Abstract - Numerical functions and their properties. Direct and inverse proportional relationships - file n1.doc

They have many properties:

1. The function is called monotonous on a certain interval A, if it increases or decreases on this interval

2. The function is called increasing on a certain interval A, if for any numbers of their set A the following condition is satisfied:.

The graph of an increasing function has a special feature: when moving along the x-axis from left to right along the interval A the ordinates of the graph points increase (Fig. 4).

3. The function is called decreasing at some interval A, if for any numbers there are many of them A the condition is met:.

The graph of a decreasing function has a special feature: when moving along the x-axis from left to right along the interval A the ordinates of the graph points decrease (Fig. 4).

4. The function is called even on some set X, if the condition is met: .

The graph of an even function is symmetrical about the ordinate axis (Fig. 2).

5. The function is called odd on some set X, if the condition is met: .

The graph of an odd function is symmetrical about the origin (Fig. 2).

6. If the function y = f(x)
f(x) f(x), then they say that the function y = f(x) accepts smallest value at=f(x) at X= x(Fig. 2, the function takes the smallest value at the point with coordinates (0;0)).

7. If the function y = f(x) is defined on the set X and there exists such that for any the inequality f(x) f(x), then they say that the function y = f(x) accepts highest value at=f(x) at X= x(Fig. 4, the function does not have the largest and smallest values) .

If for this function y = f(x) all the listed properties have been studied, then they say that study functions.

SUMMARY LESSON ON THE TOPIC “FUNCTIONS AND THEIR PROPERTIES”.

Lesson Objectives:

Methodical: increasing the active-cognitive activity of students through individual-independent work and the use of developmental type test tasks.

Educational: repeat elementary functions, their basic properties and graphs. Introduce the concept of mutually inverse functions. Systematize students’ knowledge on the topic; contribute to the consolidation of skills in calculating logarithms, in applying their properties when solving tasks of a non-standard type; repeat the construction of graphs of functions using transformations and test your skills and abilities when solving exercises on your own.

Educational: fostering accuracy, composure, responsibility, and the ability to make independent decisions.

Developmental: develop intellectual abilities, mental operations, speech, memory. Develop a love and interest in mathematics; During the lesson, ensure that students develop independent thinking in learning activities.

Lesson type: generalization and systematization.

Equipment: board, computer, projector, screen, educational literature.

Lesson epigraph:“Mathematics must be taught later because it puts the mind in order.”

(M.V. Lomonosov).

DURING THE CLASSES

Checking homework.

Repetition of exponential and logarithmic functions with base a = 2, construction of their graphs in the same coordinate plane, analysis of their relative position. Consider the interdependence between the main properties of these functions (OOF and OFP). Give the concept of mutually inverse functions.

Consider exponential and logarithmic functions with base a = ½ c

in order to ensure that the interdependence of the listed properties is observed and for

decreasing mutually inverse functions.

Organization of independent test-type work for the development of thinking skills

systematization operations on the topic “Functions and their properties.”

FUNCTION PROPERTIES:

1). y = ‌│х│ ;

2). Increases throughout the entire definition area;

3). OOF: (- ∞; + ∞) ;

4). y = sin x;

5). Decreases at 0< а < 1 ;

6). y = x³;

7). OPF: (0; + ∞) ;

8). General function;

9). y = √ x;

10). OOF: (0; + ∞) ;

eleven). Decreases over the entire definition area;

12). y = kx + b;

13). OSF: (- ∞; + ∞) ;

14). Increases at k > 0;

15). OOF: (- ∞; 0) ; (0; + ∞) ;

16). y = cos x;

17). Has no extremum points;

18). OSF: (- ∞; 0) ; (0; + ∞) ;

19). Decreases at k< 0 ;

20). y = x²;

21). OOF: x ≠ πn;

22). y = k/x;

23). Even;

25). Decreases for k > 0;

26). OOF: [ 0; + ∞) ;

27). y = tan x;

28). Increases with k< 0;

29). OSF: [ 0; + ∞) ;

thirty). Odd;

31). y = log x ;

32). OOF: x ≠ πn/2;

33). y = ctg x ;

34). Increases when a > 1.

During this work, survey students on individual assignments:

No. 1. a) Graph the function

b) Graph the function

No. 2. a) Calculate:

b) Calculate:

No. 3. a) Simplify the expression
and find its value at

b) Simplify the expression
and find its value at
.

Homework: No. 1. Calculate: a)
;

V)
;

G)
.

No. 2. Find the domain of definition of the function: a)
;

V)
; G)
.

Numeric function This correspondence between a number set is called X and many R real numbers, in which each number from the set X matches a single number from a set R. A bunch of X called domain of the function . Functions are indicated by letters f, g, h etc. If f– function defined on the set X, then real number y, corresponding to the number X there are many of them X, often denoted f(x) and write
y = f(x). Variable X this is called an argument. Set of numbers of the form f(x) called function range

The function is specified using a formula. For example , y = 2X - 2. If, when specifying a function using a formula, its domain of definition is not indicated, then it is assumed that the domain of definition of the function is the domain of definition of the expression f(x).

1. The function is called monotonous on a certain interval A, if it increases or decreases on this interval

2. The function is called increasing on a certain interval A, if for any numbers of their set A the following condition is satisfied: .

The graph of an increasing function has a special feature: when moving along the x-axis from left to right along the interval A the ordinates of the graph points increase (Fig. 4).

3. The function is called decreasing at some interval A, if for any numbers there are many of them A the condition is met: .

The graph of a decreasing function has a special feature: when moving along the x-axis from left to right along the interval A the ordinates of the graph points decrease (Fig. 4).

4. The function is called even on some set X, if the condition is met: .

The graph of an even function is symmetrical about the ordinate axis (Fig. 2).

5. The function is called odd on some set X, if the condition is met: .

The graph of an odd function is symmetrical about the origin (Fig. 2).

6. If the function y = f(x)
f(x) f(x), then they say that the function y = f(x) accepts smallest value at =f(x) at X= x(Fig. 2, the function takes the smallest value at the point with coordinates (0;0)).

7. If the function y = f(x) is defined on the set X and there exists such that for any the inequality f(x) f(x), then they say that the function y = f(x) accepts highest value at =f(x) at X= x(Fig. 4, the function does not have the largest and smallest values) .

If for this function y = f(x) all the listed properties have been studied, then they say that study functions.

Limits.

A number A is called the limit of a function as x tends to ∞ if for any E>0, there exists δ (E)>0 such that for all x satisfies the inequality |x|>δ the inequality |F(x)-A|

A number A is called the limit of a function as X tends to X 0 if for any E>0, there exists δ (E)>0 such that for all X≠X 0 satisfies the inequality |X-X 0 |<δ выполняется неравенство |F(x)-A|

UNILATERAL LIMITS.

When defining the limit, X tends to X0 in an arbitrary manner, that is, from any direction. When X tends to X0, so that it is always less than X0, then the limit is called the limit at the point X0 on the left. Or a left-handed limit. The right-hand limit is defined similarly.

This material was compiled according to the Federal State Educational Standard

mathematics lesson in 9th grade on the topic: “Numerical functions, their properties and graphs”, textbook by A.G. Mordkovich.

Lesson of developmental control and discovery of new knowledge
lesson supplement and presentation.

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Slide captions:

Numerical functions, their properties and Graphs. Mathematics lesson in 9th grade at the final certification of IDPO subgroup No. 9 Zavodskoy district of Saratov 10/25/2013

Epigraph “The only path leading to knowledge is activity.” Bernard Show

Creative work Come up with a “piecewise” function, build a graph and read it. Solution y =

Oral work Name the function and define it analytically

Theoretical quiz Formulate the definition of a numerical function. What is called the domain of definition of a function. What is called the graph of a function. List ways to define a function. What function is called increasing (decreasing). Which function is called even (odd). What number is called the smallest (largest) value of the function. What function is called limited.

Tests in GIA format (basic level)

answers Option No. 5 Option No. 6 4 3 3142 132 2 4 3 3 2 1 3 3

Performing exercises GIA No. 1. Plot a graph of the function y = x 2 - 4 +3, using the graph, find the intervals of monotonicity. For what values ​​of a does the straight line y=a have two points in common with the graph of this function? Answer: a>3, a = -1

No. 2. Solve graphically the inequality x -2 ≤ -x 3 Answer: x≤ -1

I learned I learned I repeated I consolidated Today in class

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Annex 1.

(checking homework)

Solution

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Appendix 2

Oral work

Name the function and define it analytically

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Appendix 3

Theoretical survey

1. Formulate the definition of a numerical function.