The text of the work is placed without images and formulas.
The full version of the work is available in the "Job Files" tab in PDF format

Introduction

Transformation of graphs of a function is one of the basic mathematical concepts directly related to practical activities. The transformation of graphs of functions is first encountered in algebra grade 9 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 scope of the function, the areas of decrease or increase, asymptotes, intervals of constant sign, etc. This important question is also submitted to the GIA. It follows that the construction and transformation of function graphs is one of the main tasks of teaching mathematics at school.

However, to plot many functions, a number of methods can be used to facilitate the construction. The above defines relevance research topics.

Object of study is the study of the transformation of graphs in school mathematics.

Subject of study - the process of constructing and transforming function graphs in a secondary school.

problem question: is it possible to build a graph of an unfamiliar function, having the skill of transforming graphs of elementary functions?

Target: plotting a function 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 tools for constructing and converting function graphs. 4. Be able to apply this theory in solving problems.

Necessary basic knowledge, skills, abilities:

Determine the value of the function by the value of the argument in various ways of specifying the function;

Build graphs of the studied functions;

Describe the behavior and properties of functions from the graph and, in the simplest cases, from the formula, find the largest and smallest values ​​from the graph of the function;

Descriptions with the help of functions of various dependencies, their representation graphically, interpretation of 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 the new formula, the function values ​​(the coordinates of the graph points) are changed by the number a, compared to the "old" function value. This leads to a parallel translation of the graph of the function 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) by means of a 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 ​​(the abscissas of the graph points) are changed by the number a, compared to the "old" argument value. This leads to a parallel transfer of the graph of the function along the OX axis: to the right if a< 0, влево, если a >0.

CONCLUSION

So 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 coordinates 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 k times, if k > 1, 2) "compression" to the point (0; 0) along the OY axis by a factor of 0, if 0< k < 1.

CONCLUSION

Therefore: to build a graph of the function y = kf(x), where k > 0 and k ≠ 1, you need to multiply the ordinates 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 OY axis by k times if k > 1; contraction to the point (0; 0) along the OY axis by a factor if 0< k < 1.

4. Function y = f(kx), where k > 0 and k ≠ 1

In the new formula, the values ​​of the argument (the abscissas of the graph points) change k times compared to the "old" value of the argument. 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 abscissas 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 values ​​of the function (the coordinates of the graph points) are reversed. This change results in a symmetrical display of the original graph of the function about the x-axis.

CONCLUSION

To build a graph of the function y = - f (x), you need a graph of the function y = f (x)

reflect symmetrically about the OX axis. Such a transformation is called a symmetry transformation about the OX axis.

6. Function y = f (-x).

In this formula, the values ​​of the argument (the abscissas of the graph points) are reversed. This change results in a symmetrical display of the original function graph with respect to the OY axis.

An example for the function y \u003d - x² this transformation is not noticeable, since this function is even and the graph does not change after the transformation. This transformation is visible when the function is odd and when neither even nor odd.

7. Function y = |f(x)|.

In the new formula, the function values ​​(the coordinates of the graph points) are under the module sign. This leads to the disappearance of parts of the graph of the original function with negative ordinates (that is, those located in the lower half-plane relative to the Ox axis) and a symmetrical display of these parts relative to the Ox axis.

8. Function y= f (|x|).

In the new formula, the argument values ​​(the abscissas of the graph points) are under the module sign. This leads to the disappearance of parts of the graph of the original function with negative abscissas (that is, those located in the left half-plane relative to the OY axis) and their replacement by parts of the original graph that are symmetrical about the OY axis.

Practical part

Consider 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.

Plot the function given by the formula

Solution. We transform this formula by highlighting the square of the binomial in this square trinomial:

1) Let's build a graph of the function

2) Perform a parallel transfer of the constructed graph to the vector

EXAMPLE 3.

TASK FROM THE USE Plotting a piecewise function

Function graph Function graph y=|2(x-3)2-2|; one

Function Graph Transformation

In this article, I will introduce you to linear transformations of function graphs and show how to use these transformations from a function graph to get 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 the module of the argument and/or functions.

The following actions cause the greatest difficulties in plotting graphs using linear transformations:

1. The isolation of the base function, in fact, the graph of which we are transforming.
2. 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 a function. Let's call her basic function.

When plotting a function we make transformations of the graph of the base function .

If we were to transform the function in the same order in which its value was found for a certain value of the argument, then

Let's consider what types of linear argument and function transformations exist, and how to perform them.

Argument transformations.

1. f(x) f(x+b)

1. We build a graph of a function

2. We 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. We plot the function

2. Shift it 2 units to the right:

2. f(x) f(kx)

1. We build a graph of a function

2. Divide the abscissas of the graph points by k, leave the ordinates of the points unchanged.

Let's plot the function.

1. We plot the function

2. Divide all abscissas of the graph points by 2, leave the ordinates unchanged:

3. f(x) f(-x)

1. We build a graph of a function

2. We display it symmetrically about the OY axis.

Let's plot the function.

1. We plot the function

2. We display it symmetrically about the OY axis:

4. f(x) f(|x|)

1. We plot the function

2. We erase the part of the graph located to the left of the OY axis, the part of the graph located to the right of the OY axis We complete it symmetrically about the OY axis:

The graph of the function looks like this:

Let's plot the function

1. We build a function graph (this is a function graph shifted along the OX axis by 2 units to the left):

2. Part of the graph located to the left of the OY (x<0) стираем:

3. The part of the graph located to the right of the OY axis (x>0) is completed symmetrically with respect to the OY axis:

Important! The two main rules for argument conversion.

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. We take the module from x.

2. Add the number 2 to the modulo x.

But we did the plotting in the reverse order:

First, we performed the transformation 2. - shifted the graph by 2 units to the left (that is, the abscissas of the points were reduced by 2, as if "vice versa")

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 being made

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. We plot the function

2. Move it along the OY axis by 2 units up:

2. f(x)Af(x)

1. We plot the function y=f(x)

2. We multiply the ordinates of all points of the graph by A, we leave the abscissas unchanged.

Let's plot the function

1. Graph the function

2. We multiply the ordinates of all points of the graph by 2:

3.f(x)-f(x)

1. We plot the function y=f(x)

Let's plot the function.

1. We build a function graph.

2. We display it symmetrically about the OX axis.

4. f(x)|f(x)|

1. We plot 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 about this axis.

Let's plot the function

1. We build a function graph. It is obtained by shifting the graph of the function along the OY axis by 2 units down:

2. Now the part of the graph located below the OX axis will be displayed symmetrically with respect 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. We plot the function y=f(x)

2. We erase the part of the graph located below the OX axis, then we complete the part of the graph located above the OX axis symmetrically about this axis.

Let's build a graph of the equation

1. We build a function graph:

2. We erase the part of the graph located below the OX axis:

3. The part of the graph located above the OX axis is completed symmetrically about this axis.

And finally, I suggest you watch the VIDEO LESSON in which I show a step-by-step algorithm for plotting a function graph

The graph of this function looks like this:

Parallel transfer.

TRANSFER ALONG THE Y-AXIS

f(x) => f(x) - b
Let it be required to plot the function y \u003d 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 graph of functions y = f(x) for b>0 and |b| units more - at b 0 or up at b To plot the function y + b = f(x), plot the function y = f(x) and move the x-axis to |b| units up for b>0 or by |b| units down at b

TRANSFER ALONG THE X-AXIS

f(x) => f(x + a)
Let it be required to plot the function y = f(x + a). Consider a 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 the function. Therefore, the graph of the function y = f(x + a) can be obtained by parallel displacement of the graph of the function y = f(x) along the x-axis to the left by |a| ones for a > 0 or to the right by |a| units for a To plot the function y = f(x + a), plot the function y = f(x) and move the y-axis to |a| units to the right for a>0 or |a| units to the left for a

Examples:

1.y=f(x+a)

2.y=f(x)+b

Reflection.

GRAPHING OF A FUNCTION OF THE VIEW Y = F(-X)

f(x) => f(-x)
Obviously, 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) with negative (positive) x values ​​corresponding 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 along the y-axis. The resulting graph is the graph of the function y = f(-x)

GRAPHING OF A FUNCTION OF THE VIEW 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 the function y = - f(x), you should plot the function y = f(x) and reflect it about the x-axis.

Examples:

1.y=-f(x)

2.y=f(-x)

3.y=-f(-x)

Deformation.

DEFORMATION OF THE GRAPH ALONG THE Y-AXIS

f(x) => kf(x)
Consider a function of the form y = k f(x), where k > 0. It is easy to see that for 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 ) or decrease its ordinates by 1/k times for k
k > 1- stretching from the Ox axis
0 - compression to the OX axis

GRAPH DEFORMATION ALONG THE X-AXIS

f(x) => f(kx)
Let it be required to plot the function y = f(kx), where k>0. Consider a function y = f(x), which takes the value y1 = f(x1) at an arbitrary point x = x1. Obviously, 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 the function. Consequently, the graph of the function y = f(kx) is compressed (for k 1) along the abscissa axis relative to the graph of the function y = f(x). Thus, we get the rule.
To plot the function y = f(kx), plot the function y = f(x) and reduce its abscissa by k times for k>1 (shrink the graph along the abscissa) or increase its abscissa 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 supervision of Tkach T.V., Vyazovov S.M., Ostroverkhova I.V.
©2014

Back forward

Attention! The slide preview is for informational purposes only and may not represent the full extent 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 graphs of functions.

Tasks:

Educational:

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

Educational:

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

Developing:

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

Equipment:

• Multimedia installation: computer, projector.

Literature:

1. Bashmakov, M.I. Mathematics [Text]: textbook for institutions early. and avg. prof. education / M. I. Bashmakov. - 5th ed., corrected. - M.: Publishing Center "Academy", 2012. - 256 p.
2. Bashmakov, M. I. Mathematics. Problem book [Text]: textbook. allowance for education. institutions at the beginning and avg. 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. Summary of the lesson (3 min).
6. Homework (2 min).

During the classes

1. Org. moment (3 min).

Checking those present.

Message about the purpose of the lesson.

The main properties of functions as dependencies between variables should not change significantly when the method of measuring these quantities changes, that is, when the scale of measurement and the reference point change. However, due to a more rational choice of the method for measuring variables, it is usually possible to simplify the notation of the relationship between them, to bring this notation to some standard form. In geometric language, changing the way quantities are measured means some simple transformations of graphs, which we will now study.

2. Actualization of knowledge (7 min).

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

oral work. (Slide 2).

Given functions:

3. Describe the function graphs: , , , .

3. Explanation of new material (20 min).

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

Group work.

Each group plots the given functions and presents the result for discussion.

Function	Function Graph Transformation	Function examples	Slide
	OU on the BUT units up if A>0, and on |A| units down if BUT<0.	,	(Slide 4)
	Parallel translation along the axis Oh on the a units to the right if a>0, and on - a units to the left if a<0.	,	(Slide 5)