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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, 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)

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