We can also reflect the graph of a function over the
*x*
-axis (
*y* = 0
), the
*y*
-axis(
*x* = 0
), or the line
*y* = *x*
.

Making the output negative reflects the graph over the
*x*
-axis, or the line
*y* = 0
. Here are the graphs of
*y* = *f* (*x*)
and
*y* = - *f* (*x*)
. Note that if
(*x*, *y*
_{1})
is a point on the graph of
*f* (*x*)
and
(*x*, *y*
_{2})
is a point on the graph
of
- *f* (*x*)
, then
*y*
_{2} = - *y*
_{1}
. The range also becomes negative; if range =
*f* (*x*):*a* < *f* (*x*) < *b*
, then the new range is
*g*(*x*):-*b* < *g*(*x*) < -*a*
.

Graphs of
*f* (*x*)
and
- *f* (*x*)

Making the input negative reflects the graph over the
*y*
-axis, or the line
*x* = 0
. Here are the graphs of
*y* = *f* (*x*)
and
*y* = *f* (- *x*)
. Note that if
(*x*
_{1}, *y*)
is a point on the graph of
*f* (*x*)
and
(*x*
_{2}, *y*)
is a point on the graph of
*f* (- *x*)
, then
*x*
_{2} = - *x*
_{1}
. The domain also becomes negative; if domain =
*x*:*a* < *x* < *b*
, then the new domain is
*x*:-*b* < *x* < -*a*
.

Graphs of
*f* (*x*)
and
*f* (- *x*)

Switching
*x*
and
*y*
reflects the graph over the line
*y* = *x*
(this is
equivalent to finding the inverse). Now,
*x*
is a function of
*y*
.
Here are the graphs of
*y* = *f* (*x*)
and
*x* = *f* (*y*)
. Note that if
(*x*
_{1}, *y*
_{1})
is
a point on the graph of
*f* (*x*)
and
(*x*
_{2}, *y*
_{2})
is a point on the graph of
*f* (*y*)
, then
*y*
_{1} = *x*
_{2}
and
*y*
_{2} = *x*
_{1}
.

Graphs of
*y* = *f* (*x*)
and
*x* = *f* (*y*)

In addition, we can rotate the graph of a function
180^{
o
}
around the
origin. Making the input and the output of a function negative results in a
rotation of the function around the origin. Here is a graph of
*y* = *f* (*x*)
and
*y* = - *f* (- *x*)
. Note that if
(*x*, *y*)
is a point on the graph of
*f* (*x*)
, then
(- *x*, - *y*)
is a point on the graph of
- *f* (- *x*)
. The domain and range both become
negative.

Graphs of
*f* (*x*)
and
- *f* (- *x*)