•
Transformations of Functions (Section 2.6)
 Domains and Ranges
: Here is a list of domains and ranges of some functions.
y
=
x
:
Domain
: (
−∞
,
∞
)
,
Range
: (
−∞
,
∞
)
y
=

x

:
Domain
: (
−∞
,
∞
)
,
Range
: [0
,
∞
)
y
=
x
2
:
Domain
: (
−∞
,
∞
)
,
Range
: [0
,
∞
)
y
=
x
3
:
Domain
: (
−∞
,
∞
)
,
Range
: (
−∞
,
∞
)
y
=
√
x
:
Domain
: [0
,
∞
)
,
Range
: [0
,
∞
)
y
=
3
√
x
:
Domain
: (
−∞
,
∞
)
,
Range
: (
−∞
,
∞
)
y
=
1
x
:
Domain
: (
−∞
,
0)
∪
(0
,
∞
)
,
Range
: (
−∞
,
0)
∪
(0
,
∞
)
 Transformations
: Here is a list of transformations of a function
y
=
f
(
x
).
Tip
: These transforma
tions also change the domains and ranges of the functions respectively.
y
=
f
(
x
−
a
) : ( Assume
a >
0) Horizontal shift to right by
a
y
=
f
(
x
+
a
) : ( Assume
a >
0) Horizontal shift to left by
a
y
=
f
(
x
) +
a
: ( Assume
a >
0) Vertical shift upwards by
a
y
=
f
(
x
)
−
a
: ( Assume
a >
0) Vertical shift downwards by
a
y
=
f
(
c
·
x
) : ( Assume
c >
0) Horizontal stretching of the function by a factor of
1
c
y
=
c
·
f
(
x
) : ( Assume
c >
0) Vertical stretching of the function by a factor of
c
y
=
f
(
−
x
) : Reflection along the yaxis
y
=
−
f
(
x
) : Reflection along the xaxis
 Graphs
: When graphing transformations of functions, there are two strategies:
∗
Strategy 1: Apply transformations mentioned above one step at a time.
∗
Strategy 2: Pick reference points on the graph, and identify where the reference points move to.
Connect the reference points to draw a new graph.
For example, suppose we want to draw the following function given
y
=
f
(
x
)
y
=
2
·
f
(
3
·
x
) +
5
∗
Strategy 1: There are 3 operations to take.
·
Red
: Horizontally stretch the function by a factor of
1
3
·
Green
: Vertically shift the function upwards by 5.
·
Blue
: Vertically stretch the function by a factor of 2.
∗
Stategy 2: Suppose (
x
0
, y
0
) is a point on
y
=
f
(
x
). The point (
x
0
, y
0
) moves to a new point on
y
=
c
·
f
(
a
·
x
) +
b
via the following strategy:
·
xcoordinate: multiply
x
0
by
1
3
(
Red
)
·
ycoordinate: multiply
y
0
by 2 (
Blue
), then add by 5. (
Green
)
In particular, (
1
3
x
0
,
2
y
0
+ 5) is the new point on the transformed function.
Likewise, suppose we want to draw the following function given
y
=
f
(
x
)
y
=
2
·
f
(
x
+
3
) +
5
∗
Strategy 1: There are 3 operations to take.
·
Red
: Horizontally shift the function to the left by 3
·
Green
: Vertically shift the function upwards by 5
·
Blue
: Vertically stretch the function by a factor of 2.
∗
Stategy 2: Suppose (
x
0
, y
0
) is a point on
y
=
f
(
x
). The point (
x
0
, y
0
) moves to a new point on
y
=
c
·
f
(
a
·
x
) +
b
via the following strategy:
·
xcoordinate: subtract
x
0
by 3 (
Red
)
·
ycoordinate: multiply
y
0
by 2 (
Blue
), then add by 5. (
Green
)
In particular, (
x
0
−
3
,
2
y
0
+ 5) is the new point on the transformed function.