The square root of x is what function. "Function" root of x ", its properties and graphs". Square root as an elementary function

Square root as an elementary function.

Square root is an elementary function and a special case of a power function for. The arithmetic square root is smooth at, and at zero it is continuous on the right, but not differentiable.

As a function, a complex variable root is a two-valued function in which the leaves converge at zero.

Plotting a square root function.

1. We fill in the data table:

2. Draw the points that we got on the coordinate plane.

3. We connect these points and get a graph of the square root function:

Transforms the graph of the square root function.

Let us determine what transformations of the function need to be done in order to build graphs of functions. Let's define the types of transformations.

Often, function transformations turn out to be combined.

For example, you need to plot the function ... This is a square root plot that needs to be moved one unit down the axis OY and one unit to the right along the axis OH and simultaneously stretching it 3 times along the axis OY.

It happens immediately before the plotting of the function that preliminary identical transformations or simplifications of functions are needed.

Function graph and properties at = │Oh│ (module)

Consider the function at = │Oh│, where a- a certain number.

The scope of functions at = │Oh│, is the set of all real numbers. The figure shows respectively function graphs at = │NS│, at = │ 2x │, at = │NS/2│.

You can see that the graph of the function at = | Oh| is obtained from the function graph at = Oh if the negative part of the graph of the function at = Oh(it is below the O axis NS), reflect symmetrically this axis.

The schedule is easy to see properties functions at = │ Oh │.

At NS= 0, we get at= 0, that is, the origin of coordinates belongs to the graph of the function; at NS= 0, we get at> 0, that is, all other points of the graph lie above the O axis NS.

For opposite meanings NS, values at will be the same; axis O at this is the axis of symmetry of the graph.

For example, you can plot the function at = │NS 3 │. To compare functions at = │NS 3 │ and at = NS 3, let's compile a table of their values ​​for the same argument values.

From the table we see that in order to plot the function at = │NS 3 │, you can start by plotting the function at = NS 3. After that, it stands symmetrically to the O axis NS display that part of it that is below this axis. As a result, we get the graph shown in the figure.

Function graph and properties at = x 1/2 (root)

Consider the function at = x 1/2 .

The scope of this function is the set of non-negative real numbers, since the expression x 1/2 is meaningful only for NS > 0.

Let's build a graph. To compile a table of its values, we use a microcalculator, rounding the function values ​​to tenths.

After drawing points on the coordinate plane, and smoothly connecting them, we get function graph at = x 1/2 .

The plotted graph allows us to formulate some properties functions at = x 1/2 .

At NS= 0, we get at= 0; at NS> 0, we get at> 0; the graph goes through the origin; the rest of the graph points are located in the first coordinate quarter.

Theorem... Function graph at = x 1/2 symmetrical graph function at = NS 2, where NS> 0, relatively straight at = NS.

Proof... Function graph at = NS 2, where NS> 0, is the branch of the parabola located in the first coordinate quarter. Let the point R (a; b) is an arbitrary point of this graph. Then the equality is true b = a 2. Since, by condition, the number a is nonnegative, then the equality a= b 1/2. This means that the coordinates of the point Q (b; a) transform the formula at = x 1/2 to true equality, or otherwise, point Q (b; a at= x 1/2 .

It is also proved that if the point M (with; d) belongs to the graph of the function at = x 1/2 then point N (d; with) belongs to the graph at = NS 2, where NS > 0.

It turns out that every point R(a; b) function graph at = NS 2, where NS> 0, there is a single point Q (b; a) function graph at = x 1/2 and vice versa.

It remains to prove that the points R (a; b) and Q (b; a) are symmetric with respect to the straight line at = NS... By dropping perpendiculars to the coordinate axes from points R and Q, we obtain on these axes the points E(a; 0), D (0; b), F (b; 0), WITH (0; a). Point R intersection of perpendiculars PE and QC has coordinates ( a; a) and therefore belongs to the straight line at = NS... Triangle PRQ is isosceles, since its sides RP and RQ equal │ b – a│ each. Straight at = NS halves as a corner DOF and angle PRQ and crosses the segment PQ at a certain point S... Therefore, the segment Rs is the bisector of the triangle PRQ... Since the bisector of an isosceles triangle is its height and median, then PQ ┴ Rs and PS = QS... This means that the points R (a; b) and Q (b; a) symmetric with respect to the straight line at = NS.

Since the graph of the function at = x 1/2 symmetrical graph function at = NS 2, where NS> 0, relatively straight at= NS, then the graph of the function at = x 1/2 is the branch of the parabola.

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.

Consolidating the knowledge gained and practicing the skills of transforming graphs -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 performed 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 - 7min).

Problematic situation: describe the properties of an unknown function.

Divide the class into teams of 4-5 people, distribute forms for answering the questions posed

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 depict the graph of the function y = on the forms, then exchange the forms with each other for self-and cross-checking.

Fizminutka (Clownery)

Plotting a graph together with the class with the design of plotting 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:

Output: 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 blackboard, and students evaluate their work by comparing with the blackboard.

Card number 1

Card number 2

Output: 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) to 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 getting out of a difficulty

Stage goal:

1) organize communicative interaction to build a new method 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 for 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 mistakes.

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 the 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. (Cards with reflection)

Homework

p. 13 (before example 2) № 13.3, 13.4

Solve the equation graphically.