Hypothesis: If you study the movement of the graph during the formation of an equation of functions, you will notice that all graphs obey general laws, so it is possible to formulate general laws regardless of the functions, which will not only facilitate the construction of graphs of various functions, but also use them in solving problems.

Goal: To study the movement of graphs of functions:

1) The task is to study literature

2) Learn to build graphs of various functions

3) Learn to transform graphs of linear functions

4) Consider the issue of using graphs when solving problems

Object of study: Function graphs

Subject of research: Movements of function graphs

Relevance: Constructing graphs of functions, as a rule, takes a lot of time and requires attentiveness on the part of the student, but knowing the rules for converting graphs of functions and graphs of basic functions, you can quickly and easily construct graphs of functions, which will allow you not only to complete tasks for constructing graphs of functions, but also solve problems related to it (to find the maximum (minimum height of time and meeting point))

This project is useful to all students at the school.

Literature review:

The literature discusses methods for constructing graphs of various functions, as well as examples of transforming graphs of these functions. Graphs of almost all main functions are used in various technical processes, which allows you to more clearly visualize the flow of the process and program the result

Permanent function. This function is given by the formula y = b, where b is a certain number. The graph of a constant function is a straight line parallel to the abscissa and passing through the point (0; b) on the ordinate. The graph of the function y = 0 is the x-axis.

Types of function 1Direct proportionality. This function is given by the formula y = kx, where the coefficient of proportionality k ≠ 0. The graph of direct proportionality is a straight line passing through the origin.

Linear function. Such a function is given by the formula y = kx + b, where k and b are real numbers. The graph of a linear function is a straight line.

Graphs of linear functions can intersect or be parallel.

Thus, the lines of the graphs of linear functions y = k 1 x + b 1 and y = k 2 x + b 2 intersect if k 1 ≠ k 2 ; if k 1 = k 2, then the lines are parallel.

2Inverse proportionality is a function that is given by the formula y = k/x, where k ≠ 0. K is called the inverse proportionality coefficient. The graph of inverse proportionality is a hyperbola.

The function y = x 2 is represented by a graph called a parabola: on the interval [-~; 0] the function decreases, on the interval the function increases.

The function y = x 3 increases along the entire number line and is graphically represented by a cubic parabola.

Power function with natural exponent. This function is given by the formula y = x n, where n is a natural number. Graphs of a power function with a natural exponent depend on n. For example, if n = 1, then the graph will be a straight line (y = x), if n = 2, then the graph will be a parabola, etc.

A power function with a negative integer exponent is represented by the formula y = x -n, where n is a natural number. This function is defined for all x ≠ 0. The graph of the function also depends on the exponent n.

Power function with a positive fractional exponent. This function is represented by the formula y = x r, where r is a positive irreducible fraction. This function is also neither even nor odd.

A line graph that displays the relationship between the dependent and independent variables on the coordinate plane. The graph serves to visually display these elements

An independent variable is a variable that can take any value in the domain of function definition (where the given function has meaning (cannot be divided by zero))

To build a graph of functions you need

1) Find the VA (range of acceptable values)

2) take several arbitrary values ​​for the independent variable

3) Find the value of the dependent variable

4) Construct a coordinate plane and mark these points on it

5) Connect their lines, if necessary, examine the resulting graph. Transformation of graphs of elementary functions.

Converting graphs

In their pure form, basic elementary functions are, unfortunately, not so common. Much more often you have to deal with elementary functions obtained from basic elementary ones by adding constants and coefficients. Graphs of such functions can be constructed by applying geometric transformations to the graphs of the corresponding basic elementary functions (or switch to a new coordinate system). For example, the quadratic function formula is a quadratic parabola formula, compressed three times relative to the ordinate axis, symmetrically displayed relative to the abscissa axis, shifted against the direction of this axis by 2/3 units and shifted along the ordinate axis by 2 units.

Let's understand these geometric transformations of the graph of a function step by step using specific examples.

Using geometric transformations of the graph of the function f(x), a graph of any function of the form formula can be constructed, where the formula is the compression or stretching coefficients along the oy and ox axes, respectively, the minus signs in front of the formula and formula coefficients indicate a symmetrical display of the graph relative to the coordinate axes , a and b determine the shift relative to the abscissa and ordinate axes, respectively.

Thus, there are three types of geometric transformations of the graph of a function:

The first type is scaling (compression or stretching) along the abscissa and ordinate axes.

The need for scaling is indicated by formula coefficients other than one; if the number is less than 1, then the graph is compressed relative to oy and stretched relative to ox; if the number is greater than 1, then we stretch along the ordinate axis and compress along the abscissa axis.

The second type is a symmetrical (mirror) display relative to the coordinate axes.

The need for this transformation is indicated by the minus signs in front of the coefficients of the formula (in this case, we display the graph symmetrically about the ox axis) and the formula (in this case, we display the graph symmetrically about the oy axis). If there are no minus signs, then this step is skipped.

Depending on the conditions of physical processes, some quantities take on constant values ​​and are called constants, others change under certain conditions and are called variables.

A careful study of the environment shows that physical quantities are dependent on each other, that is, a change in some quantities entails a change in others.

Mathematical analysis deals with the study of quantitative relationships between mutually varying quantities, abstracting from the specific physical meaning. One of the basic concepts of mathematical analysis is the concept of function.

Consider the elements of the set and the elements of the set
(Fig. 3.1).

If some correspondence is established between the elements of the sets
And in the form of a rule , then they note that the function is defined
.

Definition 3.1. Correspondence , which associates with each element not empty set
some well-defined element not empty set ,called a function or mapping
V .

Symbolically display
V is written as follows:

.

At the same time, many
is called the domain of definition of the function and is denoted
.

In turn, many is called the range of values ​​of the function and is denoted
.

In addition, it should be noted that the elements of the set
are called independent variables, the elements of the set are called dependent variables.

Methods for specifying a function

The function can be specified in the following main ways: tabular, graphical, analytical.

If, based on experimental data, tables are compiled that contain the values ​​of the function and the corresponding argument values, then this method of specifying the function is called tabular.

At the same time, if some studies of the experimental result are displayed on a recorder (oscilloscope, recorder, etc.), then it is noted that the function is specified graphically.

The most common is the analytical way of specifying a function, i.e. a method in which an independent and dependent variable is linked using a formula. In this case, the domain of definition of the function plays a significant role:

different, although they are given by the same analytical relations.

If you only specify the function formula
, then we consider that the domain of definition of this function coincides with the set of those values ​​of the variable , for which the expression
has the meaning. In this regard, the problem of finding the domain of definition of a function plays a special role.

Task 3.1. Find the domain of a function

Solution

The first term takes real values ​​when
, and the second at. Thus, to find the domain of definition of a given function, it is necessary to solve the system of inequalities:

As a result, the solution to such a system is obtained. Therefore, the domain of definition of the function is the segment
.

The simplest transformations of function graphs

The construction of function graphs can be significantly simplified if you use the well-known graphs of basic elementary functions. The following functions are called the main elementary functions:

1)power function
Where
;

2) exponential function
Where
And
;

3) logarithmic function
, Where - any positive number other than one:
And
;

4) trigonometric functions




;
.

5) inverse trigonometric functions
;
;
;
.

Elementary functions are functions that are obtained from basic elementary functions using four arithmetic operations and superpositions applied a finite number of times.

Simple geometric transformations also make it possible to simplify the process of constructing a graph of functions. These transformations are based on the following statements:

The graph of the function y=f(x+a) is the graph y=f(x), shifted (for a >0 to the left, for a< 0 вправо) на |a| единиц параллельно осиOx.

The graph of the function y=f(x) +b is the graph of y=f(x), shifted (at b>0 up, at b< 0 вниз) на |b| единиц параллельно осиOy.

The graph of the function y = mf(x) (m0) is the graph of y = f(x), stretched (at m>1) m times or compressed (at 0

The graph of the function y = f(kx) is the graph of y = f(x), compressed (for k >1) k times or stretched (for 0< k < 1) вдоль оси Ox. При –< k < 0 график функции y = f(kx) есть зеркальное отображение графика y = f(–kx) от оси Oy.

Which of these functions have an inverse? For such functions, find inverse functions:

4.12. A)	y = x ;					b) y = 6 −3 x ;
d) y =						e) y = 2 x 3 +5;
4.13. A)	y = 4 x − 5 ;						y = 9 − 2 x − x 2 ;
	y = sign x ;						y =1 + log(x + 2) ;

	y = 2 x 2 +1 ;
			x − 2
						at x< 0
c) y =		−x
						for x ≥ 0

Find out which of these functions are monotonic, which are strictly monotonic, and which are limited:

4.14. A)

f (x) = c, c R ;

b) f (x) = cos 2 x;

c) f (x) = arctan x;

d) f (x) = e 2 x;

e) f (x) = −x 2 + 2 x;

e) f (x) =

2x+5

y = ctg7 x .

4.15. A)

f(x) = 3− x

b) f(x) =

f(x)=

x+3

x+6

x< 0,

3x+5

d) f (x) = 3 x 3 − x;

− 10 at

f(x)=

e) f (x) =

x 2 at

x ≥ 0;

x+1

f (x) = tan(sin x).

4.2. Elementary functions. Converting Function Graphs

Recall that the graph of the function f (x) in the Cartesian rectangular coordinate system Oxy is the set of all points of the plane with coordinates (x, f (x)).

Often the graph of the function y = f (x) can be constructed using transformations (shift, stretching) of the graph of some already known function.

In particular, from the graph of the function y = f (x) the graph of the function is obtained:

1) y = f (x) + a – shift along the Oy axis by a units (up if a > 0, and down if a< 0 ;

2) y = f (x −b) – shift along the Ox axis by b units (to the right, if b > 0,

and left if b< 0 ;

3) y = kf (x) – stretching along the Oy axis k times;

4) y = f (mx) – compression along the Ox axis by m times;

5) y = − f (x) – symmetric reflection relative to the Ox axis;

6) y = f (−x) – symmetrical reflection relative to the Oy axis;

7) y = f (x), as follows: part of the graph located not

below the Ox axis, remains unchanged, and the “lower” part of the graph is symmetrically reflected relative to the Ox axis;

8) y = f (x), as follows: right side of the graph (for x ≥ 0)

remains unchanged, and instead of the “left” one, a symmetrical reflection of the “right” one is constructed relative to the Oy axis.

The main elementary functions are called:

1) constant function y = c;

2) power function y = x α , α R ;

3) exponential function y = a x, a ≠ 0, a ≠1;

4) logarithmic function y = log a x , a > 0, a ≠ 1 ;

5) trigonometric functions y = sin x, y = cos x, y = tan x,

y = ctg x, y = sec x (where sec x = cos 1 x), y = cosec x (where cosec x = sin 1 x);

6) inverse trigonometric functions y = arcsin x, y = arccos x, y = arctan x, y = arcctg x.

Elementary functions are called functions obtained from basic elementary functions using a finite number of arithmetic operations (+, −, ÷) and compositions (i.e. the formation of complex functions f g).

Example 4.6. Graph the function

1) y = x 2 + 6 x + 7 ; 2) y = −2sin 4 x .

Solution: 1) by selecting a complete square, the function is transformed to the form y = (x +3) 2 − 2, therefore the graph of this function can be obtained from the graph of the function y = x 2. It is enough to first shift the parabola y = x 2 three units to the left (we obtain a graph of the function y = (x +3) 2), and then two units down (Fig. 4.1);

	standard	sinusoid						y = sinx	four times along the axis		Ox,
we obtain a graph of the function y = sin 4 x (Fig. 4.2).
										y=sin4x
										y=sin x

By stretching the resulting graph twice along the Oy axis, we obtain a graph of the function y = 2sin 4 x (Fig. 4.3). It remains to display the last graph relative to the Ox axis. The result will be the desired graph (see Fig. 4.3).

						y=2sin4x

y=– 2sin4 x

Problems to solve independently

Construct graphs of the following functions based on the graphs of basic elementary functions:

4.16. a) y = x 2 −6 x +11 ;

4.17. a) y = −2sin(x −π ) ;

4.18. a) y = − 4 x −1 ;

4.19. a) y = log 2 (−x);

4.20. a) y = x +5 ;

4.21. a) y = tg x ;

4.22. a) y = sign x;

4.23. a) y = x x + + 4 2 ;

y = 3 − 2 x − x 2 .

y = 2cos 2 x .

, Competition "Presentation for the lesson"

Presentation for the lesson

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The purpose of the lesson: Determine the patterns of transformation of function graphs.

Tasks:

Educational:

• Teach students to construct graphs of functions by transforming the graph of a given function, using parallel translation, compression (stretching), and various types of symmetry.

Educational:

• To cultivate the personal qualities of students (the ability to listen), goodwill towards others, attentiveness, accuracy, discipline, and the ability to work in a group.
• Cultivate interest in the subject and the need to acquire knowledge.

Developmental:

• To develop spatial imagination and logical thinking of students, the ability to quickly navigate the environment; develop intelligence, resourcefulness, and train memory.

Equipment:

• Multimedia installation: computer, projector.

Literature:

1. Bashmakov, M. I. Mathematics [Text]: textbook for institutions beginning. and Wednesday prof. education / M.I. Bashmakov. - 5th ed., revised. – M.: Publishing Center “Academy”, 2012. – 256 p.
2. Bashmakov, M. I. Mathematics. Problem book [Text]: textbook. allowance for education institutions early and Wednesday prof. education / M. I. Bashmakov. – M.: Publishing Center “Academy”, 2012. – 416 p.

Lesson plan:

1. Organizational moment (3 min).
2. Updating knowledge (7 min).
3. Explanation of new material (20 min).
4. Consolidation of new material (10 min).
5. Lesson summary (3 min).
6. Homework (2 min).

During the classes

1. Org. moment (3 min).

Checking those present.

Communicate the purpose of the lesson.

The basic properties of functions as dependencies between variable quantities should not change significantly when changing the method of measuring these quantities, i.e., when changing the measurement scale and reference point. However, due to a more rational choice of the method of measuring variable quantities, it is usually possible to simplify the recording of the relationship between them and bring this recording to some standard form. In geometric language, changing the way values ​​are measured means some simple transformations of graphs, which we will study today.

2. Updating knowledge (7 min).

Before we talk about graph transformations, let's review the material we covered.

Oral work. (Slide 2).

Functions given:

3. Describe the graphs of functions: , , , .

3. Explanation of new material (20 min).

The simplest transformations of graphs are their parallel transfer, compression (stretching) and some types of symmetry. Some transformations are presented in the table (Annex 1), (Slide 3).

Work in groups.

Each group constructs graphs of given functions and presents the result for discussion.

Function	Transforming the graph of a function	Function examples	Slide
	OU on A units up if A>0, and on |A| units down if A<0.	,	(Slide 4)
	Parallel transfer along the axis Oh on A units to the right if A>0, and on - A units to the left if A<0.	,	(Slide 5)
,

Exponential function is a generalization of the product of n numbers equal to a:
y (n) = a n = a·a·a···a,
to the set of real numbers x:
y (x) = a x.
Here a is a fixed real number, which is called basis of the exponential function.
An exponential function with base a is also called exponent to base a.

The generalization is carried out as follows.
For natural x = 1, 2, 3,... , the exponential function is the product of x factors:
.
Moreover, it has properties (1.5-8) (), which follow from the rules for multiplying numbers. For zero and negative values ​​of integers, the exponential function is determined using formulas (1.9-10). For fractional values ​​x = m/n rational numbers, , it is determined by formula (1.11). For real , the exponential function is defined as the limit of the sequence:
,
where is an arbitrary sequence of rational numbers converging to x: .
With this definition, the exponential function is defined for all , and satisfies properties (1.5-8), as for natural x.

A rigorous mathematical formulation of the definition of an exponential function and the proof of its properties is given on the page “Definition and proof of the properties of an exponential function”.

Properties of the Exponential Function

The exponential function y = a x has the following properties on the set of real numbers ():
(1.1) defined and continuous, for , for all ;
(1.2) for a ≠ 1 has many meanings;
(1.3) strictly increases at , strictly decreases at ,
is constant at ;
(1.4) at ;
at ;
(1.5) ;
(1.6) ;
(1.7) ;
(1.8) ;
(1.9) ;
(1.10) ;
(1.11) , .

Other useful formulas.
.
Formula for converting to an exponential function with a different exponent base:

When b = e, we obtain the expression of the exponential function through the exponential:

Private values

, , , , .

The figure shows graphs of the exponential function
y (x) = a x
for four values degree bases: a = 2 , a = 8 , a = 1/2 and a = 1/8 . It can be seen that for a > 1 the exponential function increases monotonically. The larger the base of the degree a, the stronger the growth. At 0 < a < 1 the exponential function decreases monotonically. The smaller the exponent a, the stronger the decrease.

Ascending, descending

The exponential function for is strictly monotonic and therefore has no extrema. Its main properties are presented in the table.

	y = a x , a > 1	y = ax, 0 < a < 1
Domain	- ∞ < x < + ∞	- ∞ < x < + ∞
Range of values	0 < y < + ∞	0 < y < + ∞
Monotone	monotonically increases	monotonically decreases
Zeros, y = 0	No	No
Intercept points with the ordinate axis, x = 0	y = 1	y = 1
	+ ∞	0
	0	+ ∞

Inverse function

The inverse of an exponential function with base a is the logarithm to base a.

If , then
.
If , then
.

Differentiation of an exponential function

To differentiate an exponential function, its base must be reduced to the number e, apply the table of derivatives and the rule for differentiating a complex function.

To do this you need to use the property of logarithms
and the formula from the derivatives table:
.

Let an exponential function be given:
.
We bring it to the base e:

Let's apply the rule of differentiation of complex functions. To do this, introduce the variable

Then

From the table of derivatives we have (replace the variable x with z):
.
Since is a constant, the derivative of z with respect to x is equal to
.
According to the rule of differentiation of a complex function:
.

Derivative of an exponential function

.
Derivative of nth order:
.
Deriving formulas > > >

An example of differentiating an exponential function

Find the derivative of a function
y = 3 5 x

Solution

Let's express the base of the exponential function through the number e.
3 = e ln 3
Then
.
Enter a variable
.
Then

From the table of derivatives we find:
.
Because the 5ln 3 is a constant, then the derivative of z with respect to x is equal to:
.
According to the rule of differentiation of a complex function, we have:
.

Answer

Integral

Expressions using complex numbers

Consider the complex number function z:
f (z) = a z
where z = x + iy; i 2 = - 1 .
Let us express the complex constant a in terms of modulus r and argument φ:
a = r e i φ
Then


.
The argument φ is not uniquely defined. In general
φ = φ 0 + 2 πn,
where n is an integer. Therefore the function f (z) is also not clear. Its main significance is often considered
.

Series expansion

.

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.