Parallel transfer.
TRANSLATION ALONG THE Y-AXIS
f(x) => f(x) - b
Suppose you want to build a graph of the function y = f(x) - b. It is easy to see that the ordinates of this graph for all values of x on |b| units less than the corresponding ordinates of the function graph y = f(x) for b>0 and |b| units more - at b 0 or up at b To plot the graph of the function y + b = f(x), you should construct a graph of the function y = f(x) and move the x-axis to |b| units up at b>0 or by |b| units down at b
TRANSFER ALONG THE ABSCISS AXIS
f(x) => f(x + a)
Suppose you want to plot the function y = f(x + a). Consider the function y = f(x), which at some point x = x1 takes the value y1 = f(x1). Obviously, the function y = f(x + a) will take the same value at the point x2, the coordinate of which is determined from the equality x2 + a = x1, i.e. x2 = x1 - a, and the equality under consideration is valid for the totality of all values from the domain of definition of the function. Therefore, the graph of the function y = f(x + a) can be obtained by parallel moving the graph of the function y = f(x) along the x-axis to the left by |a| units for a > 0 or to the right by |a| units for a To construct a graph of the function y = f(x + a), you should construct a graph of the function y = f(x) and move the ordinate axis to |a| units to the right when a>0 or by |a| units to the left at a
Examples:
1.y=f(x+a)
2.y=f(x)+b
Reflection.
CONSTRUCTION OF A GRAPH OF A FUNCTION OF THE FORM Y = F(-X)
f(x) => f(-x)
It is obvious that the functions y = f(-x) and y = f(x) take equal values at points whose abscissas are equal in absolute value, but opposite in sign. In other words, the ordinates of the graph of the function y = f(-x) in the region of positive (negative) values of x will be equal to the ordinates of the graph of the function y = f(x) for the corresponding negative (positive) values of x in absolute value. Thus, we get the following rule.
To plot the function y = f(-x), you should plot the function y = f(x) and reflect it relative to the ordinate. The resulting graph is the graph of the function y = f(-x)
CONSTRUCTION OF A GRAPH OF A FUNCTION OF THE FORM Y = - F(X)
f(x) => - f(x)
The ordinates of the graph of the function y = - f(x) for all values of the argument are equal in absolute value, but opposite in sign to the ordinates of the graph of the function y = f(x) for the same values of the argument. Thus, we get the following rule.
To plot a graph of the function y = - f(x), you should plot a graph of the function y = f(x) and reflect it relative to the x-axis.
Examples:
1.y=-f(x)
2.y=f(-x)
3.y=-f(-x)
Deformation.
GRAPH DEFORMATION ALONG THE Y-AXIS
f(x) => k f(x)
Consider a function of the form y = k f(x), where k > 0. It is easy to see that with equal values of the argument, the ordinates of the graph of this function will be k times greater than the ordinates of the graph of the function y = f(x) for k > 1 or 1/k times less than the ordinates of the graph of the function y = f(x) for k To construct a graph of the function y = k f(x), you should construct a graph of the function y = f(x) and increase its ordinates by k times for k > 1 (stretch the graph along the ordinate axis ) or reduce its ordinates by 1/k times at k
k > 1- stretching from the Ox axis
0 - compression to the OX axis
GRAPH DEFORMATION ALONG THE ABSCISS AXIS
f(x) => f(k x)
Let it be necessary to construct a graph of the function y = f(kx), where k>0. Consider the function y = f(x), which in arbitrary point x = x1 takes the value y1 = f(x1). It is obvious that the function y = f(kx) takes the same value at the point x = x2, the coordinate of which is determined by the equality x1 = kx2, and this equality is valid for the totality of all values of x from the domain of definition of the function. Consequently, the graph of the function y = f(kx) turns out to be compressed (for k 1) along the abscissa axis relative to the graph of the function y = f(x). Thus, we get the rule.
To construct a graph of the function y = f(kx), you should construct a graph of the function y = f(x) and reduce its abscissas by k times for k>1 (compress the graph along the abscissa axis) or increase its abscissas by 1/k times for k
k > 1- compression to the Oy axis
0 - stretching from the OY axis
The work was carried out by Alexander Chichkanov, Dmitry Leonov under the guidance of T.V. Tkach, S.M. Vyazov, I.V. Ostroverkhova.
©2014
Depending on the conditions of physical processes, some quantities take constant values and are called constants, others change under certain conditions and are called variables.
Careful Study environment shows that physical quantities 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 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 the main 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, different from unity: 
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) (m0) 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.
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Introduction
Transformation of function graphs is one of the basic mathematical concepts directly related to practical activities. Transformation of graphs of functions is first encountered in 9th grade algebra when studying the topic “Quadratic Function”. The quadratic function is introduced and studied in close connection with quadratic equations and inequalities. Also, many mathematical concepts are considered by graphical methods, for example, in grades 10 - 11, the study of a function makes it possible to find the domain of definition and the domain of value of the function, domains of decreasing or increasing, asymptotes, intervals of constant sign, etc. This important issue is also brought up at the GIA. It follows that constructing and transforming graphs of functions is one of the main tasks of teaching mathematics at school.
However, to plot graphs of many functions, you can use a number of methods that make plotting easier. The above determines relevance research topics.
Object of study is to study the transformation of graphs in school mathematics.
Subject of study - the process of constructing and transforming function graphs in a secondary school.
Problematic question: Is it possible to construct a graph of an unfamiliar function if you have the skill of converting graphs of elementary functions?
Target: plotting functions in an unfamiliar situation.
Tasks:
1. Analyze the educational material on the problem under study. 2. Identify schemes for transforming function graphs in a school mathematics course. 3. Select the most effective methods and means for constructing and transforming function graphs. 4.Be able to apply this theory in solving problems.
Required initial knowledge, skills and abilities:
Determine the value of a function by the value of the argument in different ways of specifying the function;
Build graphs of the studied functions;
Describe the behavior and properties of functions using a graph and, in the simplest cases, using a formula; find the largest and smallest values from a graph of a function;
Descriptions using functions of various dependencies, representing them graphically, interpreting graphs.
Main part
Theoretical part
As the initial graph of the function y = f(x), I will choose a quadratic function y = x 2 . I will consider cases of transformation of this graph associated with changes in the formula that defines this function and draw conclusions for any function.
1. Function y = f(x) + a
IN new formula the function values (the ordinates of the graph points) change by the number a, compared to the “old” function value. This leads to a parallel transfer of the function graph along the OY axis:
up if a > 0; down if a< 0.
CONCLUSION
Thus, the graph of the function y=f(x)+a is obtained from the graph of the function y=f(x) using parallel translation along the ordinate axis by a units up if a > 0, and by a units down if a< 0.
2. Function y = f(x-a),
In the new formula, the argument values (abscissas of the graph points) change by the number a, compared to the “old” argument value. This leads to a parallel transfer of the function graph along the OX axis: to the right, if a< 0, влево, если a >0.
CONCLUSION
This means that the graph of the function y= f(x - a) is obtained from the graph of the function y=f(x) by parallel translation along the abscissa axis by a units to the left if a > 0, and by a units to the right if a< 0.
3. Function y = k f(x), where k > 0 and k ≠ 1
In the new formula, the function values (the ordinates of the graph points) change k times compared to the “old” function value. This leads to: 1) “stretching” from the point (0; 0) along the OY axis by a factor of k, if k > 1, 2) “compression” to the point (0; 0) along the OY axis by a factor of, if 0< k < 1.
CONCLUSION
Consequently: to plot a graph of the function y = kf(x), where k > 0 and k ≠ 1, you need the ordinates of the points given schedule function y = f(x) multiplied by k. Such a transformation is called stretching from the point (0; 0) along the OY axis k times if k > 1; compression to the point (0; 0) along the OY axis times if 0< k < 1.
4. Function y = f(kx), where k > 0 and k ≠ 1
In the new formula, the argument values (abscissas of the graph points) change k times compared to the “old” argument value. This leads to: 1) “stretching” from the point (0; 0) along the OX axis by 1/k times, if 0< k < 1; 2) «сжатию» к точке (0; 0) вдоль оси OX. в k раз, если k > 1.
CONCLUSION
And so: to build a graph of the function y = f(kx), where k > 0 and k ≠ 1, you need to multiply the abscissa of the points of the given graph of the function y=f(x) by k. Such a transformation is called stretching from the point (0; 0) along the OX axis by 1/k times, if 0< k < 1, сжатием к точке (0; 0) вдоль оси OX. в k раз, если k > 1.
5. Function y = - f (x).
In this formula, the function values (the ordinates of the graph points) are reversed. This change leads to a symmetrical display of the original graph of the function relative to the Ox axis.
CONCLUSION
To plot a graph of the function y = - f (x), you need a graph of the function y= f(x)
reflect symmetrically about the OX axis. This transformation is called a symmetry transformation about the OX axis.
6. Function y = f (-x).
In this formula, the values of the argument (abscissa of the graph points) are reversed. This change leads to a symmetrical display of the original graph of the function relative to the OY axis.
Example for the function y = - x² this transformation is not noticeable, because this function even and the graph does not change after the transformation. This transformation is visible when the function is odd and when it is neither even nor odd.
7. Function y = |f(x)|.
In the new formula, the function values (the ordinates of the graph points) are under the modulus sign. This leads to the disappearance of parts of the graph of the original function with negative ordinates (i.e., those located in the lower half-plane relative to the Ox axis) and the symmetrical display of these parts relative to the Ox axis.
8. Function y= f (|x|).
In the new formula, the argument values (abscissas of the graph points) are under the modulus sign. This leads to the disappearance of parts of the graph of the original function with negative abscissas (i.e., located in the left half-plane relative to the OY axis) and their replacement by parts of the original graph that are symmetrical relative to the OY axis.
Practical part
Let's look at a few examples of the application of the above theory.
EXAMPLE 1.
Solution. Let's transform this formula:
1) Let's build a graph of the function
EXAMPLE 2.
Graph the function given by the formula
Solution. Let's transform this formula by highlighting in this quadratic trinomial square of the binomial:
1) Let's build a graph of the function
2) Let's do it parallel transfer constructed graphics onto vector
EXAMPLE 3.
TASK FROM the Unified State Exam Graphing a Piecewise Function
Graph of the function Graph of the function y=|2(x-3)2-2|; 1
Converting Function Graphs
In this article I will introduce you to linear transformations of function graphs and show you how to use these transformations to obtain a function graph from a function graph 
A linear transformation of a function is a transformation of the function itself and/or its argument to the form 
, as well as a transformation containing an argument and/or function module.
The greatest difficulties when constructing graphs using linear transformations are caused by the following actions:
- Isolation basic function, in fact, the graph of which we are transforming.
- Definitions of the order of transformations.
AND It is on these points that we will dwell in more detail.
Let's take a closer look at the function
It is based on the function . Let's call her basic function.
When plotting a function we perform transformations on the graph of the base function.
If we were to perform function transformations in the same order in which its value was found when a certain value argument, then
Let's consider what types of linear transformations of argument and function exist, and how to perform them.
Argument transformations.
1. f(x) f(x+b)
1. Build a graph of the function
2. Shift the graph of the function along the OX axis by |b| units
- left if b>0
- right if b<0
Let's plot the function
1. Build a graph of the function
2. Shift it 2 units to the right:
2. f(x) f(kx)
1. Build a graph of the function
2. Divide the abscissas of the graph points by k, leaving the ordinates of the points unchanged.
Let's build a graph of the function.
1. Build a graph of the function
2. Divide all abscissas of the graph points by 2, leaving the ordinates unchanged:
3. f(x) f(-x)
1. Build a graph of the function
2. Display it symmetrically relative to the OY axis.
Let's build a graph of the function.
1. Build a graph of the function
2. Display it symmetrically relative to the OY axis:
4. f(x) f(|x|)
1. Build a graph of the function
2. The part of the graph located to the left of the OY axis is erased, the part of the graph located to the right of the OY axis is completed symmetrically relative to the OY axis:
The function graph looks like this:
Let's plot the function
1. We build a graph of the function (this is a graph of the function, shifted along the OX axis by 2 units to the left):
2. Part of the graph located to the left of the OY (x) axis<0) стираем:
3. We complete the part of the graph located to the right of the OY axis (x>0) symmetrically relative to the OY axis:
Important! Two main rules for transforming an argument.
1. All argument transformations are performed along the OX axis
2. All transformations of the argument are performed “vice versa” and “in reverse order”.
For example, in a function the sequence of argument transformations is as follows:
1. Take the modulus of x.
2. Add the number 2 to modulo x.
But we constructed the graph in reverse order:
First, transformation 2 was performed - the graph was shifted by 2 units to the left (that is, the abscissas of the points were reduced by 2, as if “in reverse”)
Then we performed the transformation f(x) f(|x|).
Briefly, the sequence of transformations is written as follows:
Now let's talk about function transformation . Transformations are taking place
1. Along the OY axis.
2. In the same sequence in which the actions are performed.
These are the transformations:
1. f(x)f(x)+D
2. Shift it along the OY axis by |D| units
- up if D>0
- down if D<0
Let's plot the function
1. Build a graph of the function
2. Shift it along the OY axis 2 units up:
2. f(x)Af(x)
1. Build a graph of the function y=f(x)
2. We multiply the ordinates of all points of the graph by A, leaving the abscissas unchanged.
Let's plot the function
1. Let's build a graph of the function
2. Multiply the ordinates of all points on the graph by 2:
3.f(x)-f(x)
1. Build a graph of the function y=f(x)
Let's build a graph of the function.
1. Build a graph of the function.
2. We display it symmetrically relative to the OX axis.
4. f(x)|f(x)|
1. Build a graph of the function y=f(x)
2. The part of the graph located above the OX axis is left unchanged, the part of the graph located below the OX axis is displayed symmetrically relative to this axis.
Let's plot the function
1. Build a graph of the function. It is obtained by shifting the function graph along the OY axis by 2 units down:
2. Now we will display the part of the graph located below the OX axis symmetrically relative to this axis:
And the last transformation, which, strictly speaking, cannot be called a function transformation, since the result of this transformation is no longer a function:
|y|=f(x)
1. Build a graph of the function y=f(x)
2. We erase the part of the graph located below the OX axis, then complete the part of the graph located above the OX axis symmetrically relative to this axis.
Let's plot the equation
1. We build a graph of the function:
2. We erase the part of the graph located below the OX axis:
3. We complete the part of the graph located above the OX axis symmetrically relative to this axis.
And finally, I suggest you watch a VIDEO TUTORIAL in which I show a step-by-step algorithm for constructing a graph of a function
The graph of this function looks like this:
Back forward
Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested in this work, please download the full version.
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:
- 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.
- 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:
- Organizational moment (3 min).
- Updating knowledge (7 min).
- Explanation of new material (20 min).
- Consolidation of new material (10 min).
- Lesson summary (3 min).
- 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)
|
