Definition, graph and properties of the function: the structure of the course of mathematical analysis in school

For the first time with the concept of function, students in educational schools usually meet in grade 7, when they begin to study the course of algebra as a separate direction of mathematics. Beginning the study of functions, as a rule, without entering complex definitions and terms, which is quite logical. The most important thing at the introductory stage is to provide students with an opportunity to get a general overview of elementary examples with a new and previously unknown mathematical object.

Begins to study functions with linear dependencies, the graph of which is a straight line. Students learn the mathematical notation of the dependence of one variable on another and understand which variable in the function is independent and which is independent. Parallel to this, students begin to construct graphs on the coordinate plane on which they previously marked only points.

The next function with which students are acquainted is direct proportionality. Initially, in the course of algebra, the authors of many manuals distinguish this dependence separately from the linear function, noting some important properties of the function that are inherent in this dependence.

After considering elementary functions, students are introduced to generalized concepts that characterize numerical dependencies. First of all, it is work with the record y = f (x). Further, several lessons are necessarily devoted to the practical application of the theoretical knowledge obtained, within the framework of which the applied nature of the definition and any particular property of the function characterizing a particular process is considered.

In the 8th grade, students first encounter square equations. After mastering the skills of solving equations of this type, the program provides for the study of the quadratic function and its main characteristics. Students learn not only to build a graph of dependence on the presented equation, but also to analyze the presented image, revealing the basic properties of the function and forming its mathematical description.

The course of class 9 algebra extends the set of known functions to students. Possessing a sufficiently significant theoretical base devoted to mathematical analysis, students are introduced to the inverse proportionality and fractional-linear function, and also study the differences in the representation on the graphical plane of the equation and function. In the latter case, attention is drawn to the fact that the equation graph can have several values of the dependent variable for one argument - an independent variable. The functional dependence is characterized by a single-valued correspondence of the independent and dependent variables.

At the senior level of the school, students learn complex functional dependencies and learn how to build graphics, drawing not on the table of values "argument-function", but on the properties of the function. This is due to the fact that the behavior of complex functions is difficult enough to predict "headaches", and it is quite difficult to calculate a certain set of values. Therefore, to determine the nature of the behavior of a function, its main characteristics are described: domains of definition and values, asymptotes, monotonicity, points of maximum and minimum, convexity, etc. Particular attention should be paid to such a property as parity. The even and odd functions have a special behavior: the first characteristic means that the graph of the function is symmetrical about the ordinate axis, the second is relative to the origin point.

This concludes the study of the foundations of mathematical analysis in the course of secondary school. Further study of numerical dependencies will necessarily be presented in the course of higher mathematics, as well as within the disciplines devoted to statistical data processing. The latter often use such an element as the distribution function.