The graph of the function y is the square root of x. Graph of square root function, transformation of graphs. Homework
Basic goals:
1) to form an idea of the advisability of a generalized study of the dependences of real quantities on the example of quantities related by the relation y =
2) form the ability to construct a graph y = and its properties;
3) repeat and consolidate the techniques of oral and written calculations, squaring, extraction square root.
Equipment, demonstration material: handouts.
1. Algorithm:
2. An example for performing an assignment in groups:
3. Sample for self-test:
4. Card for the stage of reflection:
1) I figured out how to plot the function y =.
2) I can list its properties according to the schedule.
3) I did not make mistakes in independent work.
4) I made mistakes in independent work (list these mistakes and indicate their reason).
During the classes
1. Self-determination for learning activities
Stage goal:
1) include students in educational activities;
2) determine the meaningful framework of the lesson: we continue to work with real numbers.
Organization of the educational process at stage 1:
- What did we learn in the last lesson? (We studied a set of real numbers, actions with them, built an algorithm to describe the properties of a function, repeated the functions learned in grade 7).
- Today we will continue to work with a set of real numbers, a function.
2. Updating knowledge and fixing difficulties in activities
Stage goal:
1) update the educational content necessary and sufficient for the perception of new material: function, independent variable, dependent variable, graphs
y = kx + m, y = kx, y = c, y = x 2, y = - x 2,
2) update the mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;
3) fix all repeated concepts and algorithms in the form of diagrams and symbols;
4) to fix the individual difficulty in the activity, demonstrating the insufficiency of the available knowledge at the personally significant level.
Organization of the educational process at stage 2:
1. Let's remember how you can set the relationship between the quantities? (Via text, formula, table, graphics)
2. What is called a function? (The relationship between two quantities, where each value of one variable corresponds to a single value of the other variable y = f (x)).
What is x called? (Independent variable is an argument)
What is the name of y? (Dependent variable).
3. In 7th grade, did we learn about functions? (y = kx + m, y = kx, y = c, y = x 2, y = - x 2,).
Individual task:
What is the graph of the functions y = kx + m, y = x 2, y =?
3. Identifying the causes of difficulties and setting the goal of the activity
Stage goal:
1) organize communicative interaction, during which the distinctive property of the task that caused difficulty in educational activity is revealed and fixed;
2) agree on the purpose and topic of the lesson.
Organization of the educational process at stage 3:
- What is special about this assignment? (The dependency is given by the formula y = which we have not met yet).
- What is the purpose of the lesson? (Get acquainted with the function y =, its properties and graph. Using the function in the table to define the type of dependence, build a formula and a graph.)
- Can you formulate the topic of the lesson? (Function y =, its properties and graph).
- Write the topic in a notebook.
4. Building a project for a way out of a difficulty
Stage goal:
1) organize communicative interaction to build a new way of action that eliminates the cause of the identified difficulty;
2) to fix a new way of action in a sign, verbal form and with the help of a standard.
Organization of the educational process at stage 4:
The work at the stage can be organized into groups by asking the groups to build a graph of y =, then analyze the resulting results. Groups can also be asked to describe the properties of this function using an algorithm.
5. Primary reinforcement in external speech
The purpose of the stage: to fix the studied educational content in external speech.
Organization of the educational process at stage 5:
Plot y = - and describe its properties.
Properties y = -.
1. The area of definition of the function.
2. The area of values of the function.
3.y = 0, y> 0, y<0.
y = 0 if x = 0.
y<0, если х(0;+)
4. Increase, decrease of function.
The function decreases at x.
Let's build a graph for y =.
Let's select part of it on the segment. Note that at naim. = 1 at x = 1, and y naib. = 3 at x = 9.
Answer: at naim. = 1, at naib. = 3
6. Independent work with self-test according to the standard
The purpose of the stage: to test your ability to apply new educational content in standard conditions on the basis of comparing your solution with a standard for self-examination.
Organization of the educational process at stage 6:
Students complete the task on their own, conduct a self-test according to the standard, analyze, and correct errors.
Let's build a graph for y =.
Using the graph, find the smallest and largest values of the function on the segment.
7. Incorporation and repetition
The purpose of the stage: to train the skills of using new content in conjunction with previously studied: 2) to repeat the educational content, which will be required in the next lessons.
Organization of the educational process at stage 7:
Solve the equation graphically: = x - 6.
One student at the blackboard, the rest in notebooks.
8. Reflection of activity
Stage goal:
1) fix the new content learned in the lesson;
2) evaluate their own activities in the lesson;
3) thank classmates who helped to get the result of the lesson;
4) fix unresolved difficulties as directions for future educational activities;
5) discuss and write down homework.
Organization of the educational process at stage 8:
- Guys, what was our goal today? (Examine the function y =, its properties and graph).
- What knowledge helped us to achieve the goal? (The ability to look for patterns, the ability to read graphs.)
- Analyze your activities in the lesson. (Reflection cards)
Homework
p. 13 (before example 2) № 13.3, 13.4
Solve the equation graphically.
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Municipal educational institution
secondary school №1
Art. Bryukhovetskaya
Municipal Formation Bryukhovetsky District
Mathematic teacher
Guchenko Angela Viktorovna
year 2014
Function y =
, its properties and graph
Lesson type: learning new material
Lesson objectives:
Tasks solved in the lesson:
teach students to work independently;
make assumptions and guesses;
be able to generalize the studied factors.
Equipment: board, chalk, multimedia projector, handouts
Duration of the lesson.
Determining the topic of the lesson together with the students -1 minute.
Determining the goals and objectives of the lesson together with the students -1 minute.
Knowledge update (frontal survey) -3 min.
Oral work -3 min.
Explanation of the new material based on the creation of problem situations -7min.
Physical minute -2 minutes.
Plotting a graph together with the class with the design of the plot in notebooks and defining the properties of the function, working with the textbook -10 min.
Consolidation of the acquired knowledge and development of graph transformation skills -9min .
Summing up the lesson, establishing feedback -3 min.
Homework -1 minute.
Total 40 minutes.
During the classes.
Determination of the topic of the lesson together with the students (1min).
The topic of the lesson is determined by students using leading questions:
function- work done by an organ, an organism as a whole.
function- possibility, option, skill of the program or device.
function- duty, range of activities.
function character in a literary work.
function- kind of subroutine in computer science
function in mathematics - the law of dependence of one quantity on another.
Determination of the goals and objectives of the lesson together with the students (1min).
The teacher, with the help of the students, formulates and articulates the goals and objectives of this lesson.
Knowledge update (frontal survey - 3 min).
Oral work - 3 min.
Frontal work.
(A and B belong, C does not)
Explanation of the new material (based on the creation of problem situations - 7 min).
Problematic situation: describe the properties of an unknown function.
Divide the class into teams of 4-5 people, distribute forms for answering the questions
Form No. 1
y = 0, at x =?
Function definition area.
The set of function values.
Each question is answered by one of the team representatives, the rest of the teams vote "for" or "against" with signal cards and, if necessary, supplement the answers of classmates.
Together with the class, draw a conclusion about the domain of definition, the set of values, the zeros of the function y =.
Problem situation : try to plot an unknown function (there is a discussion in teams, search for a solution).
With the teacher, I recall the algorithm for plotting the graphs of functions. Students use teams to try to plot the function y = on the forms, then exchange the forms with each other for self-and cross-checking.
Fizminutka (Clownery)
Building a graph together with the class with the design of the building in notebooks - 10 min.
After a general discussion, the task of constructing a graph of the function y = is performed individually by each student in a notebook. The teacher at this time provides differentiated assistance to students. After students complete the assignment, a function graph is shown on the board and students are asked to answer the following questions:
Conclusion: together with the students, draw again the conclusion about the properties of the function and read them according to the textbook:
Consolidation of the acquired knowledge and development of the skills of transformation of the schedule - 9 min.
Students work according to their card (according to options), then change and check each other. After that, graphs are shown on the board, and students evaluate their work by comparing it with the board.
Card number 1
Card number 2
Conclusion: about graph transformations
1) parallel transfer along the OU axis
2) shift along the OX axis.
9. Summing up the results of the lesson, establishing feedback - 3 min.
SLIDES – insert missing words
Domain of this function, all numbers, except … (Negative).
The function graph is located in ... (I) quarters.
If the value of the argument is x = 0, the value ... (functions) y = ... (0).
The highest value of the function ... (does not exist), smallest value -… (equal to 0)
10. Assignment at home (with comments - 1 min).
According to the textbook- §13
By the book of problems- No. 13.3, No. 74 (repetition of incomplete quadratic equations)
Basic goals:
1) to form an idea of the advisability of a generalized study of the dependences of real quantities on the example of quantities related by the relation y =
2) form the ability to construct a graph y = and its properties;
3) repeat and consolidate the techniques of oral and written calculations, squaring, square root extraction.
Equipment, demonstration material: handouts.
1. Algorithm:
2. An example for performing an assignment in groups:
3. Sample for self-test:
4. Card for the stage of reflection:
1) I figured out how to plot the function y =.
2) I can list its properties according to the schedule.
3) I did not make mistakes in independent work.
4) I made mistakes in independent work (list these mistakes and indicate their reason).
During the classes
1. Self-determination for learning activities
Stage goal:
1) include students in educational activities;
2) determine the meaningful framework of the lesson: we continue to work with real numbers.
Organization of the educational process at stage 1:
- What did we learn in the last lesson? (We studied a set of real numbers, actions with them, built an algorithm to describe the properties of a function, repeated the functions learned in grade 7).
- Today we will continue to work with a set of real numbers, a function.
2. Updating knowledge and fixing difficulties in activities
Stage goal:
1) update the educational content necessary and sufficient for the perception of new material: function, independent variable, dependent variable, graphs
y = kx + m, y = kx, y = c, y = x 2, y = - x 2,
2) update the mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;
3) fix all repeated concepts and algorithms in the form of diagrams and symbols;
4) to fix the individual difficulty in the activity, demonstrating the insufficiency of the available knowledge at the personally significant level.
Organization of the educational process at stage 2:
1. Let's remember how you can set the relationship between the quantities? (Via text, formula, table, graphics)
2. What is called a function? (The relationship between two quantities, where each value of one variable corresponds to a single value of the other variable y = f (x)).
What is x called? (Independent variable is an argument)
What is the name of y? (Dependent variable).
3. In 7th grade, did we learn about functions? (y = kx + m, y = kx, y = c, y = x 2, y = - x 2,).
Individual task:
What is the graph of the functions y = kx + m, y = x 2, y =?
3. Identifying the causes of difficulties and setting the goal of the activity
Stage goal:
1) organize communicative interaction, during which the distinctive property of the task that caused difficulty in educational activity is revealed and fixed;
2) agree on the purpose and topic of the lesson.
Organization of the educational process at stage 3:
- What is special about this assignment? (The dependency is given by the formula y = which we have not met yet).
- What is the purpose of the lesson? (Get acquainted with the function y =, its properties and graph. Using the function in the table to define the type of dependence, build a formula and a graph.)
- Can you formulate the topic of the lesson? (Function y =, its properties and graph).
- Write the topic in a notebook.
4. Building a project for a way out of a difficulty
Stage goal:
1) organize communicative interaction to build a new way of action that eliminates the cause of the identified difficulty;
2) to fix a new way of action in a sign, verbal form and with the help of a standard.
Organization of the educational process at stage 4:
The work at the stage can be organized into groups by asking the groups to build a graph of y =, then analyze the resulting results. Groups can also be asked to describe the properties of this function using an algorithm.
5. Primary reinforcement in external speech
The purpose of the stage: to fix the studied educational content in external speech.
Organization of the educational process at stage 5:
Plot y = - and describe its properties.
Properties y = -.
1. The area of definition of the function.
2. The area of values of the function.
3.y = 0, y> 0, y<0.
y = 0 if x = 0.
y<0, если х(0;+)
4. Increase, decrease of function.
The function decreases at x.
Let's build a graph for y =.
Let's select part of it on the segment. Note that at naim. = 1 at x = 1, and y naib. = 3 at x = 9.
Answer: at naim. = 1, at naib. = 3
6. Independent work with self-test according to the standard
The purpose of the stage: to test your ability to apply new educational content in standard conditions on the basis of comparing your solution with a standard for self-examination.
Organization of the educational process at stage 6:
Students complete the task on their own, conduct a self-test according to the standard, analyze, and correct errors.
Let's build a graph for y =.
Using the graph, find the smallest and largest values of the function on the segment.
7. Incorporation and repetition
The purpose of the stage: to train the skills of using new content in conjunction with previously studied: 2) to repeat the educational content, which will be required in the next lessons.
Organization of the educational process at stage 7:
Solve the equation graphically: = x - 6.
One student at the blackboard, the rest in notebooks.
8. Reflection of activity
Stage goal:
1) fix the new content learned in the lesson;
2) evaluate their own activities in the lesson;
3) thank classmates who helped to get the result of the lesson;
4) fix unresolved difficulties as directions for future educational activities;
5) discuss and write down homework.
Organization of the educational process at stage 8:
- Guys, what was our goal today? (Examine the function y =, its properties and graph).
- What knowledge helped us to achieve the goal? (The ability to look for patterns, the ability to read graphs.)
- Analyze your activities in the lesson. (Reflection cards)
Homework
p. 13 (before example 2) № 13.3, 13.4
Solve the equation graphically.
The main properties of the power function, including formulas and properties of roots, are presented. The derivative, integral, power series expansion and representation by means of complex numbers of a power function are presented.
ContentThe power function, y = x p, with exponent p has the following properties:
(1.1)
is defined and continuous on the set
at ,
at ;
(1.2)
has many meanings
at ,
at ;
(1.3)
strictly increases at,
strictly decreases at;
(1.4)
at ;
at ;
(1.5)
;
(1.5*)
;
(1.6)
;
(1.7)
;
(1.7*)
;
(1.8)
;
(1.9)
.
The proof of properties is given on the page "Power function (proof of continuity and properties)"
Roots - definition, formulas, properties
The nth root of a number x is a number raised to the nth power gives x:.
Here n = 2, 3, 4, ... - a natural number greater than one.
You can also say that the nth root of x is the root (that is, the solution) of the equation
.
Note that the function is the inverse of the function.
The square root of x is the root of the power of 2:.
The cube root of x is the 3 rd root:.
Even degree
For even degrees n = 2 m, the root is defined for x ≥ 0
... Often a formula is used that is valid for both positive and negative x:
.
For the square root:
.
The order in which the operations are performed is important here - that is, first the squaring is performed, as a result of which a non-negative number is obtained, and then the root is extracted from it (you can extract the square root from a non-negative number). If we changed the order of:, then for negative x the root would be undefined, and together with it the whole expression is undefined.
Odd degree
For odd powers, the root is defined for all x:
;
.
Properties and formulas of roots
The root of x is a power function:
.
For x ≥ 0
the following formulas hold:
;
;
,
;
.
These formulas can also be applied with negative values of the variables. You just need to make sure that the radical expression of even degrees is not negative.
Private values
The root of 0 is 0:.
The root of 1 is 1:.
The square root of 0 is 0:.
The square root of 1 is 1:.
Example. Root from roots
Consider an example of a square root of roots:
.
Convert the inner square root using the formulas above:
.
Now let's transform the original root:
.
So,
.
y = x p for different values of the exponent p.
Here are the graphs of the function for non-negative values of the argument x. The power function plots defined for negative x values are given on the page "Power function, its properties and plots"
Inverse function
The inverse for a power function with exponent p is a power function with exponent 1 / p.
If, then.
Derivative of a power function
Derivative of the nth order:
;
Derivation of formulas>>>
Integral of power function
P ≠ - 1
;
.
Power series expansion
At - 1
< x < 1
the following decomposition takes place:
Expressions in terms of complex numbers
Consider a function of a complex variable z:
f (z) = z t.
Let us express the complex variable z in terms of the modulus r and the argument φ (r = | z |):
z = r e i φ.
We represent the complex number t in the form of real and imaginary parts:
t = p + i q.
We have:
Further, we will take into account that the argument φ is not uniquely defined:
,
Consider the case when q = 0
, that is, the exponent is a real number, t = p. Then
.
If p is an integer, then kp is an integer as well. Then, due to the periodicity of trigonometric functions:
.
That is, the exponential function for a whole exponent, for a given z, has only one value and therefore is unambiguous.
If p is irrational, then the products of kp do not give an integer for any k. Since k runs through an infinite series of values k = 0, 1, 2, 3, ..., then the function z p has infinitely many values. Whenever the z argument gets incremented 2 π(one turn), we move to a new branch of the function.
If p is rational, then it can be represented as:
, where m, n- integers that do not contain common divisors. Then
.
The first n quantities, for k = k 0 = 0, 1, 2, ... n-1 give n different values of kp:
.
However, subsequent values give values that differ from the previous ones by an integer. For example, for k = k 0 + n we have:
.
Trigonometric functions whose arguments differ by multiples 2 π, have equal meanings. Therefore, with a further increase in k, we obtain the same values of z p as for k = k 0 = 0, 1, 2, ... n-1.
Thus, an exponential function with a rational exponent is multivalued and has n values (branches). Whenever the z argument gets incremented 2 π(one turn), we move to a new branch of the function. After n such revolutions, we return to the first branch from which the counting began.
In particular, a root of degree n has n values. As an example, consider the nth root of a real positive number z = x. In this case φ 0 = 0, z = r = | z | = x,
.
.
So, for a square root, n = 2
,
.
For even k, (- 1) k = 1... For odd k, (- 1) k = - 1.
That is, the square root has two meanings: + and -.
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Technical Institutions, "Lan", 2009.
