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, y1) is a point on the graph of
f (x) and
(x, y2) is a point on the graph
of
- f (x), then
y2 = - y1. 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
(x1, y)
is a point on the graph of
f (x) and
(x2, y) is a point on the graph of
f (- x), then
x2 = - x1. 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
(x1, y1) is
a point on the graph of
f (x) and
(x2, y2) is a point on the graph of
f (y), then
y1 = x2 and
y2 = x1.
Graphs of y = f (x) and x = f (y)
In addition, we can
rotate the graph of a function
180o 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)
