A good definition of symmetry was given by the mathematician Hermann Weyl:
An object is symmetrical if, after you performed an operation on it, it still looks the same as it did before.
- A function can be moved in the horizontal direction so-called translation, by adding or subtracting a number. For example :
f(x) = sin(x-a) is the function f(x) = sin(x) moved a steps to the right.
- The graph of a function can be moved in vertical direction (vertical translation) by adding or subtracing a number to the function e.g. f(x) = sin(x) +4 is the function f(x) = sin(x) moved four steps upwa
- A vectorial translation i.e. a translation in both the x- and the y- direction. y= sin (x+a) +b.
- By dilatation is meant a contraction or comprimation of the function. f(x) = sin(x/m) is an extraction of the function f(x)=sin(x/m) by the scale m.
- A function may also be mirrored in the x-axis. -f(x) is the mirror -image of f(x) in the x-axis. f(x)=-sin(x) is the mirror-image of f(x) = sin(x) in the x-axis.
- f(-x) is the mirror-image in the y-axis of f(x). f(x)=sin(-x) mirrors f(x) = sin(x) in the y-axis.
- By combining both types of mirroring one obtains a mirroring in the origin of the function.
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