There are numerous ways to apply
transformations to functions to create new functions.
Let's look at some of the possibilities. Remember to utilize your
graphing calculator to compare the graphs of your functions and their
transformations.
Reflections and Functions:
Examining
f (x) and f (x) 
Reflection
over the xaxis
f (x) reflects
f (x) over the xaxis.


A reflection is a mirror image. Placing the
edge of a mirror on the xaxis will form a reflection in the
xaxis.
This can also be thought of as "folding" over the xaxis.
If the original (parent) function is
y = f (x), the reflection over the
xaxis is function f
(x).


Reflection
over the yaxis
f (x) reflects
f (x) over the yaxis.


Placing the edge of a mirror on the yaxis will form a
reflection in the yaxis. This can also be thought of as "folding"
over the yaxis.
If the original (parent) function is
y = f (x), the reflection over the yaxis is function
f
(x).


Stretch or
Compress Functions: Examining
f
(ax) and a f (x) 
Horizontal Stretch or Compress
f
(ax) stretches/compresses f (x)
horizontally

A horizontal stretching is the
stretching of the graph away from the yaxis.
A horizontal compression is the squeezing
of the graph towards the yaxis.
If the original (parent) function is
y = f
(x),
the horizontal stretching or compressing of the function
is the function
f (ax).
 if
0 < a < 1 (a
fraction), the graph is
stretched horizontally
by a factor
of a
units.


 if
a > 1,
the graph is
compressed
horizontally by a factor of
a units.
 if a should be negative,
the horizontal compression or horizontal stretching of the
graph is followed by a reflection of the graph across the yaxis.


Vertical Stretch or Compress
a f
(x) stretches/compresses f (x)
vertically

A vertical stretching is the stretching of the graph
away from the xaxis.
A vertical compression is the squeezing of
the graph towards the xaxis.
If the original (parent) function is
y = f
(x),
the vertical stretching or compressing of the function
is the function
a f(x).
 if
0 < a
< 1 (a
fraction), the graph is
compressed vertically
by a factor
of a units.
 if
a
> 1,
the graph is
stretched
vertically by a factor of
a units.


 If
a
should be negative,
then the vertical compression or vertical stretching of the graph is
followed by a reflection across the xaxis.


