(Section 1.7: Transformations)
1.83
SECTION 1.7: TRANSFORMATIONS
PART A: TRANSLATIONS
Translations are transformations that move a graph without changing its shape or
orientation.
Let
G
be the graph of
y
=
f x
( )
.
Let
c
be a positive real number
c
∈
R
+
( )
.
Vertical Shifts
The graph of
y
=
f x
( )
+c
is
G
shifted
up
by
c
units.
The graph of
y
=
f x
( )

c
is
G
shifted
down
by
c
units.
Horizontal Shifts
The graph of
y
=
f x
c
( )
is
G
shifted
right
by
c
units.
The graph of
y
=
f x
+c
( )
is
G
shifted
left
by
c
units.
Warning: Many people confuse the Horizontal Shifts. You may want to think, “It’s the
opposite
from what you expect.” Another approach: If 0 is in the domain of
f
, then the
y
coordinate at
x
=
0
for the graph of
y
=
f x
( )
becomes the
y
coordinate at
x
=
c
for the graph of
y
=
f x
c
( )
.
Both arguments for
f
are 0 in those cases.
Informally, what “happened” at
x
=
0
now “happens” at
x
=
c
.
PART B: REFLECTIONS (“MIRROR IMAGES”)
The graph of
y
=
−
f x
( )
is
G
reflected about the
x
axis.
The graph of
y
=
f
−
x
( )
is
G
reflected about the
y
axis.
If reflecting a graph about an axis yields the same graph, then that graph is
symmetric about that axis; see
Notes 1.51
. Reflections, like translations, are actions
performed on a graph, while symmetries are properties inherent to a graph.
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1.84
PART C: EXAMPLES
Example
Let
f x
( )
=
x
.
Translations:
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 Fall '11
 staff
 Calculus, Transformations, Conic section, NONRIGID TRANSFORMATIONS

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