Power function, its properties and graphs. Function. Power function The power function has the form
Lesson and presentation on the topic: "Power functions. Negative integer exponent. Graph of a power function"
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Kind of a power function with a negative exponent
Guys, we continue to study numerical functions. The topic of today's lesson will also be power functions, but not with a natural exponent, but with a negative integer.looks like this: $y=x^(-n)=\frac(1)(x^n)$.
One of these functions we know very well is hyperbole. Guys, do you remember the hyperbole graph? Build it yourself.
Let's look at one of the functions suitable for us and define properties for it. $y=x^(-2)=\frac(1)(x^2)$.
Let's start with parity. It is worth noting that the parity property greatly simplifies the construction of function graphs, since we can build a half of the graph and then just reflect it.
The domain of our function is the set of real numbers, except for zero, we all know very well that you cannot divide by zero. The domain of definition is a symmetric set, we proceed to the calculation of the value of the function from a negative argument.
$f(-x)=\frac(1)((-x)^2)=\frac(1)(x^2)=f(x)$.
Our function is even. So, we can build a graph for $x≥0$, and then reflect it along the y-axis.
Guys, this time I propose to build a function graph together, as they do in "adult" mathematics. First, we define the properties of our function, and then we build a graph based on them. We will take into account that $x>0$.
1. Domain D(y)=(0;+∞).
2. The function is decreasing. Let's check it out. Let $x1
3. The function is limited from below. It is obvious that $\frac(1)(x^2)>0$, which means that it is bounded from below.
There is no upper limit, because if we take the value of the argument very small, close to zero, then the value of the function will tend to plus infinity.
4. There is no maximum or minimum value. There is no maximum value, since the function is not bounded from above. What about the smallest value, because the function is bounded from below.
What does it mean that a function has the smallest value?
There is a point x0 such that for all x from the domain $f(x)≥f(x0)$, but our function is decreasing on the entire domain, then there is such a number $х1>x0$, but $f(x1) Plots of power functions with negative exponents
Let's build a graph of our function by points. 
The graph of our function is very similar to the graph of a hyperbola.
Let's use the parity property and reflect the graph along the y-axis. 
Let's write the properties of our function for all x values.
1) D(y)=(-∞;0)U(0;+∞).
2) An even function.
3) Increases by (-∞;0], decreases by .
Solution. The function decreases over the entire domain of definition, then it reaches its maximum and minimum values at the ends of the segment. The largest value will be on the left end of the segment $f(1)=1$, the smallest on the right end $f(3)=\frac(1)(27)$.
Answer: The largest value is 1, the smallest is 1/27.
Example. Plot the function $y=(x+2)^(-4)+1$.
Solution. The graph of our function is obtained from the graph of the function $y=x^(-4)$ by moving it two units to the left and one unit up.
Let's build a graph: 
Tasks for independent solution
1. Find the smallest and largest value of the function $y=\frac(1)(x^4)$ on the segment .2. Plot the function $y=(x-3)^(-5)+2$.
The functions y \u003d ax, y \u003d ax 2, y \u003d a / x - are special types of a power function for n = 1, n = 2, n = -1 .
If n fractional number p/
q with an even denominator q and odd numerator R, then the value 
can have two signs, and the graph has one more part at the bottom of the x-axis X, and it is symmetrical to the upper part.
We see a graph of a two-valued function y \u003d ± 2x 1/2, i.e. 
represented by a parabola with a horizontal axis.
Function Graphs y = xn at n = -0,1; -1/3; -1/2; -1; -2; -3; -10 . These graphs pass through the point (1; 1).
When n = -1 we get hyperbole. At n < - 1 the graph of the power function is first located above the hyperbola, i.e. between x = 0 and x = 1, and then below (at x > 1). If a n> -1 the graph runs in reverse. Negative values X and fractional values n similar for positive n.
All graphs approach indefinitely as to the x-axis X, as well as to the y-axis at without coming into contact with them. Because of their resemblance to a hyperbola, these graphs are called hyperbolas. n th order.
1. Analysis of educational literature on the topic: “Properties of a power function”
The study of the power function begins in the 7th grade, with special cases, and continues throughout the course of algebra. Up to grade 11, knowledge about the power function is generalized, expanded and systematized.
The analysis of educational literature must be carried out for grade 9 in order to build the content of the didactic manual based on this analysis of educational literature.
Textbook: “Algebra. Grade 9”. Mordkovich A. G., Semenov P. V. (Mnemozina, 2009)
The textbook deals with power functions with an integer exponent. Theoretical material on the topic "Power function" is included in the chapter " Numeric functions» in separate paragraphs, which consider both the functions themselves and their properties and graphs.
Presentation of the material accessible to schoolchildren, included big number examples with detailed and thorough solutions in the 1st part (in the textbook), and exercises for independent work placed in the 2nd part (in the problem book).
The structure of the study of the material:
CHAPTER 3 Numeric Functions
§12. Functions, their properties and graphs.
§13. Functions, their properties and graphs.
§fourteen. Functions, its properties and graph.
Next, power functions are defined as functions with a natural exponent (first, special cases of power functions are given, then the general formula is revealed). We consider power functions with an even exponent, their graphs, by which properties are later revealed (range of value and domain of definition of the function, even and odd, monotonicity, continuity, maximum and minimum value of the function, convexity). Next, we consider power functions with an odd exponent, as well as their graphs and properties.
In § 13 power functions with negative exponents are defined: first even functions, then odd ones. Similar to power functions with a natural exponent, special cases are given:
After that, the general formula is revealed, the graphs and properties are also considered.
In § 14 we introduce the function
its properties and graph as special case power function with rational exponent n =
The transformation of graphs (symmetry) boils down to the fact that the graph of an even function is symmetrical about the y-axis, and the graph of an odd function is about the origin. Therefore, for the steppe functions, we consider given function on a certain ray, its graph is built and, using symmetry, a graph is built on the entire number line. Next, the graph is read, i.e., according to the graph, the properties of the function are listed according to the scheme:
1) domain of definition;
2) even, odd;
3) monotony;
4) boundedness from below, from above;
5) the smallest and greatest value functions;
6) continuity;
7) range of values;
8) bulge.
a) goes to the auxiliary coordinate system with the origin at the point at which the values \u200b\u200bare obtained at x = 0 and y = 0.
b) “binds” the function to new system coordinates.
Example 3. Graph a function
Solution. Let's move on to the auxiliary coordinate system with the origin at the point (-1; -2) (dashed lines in Fig. 117) and "attach" the function to the new coordinate system. We get the required schedule (Fig. 117)
In the problem book “Algebra. Grade 9.” under the editorship of Mordkovich A. G. and Semenov P. V. a diverse system of exercises is presented. The set of exercises is divided into two blocks: the first contains tasks of two basic levels: oral (semi-oral) and tasks of medium difficulty; the second block contains tasks of a level above average or increased difficulty. Most of the tasks of the second and third levels are answered. The taskbook contains a large number of various tasks for plotting graphs. various kinds power function and determining the properties of a function from its graph. For example:
No. 12.10. Plot the function:
No. 12.15. Solve the Equation Graphically
No. 12.19. Plot and Read the Graph of a Function
Plot and Read the Graph of a Function
Textbook: “Algebra. Grade 9”. Nikolsky S. M., Potapov M. K., Reshetnikov N. N., Shevkin A. V. (Enlightenment, 2006)
This textbook is also intended for general education classes, in which additional materials and complex tasks can be omitted. If there are enough hours, if the class shows interest in mathematics, then due to the additions at the end of the chapters of the textbook, as well as points and individual tasks with an asterisk, which are optional in ordinary general education classes, it is possible to expand and deepen the content of the studied material to the volume provided by the program for classes with in-depth study of mathematics. That is, the textbook can be used both in ordinary and in classes with in-depth study of mathematics.
The structure of the study of the material:
CHAPTER II. Degree of
§four. degree root
4.1 Function properties
4.2 Graph of a function
4.3 The concept of the root of a degree
4.4 Even and odd roots
4.5 Arithmetic root
4.6 Properties of roots
4.7 *Root of a natural number
4.8 *Function
The study of the topic begins with the properties of the function (for example, n = 2 and n = 3) and its graph. Then we study the n-th root, the arithmetic root, and the properties of n-th roots, and how they apply to transforming expressions. In classes with an in-depth study of mathematics, the following topics are additionally considered: "Function", "Power with a rational exponent and its properties."
It is stated that the functions have a number of identical properties (domain, zeros of the function, evenness, oddness, continuity, intervals of monotonicity). Therefore, it is advisable to consider in the general case a function, where is some natural number, . The introduction of the definition of the graph of a function is carried out through the definition of a parabola. That is, according to known fact that the graph of a function is a parabola, then this graph is called a parabola of the second degree, the graph of a function is called a parabola of the th degree or, briefly, a parabola. Function properties are considered only for non-negative ones with some proofs.
The study of building a graph of a function begins with the display of graphs of functions on one coordinate plane only for non-negative values.
The study of the function is based on previously acquired knowledge about the arithmetic root of the degree. The construction of the graph of the function is carried out in the Cartesian coordinate system. To begin with, a power function and the construction of its graph in the O coordinate system are considered. Thus, it is proved that the function graph is part of a degree parabola.
1) If x = 0, then y = 0.
2) If, then.
3) The function is increasing.
4) If, then.
5) The function is continuous.
The system of exercises on the topic "Power function" is diverse. It contains training tasks both oral and written. For example:
No. 316. Given a function
Explore this function and plot its graph.
#318 Graph the function
№ 321. In one coordinate system, build graphs of functions
#441 Plot a function graph for:
#442 Plot a function graph for:
Textbook: “Algebra. Grade 9". Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova (Enlightenment, 2009)
This textbook is intended for secondary schools.
The structure of the study of the material:
CHAPTER IV. Degree with rational exponent
§9. Power function
21. Even and Odd Functions
22. Function
§ten. Root nth degree
23. Determining the root of the nth degree
24. Properties of the arithmetic root of the nth degree
§eleven. Degree with rational exponent and its properties
25. Determining the degree with a fractional exponent
26. Properties with rational exponent
27. Converting expressions containing degrees with fractional exponents
The study of a power function begins with the introduction of the concepts of even and odd functions using examples of comparing function values for two opposite values of the argument. Further, the definition of an even and odd function is given with the construction of the corresponding graphs.
It is said that the power functions at = 1, 2 and 3 (i.e. functions), their properties and graphs, have been studied earlier. Next, the properties of the power function and the features of its graph for any natural number are clarified. Functions are considered when the exponent n is an even number, then n is an odd number. Parse properties on examples, according to the scheme:
1. Domain of definition;
2. Scope of value;
3. Function zeros;
4. Parity;
5. Odd;
6. Monotonicity of a function.
The next section of the chapter is devoted to the n-th root, in which the definition is introduced and properties are considered.
The definition is repeated: square root from the number a is called such a number, the square of which is equal to a. The root of any natural degree n is defined similarly: the root of the nth degree from the number a is such a number, nth power which is equal to a. To do this, we consider a power function first with an odd exponent n and its graph, which shows that for any number a there is a unique value x, the nth power of which is equal to a. Then a power function with an even exponent n is considered, moreover, if, then there are two opposite values of x, for such a number is one (number 0), for there are no such numbers.
At the end of the chapter, a degree with a rational exponent and its properties are considered.
The system of exercises is varied. For example:
No. 503. Plot a Function
No. 508. Solve the Equation Graphically
No. 513. Using the graph of the function, solve the equation
No. 580. Plot the Function
No. 644. Plot the function f , knowing that it is odd and that its value at can be found by the formula
No. 643. Plot the Function
No. 663. Plot the function graph. Using the graph, compare the value of the roots
No. 669. Plot the Function
Textbook: “Algebra. Grade 9". Sh.A. Alimov, Yu. M. Kolyagin, Yu. V. Sidorov and others (Enlightenment, 2009)
When studying this topic, special attention is paid to the properties of functions and the display of these properties on graphs. At the same time, initial skills are formed to perform the simplest transformations of function graphs.
The structure of the study of the material:
CHAPTER III. Power function
§12. Function scope
§13. Function Ascending and Decreasing
§fourteen. Even and odd functions
§fifteen. Function
§16. Inequalities and Equations Containing Power
The main purpose of this chapter is not only to introduce students to the power function, but also to expand the known information about the properties of the function as a whole (domain, monotonicity, evenness and oddness of the function), to develop the ability to investigate functions according to a given graph,
When studying the material of this chapter, the functional representations of students are deepened and significantly expanded.
§12 formulates the definition of the function, the argument, and the scope of the function. The definition of the graph of a function is recalled, the ways of its construction, including with the help of elementary transformations.
Section 13 introduces the notion of a power function. On examples and the domain of definition is revealed; the definitions of an increasing and decreasing function are recalled, and the definitions of the increase and decrease of a power function are given.
The idea of an even and odd function is given to students at a visual level. The tutorial covers two tasks in which it is required to plot the function and. The properties of these functions are studied and, on the basis of symmetry, the concepts of even or odd functions are given.
In §15, students get an idea of a function for various values of k, learn to build a graph of a function and read it (i.e., determine the properties of a function from its graph). With the help of the function, the concept of inverse proportionality is clarified, which was only mentioned in the 8th grade algebra course.
When studying a function for k > 0, at first the function is presented as a special case of a power law: taking into account the change in the parameter k.
The paragraph deals with four problems in which it is required to plot function graphs. In Problem 1, to plot a function graph, all the properties of the function studied in the previous paragraphs are used. In problem 2, when constructing graphs of functions and, the already known stretching of the graph of the function along the abscissa axis by 2 times is used. And, based on these two problems, the properties of the function for and are formulated.
In task 4, it is required to build a function graph (based on tasks 1-2), i.e., the graph of this function can be built by shifting the graph of the function along the Ox axis to the right by one and along the Oy axis down by 2 units.
The system of exercises presents various types of tasks: both mandatory and additional tasks of increased complexity.
Among the tasks for plotting graphs of power functions, the following exercises can be distinguished:
№ 164. Draw a graph and find the intervals of increasing and decreasing functions
№ 166. Draw a sketch of the graph of the function when
№ 171. Draw a graph and find the intervals of increasing and decreasing functions
No. 174. Sketch a graph of a function
No. 179. Find out the properties of a function and build its graph
#180 Plot a Function
#191 Plot a Function
#218 Find out if a function is even or odd
Students studying the material master such concepts as the domain of definition, even and odd functions, increasing and decreasing functions on the interval.
Students met the concept of increasing and decreasing functions in the 8th grade algebra course, but only when studying this topic, definitions of these concepts are formed, and therefore, it becomes possible to analytically prove the increase or decrease of a specific function on the interval (however, such proofs are not among the required skills) . Students learn to find intervals of increase in a function using the graph of the function in question.
When studying the topic, examples of a power function with a fractional exponent are not considered, since the concept of a degree with a rational exponent is not introduced in this course.
When studying each specific function (including functions), students will be able to draw a sketch of the graph of the function in question and list its properties according to the graph.
Textbook: “Algebra. Deep Learning. Grade 9.” Mordkovich A. G. (Mnemozina, 2006)
We took the textbook for 2006, since this textbook, unlike later editions, includes the topic degree with a rational indicator. Generally speaking, at present, this topic is studied in high school, but in the multimedia manual we have included it as a propaedeutic material.
The book is intended for in-depth study of the mathematics course in the 9th grade high school. This textbook is based on a 9th grade textbook for educational institutions(A. G. Mordkovich. Algebra-9). It implements the same program, but the difference lies in a deeper study of the relevant issues of the course: simple examples are replaced by more complex and interesting ones.
The structure of the study of the material:
CHAPTER 4. Power functions. Degrees and Roots
§17. Power with a negative integer exponent
§eighteen. Functions, their properties and graphs
§19. concept root nth degrees from a real number
§twenty. Functions, their properties and graphs
§21. Properties of the nth root
§22. Converting Expressions Containing Radicals
§23. Generalization of the concept of exponent
§24. Functions, their properties and graphs
In § 18 we are talking about power functions with an integer exponent, i.e. about functions, etc. This paragraph is divided into points:
The author recalls that simplest case such a function was considered in the 7th grade - it was a function. This section begins with a discussion of the function. A graph is built and the properties of this function are listed in a certain order: 1) domain of definition; 2) even, odd; 3) monotony; 4) boundedness from below, from above; 5) the smallest and largest values of the function; 6) continuity; 7) range of values; 8) bulge.
The properties were read from the graph, now it is proposed to prove the existence of a number of these properties analytically.
The author concludes that the graph of any power function is similar to the graph of a function, only its branches are directed upwards and are more pressed to the x-axis on the segment and notes that the curve touches the x-axis at the point (0; 0).
At the end of the paragraph, an example of constructing a graph of a function is given Construction: 1) transition to an auxiliary coordinate system with the origin at the point (1; -2); 2) construction of a curve.
1) Function
The properties and graph of a power function with an odd exponent are first examined using the example of a function whose graph is a cubic parabola.
The author concludes that the graph of any power function is similar to the graph of a function, only the larger the exponent, the more steeply directed upwards (and accordingly downwards) the branches of the graph and notes that the curve touches the x-axis at the point (0; 0).
The following is an example of using a graph of a power function to solve an equation The solution takes place in 4 stages: 1) two functions are considered: and; 2) plotting a function graph; 2) plotting linear function; 4) find the intersection point and check.
2) Function
We are talking about power functions with a negative integer exponent (even). Let's look at an example function first. A graph is built and the properties of this function are listed. In particular, the property of the function decreasing as is proved.
multimedia visualization function school mathematics
3) Function
In this case, power functions with a negative integer exponent (odd) are considered: etc. The author recalls that one such function has already been studied in the 8th grade - this. Its properties and graph (hyperbola) are recalled, and it is concluded that the graph of any function is similar to a hyperbola.
In § 19, the concept of the nth root of a real number is given and, in particular, it is noted that from any non-negative number one can extract the root of any degree (second, third, fourth, etc.), and from a negative number one can extract the root of any odd degree.
In § 20, we talk about a function given at, and study its graph and properties using a particular example (at). According to the figure, which shows the graph of the function and the graph of the function, the symmetry of these graphs is determined and then confirmed analytically.
In the same paragraph, the function is considered in the case of odd for any values. We talk about the properties of this function and build a graph.
If is an even number, then the graph of the function has the form shown in Fig. one;
If is an odd number, then the graph of the function has the form shown in Fig. 2.
In § 24, we consider a function of the form, - any real number (we restrict ourselves to cases of a rational exponent).
1. If is a natural number, then we get a function (graphs and properties are known)
2. If, then we get a function, i.e. . In the case of an even graph has the form shown in Fig. 3a, in the case of an odd graph has the form shown in Fig. 3b
rice.
3. If, i.e., we are talking about a function, then this is a function, where
The situation is approximately the same for any power function of the form, where:
1. - an improper fraction (the numerator is greater than the denominator). Its graph is a curve similar to a parabola branch. The higher the index, the steeper this curve is directed upwards. A graph is built and properties are given.
2. - proper fraction () (§ 20). A graph is built and properties are given.
A graph is built and properties are given.
In the problem book “Algebra. In-depth study. Grade 9.” Zavich L. I., Ryazanovsky A. R. presents a diverse system of exercises. The complexity of tasks increases as their serial numbers increase. The task book contains a large number of various exercises for plotting graphs of various types of power functions, studying and applying its properties.
For example:
No. 17.05. Build function graphs on one drawing
Plot Functions
No. 17.35. Plot the Function
and using the graph, indicate the intervals of its monotonicity, extremum points, extrema and the number of its zeros.
Plot the function graphs:
No. 19.01. Build function graphs on one drawing
No. 19.04. Plot Functions
No. 19.22. Plot Graphs and Conduct Feature Exploration
No. 21.01. Build on one drawing graphs of functions, with and, with and list the properties of the function: a) the domain of definition D (y); b) the set of values E(y); c) function zeros; d) intervals of monotony; e) intervals of convexity; f) extremum points; g) extremes; h) even or odd; i) the largest and smallest values.
No. 21.03. Plot and explore the following features
No. 21.11. Build function graphs on one drawing
on the segment
No. 21.17. Plot Functions
No. 25.01. Build on the same drawing sketches of graphs of the following pairs of functions
No. 25.05. Plot Function Graphs and Describe Their Properties
No. 25.06. Build function graphs on neighboring drawings
No. 25.18. Plot Functions
No. 25.30. Plot Functions
Analysis of educational literature allows us to draw some conclusions
Considering the standard of the main general education in mathematics, we see that students should learn the following types of power function:
Special cases (direct, inverse proportionality, quadratic function),
With a natural indicator
With an integer
With a positive rational exponent,
With a rational indicator,
With an irrational indicator,
with real indicator.
An important role in this topic is played by the formation of the image of function graphs. Also, students should be able to: determine the properties of a function according to its graph; describe the properties of the studied functions, build their graphs. Consideration of the standard allows us to conclude that the topic “Power function” is included in the mandatory minimum of knowledge, skills and abilities of schoolchildren and, therefore, our attention to it is fully justified.
In order to form strong skills and abilities about the power function, it is necessary to study the methodology of the topic “Properties of the power function”, to which we are moving.
2. Methodological foundations for studying the topic “Properties of a power function” at school
The power function belongs to the class of elementary functions.
The purpose of its study is not only to introduce students to the power function, but also to expand the information they know about the properties of functions in general.
When studying the topic "Power function", they mainly use analytical and graphic method function research. In cases where an analytical study is difficult to perceive by students, graphic methods are used, but the latter cannot serve as evidence.
Students perform a large number of graphic works, while paying attention not only to the accuracy and accuracy of their implementation, but also to rational methods of constructing graphs.
It is possible to form strong skills in constructing and reading graphs of a power function, to ensure that each student can perform the main types of tasks independently, only if students complete a sufficient number of training exercises.
For example, in the journal “Mathematics at School” Lopatina, L.V. offers the following tutorial:
The lesson-workshop aims students to acquire knowledge by their own labor. This is the main leitmotif of developing pedagogy. The topic “Power Function” is very suitable for the creative work of the whole class, since the power function (, where is any rational number) is actually a set of functions that have different properties depending on the exponent.
The discussion of these properties is best organized in groups. To do this, it is advisable to divide the class into six groups.
First of all, the teacher needs to imagine the sequence of work in the "workshop":
Stage I - induction - appeal to previous experience;
Stage III - gap - the moment when students must realize that there are gaps in their knowledge that they themselves must fill;
Stage IV - reflection - determination of the degree of assimilation.
Let's describe in more detail each of the stages of the lesson.
Stage I - induction. The teacher reminds that the class has already studied functions, their properties and graphs. These functions can be generally defined by the formula: , where - is some integer. Such a function is called a power function. The class is given the following task: to list the questions that we must answer when learning a new function.
The class discusses these questions in groups, and then all the questions of the other groups are collected in a single list:
What properties does this function have?
· What is her schedule?
In what situations is it used?
Let's start by answering the last question. Let us give examples of several situations in which a power function appears.
Three students take it in turns to go to the blackboard and make the messages prepared at home.
The first student considers the function, where is the cross-sectional area of the wire diameter. Listeners notice that this power function is actually a quadratic function, but with restrictions on the value of the argument.
The second student tells that the force of attraction of two bodies with masses is expressed by a formula. This is a function of the distance between these bodies. There will be a student in the class who will notice that we have already plotted a function of this kind, although we did not specifically study it.
The third student analyzes the distance of the horizon from the observer: . This is a function of the height to which the observer is elevated above sea level. If the guys themselves did not notice this, then the teacher should emphasize that here the value cannot increase indefinitely. Indeed, no matter how high the observer is raised, he cannot see more than the possibilities of his vision and the bulge of the globe allow. This example is especially indicative, as it allows one to judge the appropriateness of restrictions on the values of the function. Here we must impose some restrictions on the values of the function, although the values, theoretically speaking, can increase indefinitely.
Stage II - discussion of the topic. Students are given some time to analyze the properties of one of their chosen power functions. the main problem here in the function selection. One group tends to simplify the task by limiting themselves to a view function that is well known to all students. Another group overcomplicates their work by taking up the function of the view, or even both together, although the general approach to the question is not yet clear to the students.
In the end, there are groups that have chosen functions whose graphs have already been considered earlier, although they have not been given the necessary emphasis.
The first group considered the function of the species; marked the area of its definition: and the zero value of the function at. The guys especially focused on the fact that the function increases over the entire domain of definition. We singled out the intervals on which the function is greater or less than zero. Speakers emphasized that this function is odd and has neither the largest nor the smallest value.
From this group, one student speaks to the class, who talks about the results of research in the group.
The second group chose a function to consider. The guys noticed that now they will have to exclude the number 0 from the function definition area, i.e. . Unlike the previous one, this function has no zeros. But, like the one considered above, this function is positive for and negative for. It decreases over the entire domain of definition.
The representative of this group emphasizes the differences between the functions and.
Two more students talk about functions.
During their presentations, all speakers should demonstrate graphs of the considered functions.
During the third stage of the lesson, students should summarize their knowledge. And they must do this on their own, surprised by the variety of functions considered. “Why are they given one name, if there are so many of them and they are different?” This is the question students should ask themselves. The task of the teacher is to imperceptibly bring students to this issue. There comes a moment of the so-called gap, when the guys must realize the shortcomings of their knowledge, their limitations or incompleteness. Indeed, one of the considered functions has zeros, the other does not. One increases over the entire domain of definition, the other either increases or decreases. What characterization should we give to the entire power function so that it covers as many special cases as possible?
In search of an answer to this question, one of the guys eventually guesses that it is convenient to associate the form of a power function with the even or odd exponent.
Now it is appropriate to ask the groups again to discuss the properties of the functions
where - odd;
where is even;
where is odd;
where is even.
Once again, we note the plan for the study of the function:
Specify the domain of definition.
Determine whether a function is even or odd (or note that it is neither even nor odd).
1. Find the zeros of the function, if any.
2. Mark the intervals of constancy.
3. Find intervals of increase and decrease.
4. Specify the largest or smallest value of the function.
At the end, students are presented with graphs of the considered functions, = -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. These graphs are performed by representatives of each of the groups.
Now, together with the class, we build function graphs, where is a natural number and.
noted common property of these functions: they both have a domain of definition - a span. They are both neither even nor odd. They are both greater than zero.
But these functions also have differences. The guys call them specifically: the view function increases on its domain of definition, and the view function decreases on the same domain. The form function has a zero value at, and the form function has no zeros.
At stage IV, students should do reflection, i.e. determining the degree of assimilation of the material. The whole class receives the following task according to fig. 3.
On fig. 3, a-h schematically shows the graphs of functions that are given by the formulas
Determine which formula from the given list approximately corresponds to each of the charts a-h.
In the journal "Mathematics at School" Petrov, N.P. offers the project "Studying the properties of a power function using Excel":
The educational project described in the article on the topic “Studying the properties of functions and using Excel spreadsheets” was conducted by teachers of mathematics and computer science of our lyceum in the ninth grade and was designed for five lessons.
The goal of the project was to provide students with independence and initiative in learning new topic and practical application of previously learned material.
During the project implementation, ninth graders had to show:
· ability to correctly formulate project tasks;
ability to analyze information and draw conclusions;
The ability to correctly interpret the results obtained and apply them in practice.
The students were faced with the task of investigating the behavior of graphs of functions using the Excel program, and then, based on the data obtained, describe the properties of the functions.
As a result of the project, ninth-graders had to learn general form graphs of functions and, learn how to build and "read" these graphs, as well as solve graphically equations of the form = f (x).
Note that the work on this project was intended to promote the development of schoolchildren's ability to compare, highlight common features and differences in power function graphs for different values.
Here is a step-by-step description of the project.
Stage I. Preparation (exploratory stage)
Awakening students' interest in the topic of the project occurs in the process of conversation. Students are invited to solve equations known to them
It turns out that the guys can solve the equation in two ways: analytical and graphical, the equation - in a graphical way. They find it difficult to solve the rest of the equations, but if they were familiar with the graphs of functions, they would solve the problem graphically.
The result of the conversation is the formulation of the problematic question: what do the graphs of functions look like and where? After that, directions for further work are determined, tasks are formulated:
1. Use Excel to find out what the function graph looks like for even n and describe the properties of this function.
2. Use Excel to find out what the function graph looks like for odd n and describe the properties of this function.
3. Use Excel to find out what the function graph looks like for even n and describe the properties of this function.
4. Use Excel to find out what the function graph looks like for odd n and describe the properties of this function.
Then the class is divided into working groups. The teacher invites students to independently divide into four groups (optional) and choose a leader in each group. When groups are formed, they choose one of the areas of work in the project (according to the tasks listed above).
Stage II. Planning (analytical stage)
The teacher helps the groups to draw up a work plan for solving the chosen problem and recommends sources for obtaining information. Students independently distribute roles in groups. The approximate distribution of roles in the group is shown in the following table. The number of students in a group depends on the number of students in the class.
At the same stage, the form of presenting the results of the work is discussed. In this case, a computer presentation using PowerPoint was chosen.
Stage III. Research (practical stage)
Students complete tasks according to the planned work plan. The teacher supervises their activities and advises students if necessary.
As an example, we will give the work plan of group No. 1.
1. Construction of graphs of functions using the Excel program.
2. Comparison of graphs, formulation of options for recommendations for constructing a graph of a function for a natural even n.
3. Determination of the properties of the function according to the schedule.
4. Analysis of examples of practical application of the function graph.
On the basis of the study, students conclude that the graphs of functions of the form for natural even n are curves similar to a parabola, and give recommendations for plotting: it should be borne in mind that the graph is symmetrical about the Oy axis, so it is enough to make a table of function values for positive values of the argument X.
In addition, at this stage, a general presentation script is created, which will be refined during the project. In this scenario, in particular, the number of slides, the purpose of each slide, and the main objects that should be placed on the slides are determined.
Stages IV and V. Protection of the project, evaluation of the results (presentation and control stages)
Protection of projects (in groups) takes place at the last of the planned lessons.
We now give a lesson schedule for working on this project and the content of each lesson.
Lesson 1 (Math)
· Statement of the project task. Definition of directions of work, formulation of project objectives.
· Dividing into working groups, choosing a leader in groups.
· Drawing up a work plan for solving the tasks set, the distribution of roles in groups, the choice of the form for presenting the results.
Lesson 2 (computer science)
· Talk about the purpose of Excel spreadsheets.
· Repeating the construction of graphs of various functions using Excel.
· Construction of graphs of the studied functions by means of Excel. Analysis of the received information, formulation of conclusions.
Lesson 3 (Math)
Construction and "reading" of graphs of functions and
· Solving equations of the form, where in a graphical way.
· Create a presentation script.
Lesson 4 (computer science)
Repetition of the purpose and principles of the Power Point program.
· Creating a presentation.
Lesson 5 (Math)
· Protection of projects.
We also give overall plan lesson - protection of the project.
1. Organizational moment.
2. Motivation to apply knowledge through problem identification.
Introductory speech of the teacher
In today's lesson, the main object of study is functions and, where, their properties and graphs. You already know how to solve first-degree (linear) and second-degree (square) equations using root formulas. There are also special root formulas for equations of the 3rd degree, but they are very cumbersome and rarely used in practice. For equations whose degree is higher than the third, general formulas there are no roots. The problem arises: how can such equations be solved? It turns out, if not analytically, then graphically. And in order to apply a graphical method for solving equations of the form and, one must be able to plot functions and, where.
Four groups were involved in the study of the graphs of these functions. Now each of them will acquaint us with the results of the work done.
3. Group performances.
Presentation (defense) of the project by each group, answers to questions from opponents.
4. Self-assessment and assessment of each performance by the other groups (on a five-point scale).
We list the main evaluation criteria:
correspondence of the content to the declared topic, accuracy, completeness of presentation;
The absence of errors
design (design): how the layout of the slides meets aesthetic requirements;
Is the text easy to read? whether the image matches the content, etc.;
persuasiveness, argumentativeness of the speech; literacy of speech, knowledge of terminology;
completeness of answers to questions.
Separately, the interaction in the group is assessed: sociability, respect and attention to other participants, activity.
The total number of points earned and the rating score (arithmetic mean score) are calculated; on their basis, an assessment is made for participation in the project.
5. Discussing each student's contribution to the project and grading.
6. Summing up (reflection).
7. Final word of the teacher
During the project activity on this topic, you answered the question of what the graphs of functions and are, and gave recommendations on how to build them. Now you can solve some equations of the form and graphically. We thank all students for their creative and fruitful work, which contributed to the achievement of the goals of the project.
Given the above, in our manual we tried to reflect a systematic approach to the study of the power function. In order to minimize the difficulties of working with a computer, we tried to make convenient and natural navigation and take into account the requirements for didactic software.
Are you familiar with the features y=x, y=x 2 , y=x 3 , y=1/x etc. All these functions are special cases of the power function, i.e., the function y=x p, where p is a given real number. The properties and graph of a power function essentially depend on the properties of a power with a real exponent, and in particular on the values for which x and p makes sense x p. Let us proceed to a similar consideration of various cases depending on the exponent p.
Index p=2n is an even natural number.
In this case, the power function y=x 2n, where n is a natural number, has the following
properties:
the domain of definition is all real numbers, i.e., the set R;
set of values - non-negative numbers, i.e. y is greater than or equal to 0;
function y=x 2n even, because x 2n =(-x) 2n
the function is decreasing on the interval x<0 and increasing on the interval x>0.
Function Graph y=x 2n has the same form as, for example, the graph of a function y=x 4 .
2. Indicator p=2n-1- odd natural number In this case, the power function y=x 2n-1, where is a natural number, has the following properties:
domain of definition - set R;
set of values - set R;
function y=x 2n-1 odd because (- x) 2n-1 =x 2n-1 ;
the function is increasing on the entire real axis.
Function Graph y=x2n-1 has the same form as, for example, the graph of the function y=x3.
3.Indicator p=-2n, where n- natural number.
In this case, the power function y=x -2n =1/x 2n has the following properties:
set of values - positive numbers y>0;
function y =1/x 2n even, because 1/(-x) 2n =1/x 2n ;
the function is increasing on the interval x<0 и убывающей на промежутке x>0.
Graph of the function y =1/x 2n has the same form as, for example, the graph of the function y =1/x 2 .
4.Indicator p=-(2n-1), where n- natural number. In this case, the power function y=x -(2n-1) has the following properties:
domain of definition - set R, except for x=0;
set of values - set R, except for y=0;
function y=x -(2n-1) odd because (- x) -(2n-1) =-x -(2n-1) ;
the function is decreasing on the intervals x<0 and x>0.
Function Graph y=x -(2n-1) has the same form as, for example, the graph of the function y=1/x 3 .
Inverse trigonometric functions, their properties and graphs.
Inverse trigonometric functions, their properties and graphs.Inverse trigonometric functions (circular functions, arcfunctions) are mathematical functions that are inverse to trigonometric functions.
arcsin function
Function Graph 
.
arcsine numbers m is called such an angle x, for which
The function is continuous and bounded on its entire real line. Function 
is strictly increasing.
[Edit] Properties of the arcsin function
[Edit] Getting the arcsin function
Given a function Throughout its domains she is piecewise monotonic, and hence the inverse correspondence 
is not a function. Therefore, we consider the interval on which it strictly increases and takes all values ranges- . Since for a function on the interval, each value of the argument corresponds to a single value of the function, then on this segment there exists inverse function 
whose graph is symmetrical to the graph of a function on a segment with respect to a straight line
1. Power function, its properties and graph;
2. Transformations:
Parallel transfer;
Symmetry about the coordinate axes;
Symmetry about the origin;
Symmetry about the line y = x;
Stretching and shrinking along the coordinate axes.
3. An exponential function, its properties and graph, similar transformations;
4. Logarithmic function, its properties and graph;
5. Trigonometric function, its properties and graph, similar transformations (y = sin x; y = cos x; y = tg x);
Function: y = x\n - its properties and graph.
Power function, its properties and graph
y \u003d x, y \u003d x 2, y \u003d x 3, y \u003d 1 / x etc. All these functions are special cases of the power function, i.e., the function y = xp, where p is a given real number.
The properties and graph of a power function essentially depend on the properties of a power with a real exponent, and in particular on the values for which x and p makes sense xp. Let us proceed to a similar consideration of various cases, depending on
exponent p.
- Index p = 2n is an even natural number.
y=x2n, where n is a natural number and has the following properties:
- scope - all real numbers, i.e., the set R;
- set of values - non-negative numbers, i.e. y is greater than or equal to 0;
- function y=x2n even, because x 2n = (-x) 2n
- the function is decreasing on the interval x< 0 and increasing on the interval x > 0.
Function Graph y=x2n has the same form as, for example, the graph of a function y=x4.
2. Indicator p = 2n - 1- odd natural number
In this case, the power function y=x2n-1, where is a natural number, has the following properties:
- domain of definition - set R;
- set of values - set R;
- function y=x2n-1 odd because (- x) 2n-1= x 2n-1 ;
- the function is increasing on the entire real axis.
Function Graph y=x2n-1 y=x3.
3. Indicator p=-2n, where n- natural number.
In this case, the power function y=x-2n=1/x2n has the following properties:
- set of values - positive numbers y>0;
- function y = 1/x2n even, because 1/(-x) 2n= 1/x2n;
- the function is increasing on the interval x0.
Graph of the function y = 1/x2n has the same form as, for example, the graph of the function y = 1/x2.
4. Indicator p = -(2n-1), where n- natural number.
In this case, the power function y=x-(2n-1) has the following properties:
- the domain of definition is the set R, except for x = 0;
- set of values - set R, except for y = 0;
- function y=x-(2n-1) odd because (- x)-(2n-1) = -x-(2n-1);
- the function is decreasing on the intervals x< 0 and x > 0.
Function Graph y=x-(2n-1) has the same form as, for example, the graph of the function y = 1/x3.
