بحث عن الدوال الرئيسة الام والتحويلات الهندسية

بحث عن الدوال الرئيسة الام والتحويلات الهندسية , مما لا شك فيه أن هذا الموضوع من أهم وأفضل الموضوعات التي يمكن أن أتحدث عنها اليوم، حيث أنه موضوع شيق ويتناول نقاط حيوية، تخص كل فرد في المجتمع، وأتمنى من الله عز وجل أن يوفقني في عرض جميع النقاط والعناصر التي تتعلق بهذا الموضوع.

Research on the main functions and geometric transformations is one of the important research that is closely related to the study of functions and how to analyze and represent them graphically. Learning this topic is one of the things that make it easier for the student to understand the formation of functions and graphs; The difference between them is due to the different geometric transformations. In this article from the reference website, we will include for you a research on the main parent functions and geometric transformations, including the definition of these parent functions and their properties, in addition to clarifying the geometric transformations and their impact.

Introduction to the study of the main parent functions and geometric transformations

In the name of God the Merciful Praise be to Allah, the Lord of the worlds; To Him be praise, a good and blessed praise worthy of the majesty of His countenance and the greatness of His authority, and prayers and peace be upon our Master Muhammad, the Master, the Guide and the Guide, and upon all his family and companions, and then:

I present this research on one of the most important topics in mathematics, which is the main functions and the geometric transformations that take place on them. It and the arithmetic operations that affect it, leading to a change in its place or shape on the graphic level, according to the operation being performed.

Understanding engineering transformations helps build applications and programs that help owners of companies and organizations make diagrams that show the course of events; Such as studying the reduction of the selling price on the gross profit, or studying the impact of a particular loss on the company’s revenues, and so on.

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Find the main parent functions and geometric transformations

I will start in this research by mentioning the definition of the main parent functions and their types in a general way, and then I will separate the main types of functions and their standard formulas, and explain the parent function for each of them, with an explanation of how to draw them in a simplified manner. Then a statement of the characteristics of the parent main functions, and the geometric transformations that take place on them in order, so that the order of the topics will be as follows:

• Part one: Defining the main parent functions and their types; Constant function, parent linear function, parent quadratic function, parent square root function, parent relative function, parent degree function, parent absolute value function.
• Chapter Two: Characteristics of the Parent Main Function
• Chapter Three: Defining Geometric Transformations on Parent Functions
• Chapter Four: Types of Geometric Transformations on Functions
  • The first requirement: the withdrawal of the main main functions: the vertical withdrawal of the functions, and the horizontal withdrawal of the functions.
  • The second requirement: the reflection around the two coordinate axes of the parent principal functions: the reflection around the x-axis, and the reflection around the y-axis.
  • The third requirement: the expansion of the mother main functions: the vertical expansion of the functions, and the horizontal expansion of the functions
• Chapter Five: Geometric Transformations with Absolute Value Functions

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Definition of the main parent functions

Functions consist of different families, and these families share traits and properties, and in each family there is a function known as the parent main function as the simplest function in the family, and by performing geometric transformations on it, we can find the rest of the family functions. This is certainly reflected in its graphic representation through its displacement, expansion, reflection, and other matters that differ according to the processes taking place on it.

static function

The constant function is a type of polynomial functions, in which the degree of the polynomial functions is zero, that is, the exponent on the variable (x) in it is equal to zero, and therefore this function is a constant (a certain number). And the formula of the constant function is: (f (x) = a) and the following is the graph of the constant function.

linear parent function

The linear function is one of the polynomial functions of the first degree, in which the variable is raised to the power of one, and its general formula is: (f (x) = ax + c). For an accurate result, 5 points can be taken. [1]

And the parent linear function is: (f (x) = x)

quadratic parent function

The quadratic function is one of the types of polynomials, and the degree of function is the second, that is, the largest exponent in it is 2, and its standard form is (f(x) = ax2 + bx + c). find pictures of three points; Which are the zeros of the coupling and the vertex of the curve or what is known as the segment vertex in which the coupling curve is divided into two identical parts.

Parent quadratic function: f(x) = x2, and the following figure represents the graph of the parent quadratic function.

Parent square root function

A radical function is a type of real function, written in the following standard form: f(x) = √g(x). where e is a polynomial function. In order to graph this function, we must first determine the domain of the function, then find images of a set of elements from the domain, and then project the resulting points and graph them in the Cartesian plane. The domain of the radical function is all the values ​​that make the subradical greater than or equal to zero. [2]

Parent square root function: f(x) = √x, and the following graph between the parent square root function:

Parent function

Relative functions are a type of functions, which are functions that can be written as a fraction (numerator and denominator) between polynomials, and the standard form is: (f (x) /g (x), since g (x) cannot be equal The domain of the relative function can be determined by determining the zeros of the denominator, the domain is the set of real numbers except for what makes the denominator equal to zero.The calculation of the domain of the relative function is by analyzing the polynomial in the denominator and finding and excluding zeros from the set of real numbers.

Relative parent function: f(x) = 1/x, also called the reciprocal function, the two parts of the function are the same with respect to the origin. The following figure represents the parent relative function:

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Parent function

The largest integer function, which defines the stair function because of its stair-like shape, is one of the types of real functions, and it is an association in which every integer is paired with the variable x, and symbolizes the largest integer with the symbol [x]. And if n≤x

Parent degree function: f(x) = [x]. The following image represents the parent function:

The parent absolute value function

The absolute value means the distance of the point from zero on the number line, that is, the number resulting from it is without a sign, and the general form of it is f(x) = ιg(x)ι, this means that all the answers to the absolute value function will be positive and so is the graph It will not go down to the negative space in the event that something outside the absolute value is added to it, and its curve takes the form of the letter V, and to draw it, it must be redefined by knowing to find the field so that the bifurcation point is the point from which the two curve lines branch.

The parent absolute value function: f(x) = ιxι, and the following image represents the parent degree function:

The properties of the parent main function

We can find for each of the parent functions that we mentioned earlier, the characteristics that distinguish it from others, in terms of mentioning the domain, the x- and y-section, and whether the curve has two identical halves, and whether it is continuous or not, and what is the beginning and end of the curve, i.e. the starting point And the end of its range, and describe the function in terms of increasing and decreasing. In the following, we will give you an example of the properties of the quadratic function: f(x) = x2:

• The domain of the function is the set of real numbers (h), and its range is . [0، ∞).
• ليس للمنحنى سوى مقطع سيني وصادي واحد وهو يمثل أيضًا رأس القطع، وهي النقطة (0،0).
• المنحنى متماثل من محور التماثل التي تمر من النقطة (0،0).
• الدالة زوجية.
• المنحنى متصل عند جميع قيم المجال.
• يبدأ المنحنى عند x=0 ، وتكون (∞ = ∞→limx )، أي نهاية الاقتران عندما x تؤول إلى المالانهاية هي المالانهاية.

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تعريف التحويلات الهندسية على الدوال الأم

التحويلات الهندسية على الدوال الأم هي مجموعة من العمليات التي يتم إجراؤها على الدوال الأم فتؤثر فيها مما يغير في شكل منحنى الدالة الأم، من حيث الموقع ومن حيث الشكل والأبعاد، وهي نوعان: [3]

• Standard geometric transformations: They change the location of the curve only, not its shape or dimensions.
• Standard non-geometric transformations: They change the shape or dimensions of the curve.

Types of geometric transformations on functions

As we mentioned, the standard transformations are the changes that occur in the parent function, which leads to a change in its shape, dimensions and location. Examples of standard transformations are pull-down (displacement), and reversal, and non-standard transformations include dilation and transformations from absolute value mode. We will discuss the types of geometric transformations on the parent functions in detail: [4]

Withdrawal of the parent principal functions

The withdrawal from the standard geometric transformations, which is one of the changes that move the position of the curve, is divided into two parts; Vertical withdrawal of functions, which is related to moving the curve of the function up and down, and horizontal withdrawal, which moves the curve of the function to the right and to the left. [5]

Vertical withdrawal of functions

The vertical pull is when the function curve is shifted up or down as a result of increasing a positive or negative number to the function, and its form: (g(x) = f(x) + k ), such that when k is less than zero the curve shifts to Downward in k units, but when k is greater than zero the curve shifts upward in k units. As in the following figure:

Horizontal withdrawal of functions

Horizontal retraction is when the function curve is shifted to the right or to the left as a result of increasing a positive or negative number to the variable, and its form: (g(x) = f(xh)) , so that when the value of h is less than zero, the curve will shift The curve will shift to the left in h units, but when h is greater than zero the curve will shift to the right in h units. As in the following figure:

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Reflection about the two coordinate axes of the parent principal functions

Reflection is one of the standard geometric transformations, in which the function curve has an inverse image like a mirror with respect to a specified line, and it is divided into two parts; They are the reflection about the x axis, and the reflection about the y axis, and we will show these two types in the following:

Reflection about the x-axis

The curve of the function g(x) = – f(x) is a reflection of f(x) about the x axis. The following figure represents an illustration of this type of reflection:

Reflection about the y-axis

The curve of the function g(x) = f(-x) is a reflection of the curve of the function f(x) about the y-axis. The following figure represents an illustration of this type of reflection:

Dilation of the parent principal functions

Expansion enumerates a kind of non-standard geometric transformation of the parent functions, and it leads to a narrowing or expansion of the curve of the function in a vertical or horizontal manner, and it is divided into two types of exponential expansion, and the other is horizontal expansion.

Vertical expansion of the functions

If the number a is a positive real number, then the curve of the function g(x) = af(x) is the vertical expansion and stretching of the f(x) curve if the constant a has a value greater than one, and if the value of the constant is less than one and greater than zero in between, then the curve g(x) will represent the vertical narrowing i.e. pressure of the function f(x) curve.

Horizontal expansion of functions

If the number a is a positive real number, then the function curve g(x) = f(ax) will have a horizontal narrowing and compression of the f(x) curve if the constant has a value greater than 1, but if the value of the constant a between the numbers one and zero; i.e. less than one and greater than zero then the curve g(x) will represent the horizontal expansion and stretching of the function curve f(x).

Geometric transformations with absolute value functions

Absolute value transformations are geometric transformations that are performed on some functions and they are non-standard transformations, and they are divided into two types as follows:

• g(x) = ιf(x)ι : This type of transformation changes the under-positive x-axis portion of the coupling curve to be inverted and above the curve, so it is a reflection of that portion about the positive x-axis. It is the first picture below.
• g(x) =f(ιxι): This type of geometric transformation occurs on a part of the coupling that lies to the left of the y-curve, so its place becomes an image of the part that lies to the right of the y-axis of the curve, so it is a reflection about the y-axis of the part on the right. The second image is at the bottom.

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Conclusion of a search for the main parent functions and geometric transformations

To conclude the topic of searching for the main parent functions and geometric transformations, it is necessary to emphasize the importance of studying this topic because of its great applications that can be benefited from during various works through the development of programs and graphic representations that depend in particular on the geometric transformations that occur on the parent functions.

One of the life images to benefit from engineering transformations in public life is to study the difference between the production price and costs after reducing or raising them by making adjustments to the basic production function and then studying the two curves before and after in order to understand the effect. In addition, there are many other applications that can be used.

Research on parent principal functions and geometric transformations pdf

Functions are a relationship between variables and numbers, and what distinguishes them from relationships is that an element in the field is related to one element in the range, and the study of functions is useful in many sciences and applications in life. Its importance in the study of integration sciences and its applications, in addition to its importance in many physical, chemical, and even life and human sciences. And you can “from here” download a research paper on the main parent functions and geometric transformations in pdf.

Search for the main parent functions and geometric transformations clear doc

Writing a paper on the main parent functions and geometric transformations is very important in order to understand the functions and the geometric transformations that occur on them, and the applications related to this type of science. The Word program is the best program for writing research papers for students, due to the ability this program allows for writing and modification, in addition to the ease of writing equations through ready-made equations images included in the program, as well as the ability to easily insert images as well. And you can download a ready-made search for the main parent functions and geometric transformations “from here” in doc format.

At the conclusion of this article, we have included for you a research on the parent principal functions and geometric transformations in both doc and pdf formats, including a set of mathematical information supported by images about the parent principal functions and the effect of geometric transformations on them.

خاتمة لموضوعنا بحث عن الدوال الرئيسة الام والتحويلات الهندسية ,وفي نهاية الموضوع، أتمنى من الله تعالى أن أكون قد استطعت توضيح كافة الجوانب التي تتعلق بهذا الموضوع، وأن أكون قدمت معلومات مفيدة وقيمة.