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Reflections and Rotations

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.

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.

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}.

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.