log
(
x
+
c
)
+
d
Find
a
,
c
and
d
for
g
(
x
)
=
log
(
x
)
.
a
1
=
1
c
1
=
0
d
1
=
0
Find
a
,
c
and
d
for
f
(
x
)
=
4
log
(
x
+
2
)
−
3
.
a
2
=
4
c
2
=
2
d
2
=
−
3
The horizontal shift depends on the value of
c
. When
c
>
0
, the horizontal shift is described as:
f
(
x
)
=
a
log
(
x
+
c
)
+
d
- The graph is shifted to the left
c
units.
f
(
x
)
=
a
log
(
x
−
c
)
+
d
- The graph is shifted to the right
c
units.
Horizontal Shift: Left
2
Units
The vertical shift depends on the value of
d
. When
d
>
0
, the vertical shift is described as:
f
(
x
)
=
a
log
(
x
+
c
)
+
d
- The graph is shifted up
d
units.
f
(
x
)
=
a
log
(
x
+
c
)
−
d
- The graph is shifted down
d
units.
Vertical Shift: Down
3
Units
The sign of
y
describes the reflection across the x-axis.
−
y
means the graph is reflected across the x-axis.
Reflection about the x-axis: None
The sign of
x
describes the reflection across the y-axis.
−
x
means the graph is reflected across the y-axis.
Reflection about the y-axis: None
The value of
a
describes the vertical stretch or compression of the graph.
|
a
2
|
>
|
a
1
|
is a vertical stretch (makes it narrower)
|
a
2
|
<
|
a
1
|
is a vertical compression (makes it wider)
Vertical Compression or Stretch: Stretched
To find the transformation, compare the two functions and check to see if there is a horizontal or vertical shift, reflection about the x-axis, reflection about the y-axis, and if there is a vertical stretch or compression.
Parent Function:
g
(
x
)
=
log
(
x
)
Horizontal Shift: Left
2
Units
Vertical Shift: Down
3
Units
Reflection about the x-axis: None
Reflection about the y-axis: None
Vertical Compression or Stretch: Stretched
