The parent function is the simplest form of the type of function given.
f
(
x
)
=
|
x
|
f
(
x
)
=
|
x
|
The transformation from the first equation to the second one can be found by finding
a
a
,
h
h
, and
k
k
for each equation.
y
=
a
|
x
−
h
|
+
k
y
=
a
|
x
-
h
|
+
k
Factor a
1
1
out of the absolute value to make the coefficient of
x
x
equal to
1
1
.
y
=
|
x
|
y
=
|
x
|
Factor a
1
1
out of the absolute value to make the coefficient of
x
x
equal to
1
1
.
y
=
|
x
−
5
|
y
=
|
x
-
5
|
Find
a
a
,
h
h
, and
k
k
for
y
=
|
x
−
5
|
y
=
|
x
-
5
|
.
a
=
1
a
=
1
h
=
5
h
=
5
k
=
0
k
=
0
The horizontal shift depends on the value of
h
h
. When
h
>
0
h
>
0
, the horizontal shift is described as:
g
(
x
)
=
f
(
x
+
h
)
g
(
x
)
=
f
(
x
+
h
)
- The graph is shifted to the left
h
h
units.
g
(
x
)
=
f
(
x
−
h
)
g
(
x
)
=
f
(
x
-
h
)
- The graph is shifted to the right
h
h
units.
Horizontal Shift: Right
5
5
Units
The vertical shift depends on the value of
k
k
. When
k
>
0
k
>
0
, the vertical shift is described as:
g
(
x
)
=
f
(
x
)
+
k
g
(
x
)
=
f
(
x
)
+
k
- The graph is shifted up
k
k
units.
g
(
x
)
=
f
(
x
)
−
k
g
(
x
)
=
f
(
x
)
-
k
- The graph is shifted down
k
k
units.
Vertical Shift: None
The sign of
a
a
describes the reflection across the x-axis.
−
a
-
a
means the graph is reflected across the x-axis.
Reflection about the x-axis: None
The value of
a
a
describes the vertical stretch or compression of the graph.
a
>
1
a
>
1
is a vertical stretch (makes it narrower)
0
<
a
<
1
0
<
a
<
1
is a vertical compression (makes it wider)
Vertical Compression or Stretch: None
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, and if there is a vertical stretch.
Parent Function:
f
(
x
)
=
|
x
|
f
(
x
)
=
|
x
|
Horizontal Shift: Right
5
5
Units
Vertical Shift: None
Reflection about the x-axis: None
Vertical Compression or Stretch: None
image of graph
g
(
x
)
=
|
x
−
5
|
g
(
x
)
=
|
x
-
5
|