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How to teach mono-unary algebras and functional graphs with the use of computers in secondary schools

In this paper, alternative descriptions of functions are demonstrated with the use of a computer. If we understand functions as mono-unary algebraic functions or functional graphs, it is possible, even at the school level, to suitably present many of their characteristics. First, we describe cyclic graphs of constant and linear functions, which are a part of the upper-secondary level educational curriculum. Students are usually surprised by the unexpected characteristics of such simple functions which cannot be revealed using the traditional Cartesian graphing. The next part of the paper deals with the characteristics of functional graphs of quadratic functions, which play an important role in school mathematics and in applications, for instance, in the description of non-linear processes. We show that their description is much more complicated. In contrast to functional graphs of linear functions, it is necessary to use computers. Students can find space for their own individual exploration to reveal lines of interesting characteristics of quadratic functions, which give students a new view on this part of school mathematics.