Construct a graph of the function y root x. Square root

8th grade

Teacher: Melnikova T.V.

Lesson objectives:

Equipment:

Computer, interactive whiteboard, handouts.

Presentation for the lesson.

DURING THE CLASSES

Lesson plan.

Teacher's opening speech.

Repetition of previously studied material.

Learning new material (group work).

Function study. Chart properties.

Discussion of the schedule (front work).

Game of math cards.

Lesson summary.

I. Updating of basic knowledge.

Greeting from the teacher.

Teacher :

The dependence of one variable on another is called a function. So far you have studied the functions y = kx + b; y =k/x, y=x 2. Today we will continue to study functions. In today's lesson you will learn what a graph of a square root function looks like, and learn how to build graphs of square root functions yourself.

Write down the topic of the lesson (slide1).

2. Repetition of the studied material.

1. What are the names of the functions specified by the formulas:

a) y=2x+3; b) y=5/x; c) y = -1/2x+4; d) y=2x; e) y = -6/x f) y = x 2?

2. What is their graph? How is it located? Indicate the domain of definition and domain of value of each of these functions ( in Fig. graphs of functions given by these formulas are shown; for each function, indicate its type) (slide2).

3. What is the graph of each function, how are these graphs constructed?

(Slide 3, schematic graphs of functions are constructed).

3. Studying new material.

Teacher:

So today we are studying the function
and her schedule.

We know that the graph of the function y=x2 is a parabola. What will be the graph of the function y=x2 if we take only x ≥ 0 ? Part of the parabola is its right branch. Let us now plot the function
.

Let us repeat the algorithm for constructing graphs of functions ( slide 4, with algorithm)

Question : Looking at the analytical notation of the function, do you think we can say what values X acceptable? (Yes, x≥0). Since the expression
makes sense for all x greater than or equal to 0.

Teacher: In natural phenomena and human activity, dependencies between two quantities are often encountered. How can this relationship be represented by a graph? ( group work)

The class is divided into groups. Each group receives a task: build a graph of the function
on graph paper, performing all points of the algorithm. Then a representative from each group comes out and shows the group's work. (Slad 5 opens, a check is carried out, then the schedule is built in notebooks)

4. Study of the function (work in groups continues)

Teacher:

find the domain of the function;

find the range of the function;

determine the intervals of decrease (increase) of the function;

y>0, y<0.

Write down the results for you (slide 6).

Teacher: Let's analyze the graph. The graph of a function is a branch of a parabola.

Question : Tell me, have you seen this graph somewhere before?

Look at the graph and tell me if it intersects the line OX? (No) OU? (No). Look at the graph and tell me whether the graph has a center of symmetry? Axis of symmetry?

Let's summarize:

Now let’s see how we learned a new topic and repeated the material we covered. A game of mathematical cards. (rules of the game: each group of 5 people is offered a set of cards (25 cards). Each player receives 5 cards with questions written on them. The first student gives one of the cards to the second student, who must answer the question from the card If the student answers the question, then the card is broken, if not, then the student takes the card for himself and moves on, etc. for a total of 5 moves. If the student has no cards left, then the score is -5, 1 card remains - score 4, 2 cards – score 3, 3 cards – score 2)

5. Lesson summary.(students are graded on checklists)

Homework assignment.

Study paragraph 8.

Solve No. 172, No. 179, No. 183.

Prepare reports on the topic “Application of functions in various fields of science and literature.”

Reflection.

Show your mood with pictures on your desk.

Today's lesson

I like it.

I did not like.

Lesson material I ( understood, did not understand).

I looked again at the sign... And, let's go!

Let's start with something simple:

Just a minute. this, which means we can write it like this:

Got it? Here's the next one for you:

Are the roots of the resulting numbers not exactly extracted? No problem - here are some examples:

What if there are not two, but more multipliers? The same! The formula for multiplying roots works with any number of factors:

Now completely on your own:

Answers: Well done! Agree, everything is very easy, the main thing is to know the multiplication table!

Root division

We've sorted out the multiplication of roots, now let's move on to the property of division.

Let me remind you that the general formula looks like this:

Which means that the root of the quotient is equal to the quotient of the roots.

Well, let's look at some examples:

That's all science is. Here's an example:

Everything is not as smooth as in the first example, but, as you can see, there is nothing complicated.

What if you come across this expression:

You just need to apply the formula in the opposite direction:

And here's an example:

You may also come across this expression:

Everything is the same, only here you need to remember how to translate fractions (if you don’t remember, look at the topic and come back!). Do you remember? Now let's decide!

I am sure that you have coped with everything, now let’s try to raise the roots to degrees.

Exponentiation

What happens if the square root is squared? It's simple, remember the meaning of the square root of a number - this is a number whose square root is equal to.

So, if we square a number whose square root is equal, what do we get?

Well, of course, !

Let's look at examples:

It's simple, right? What if the root is to a different degree? It's OK!

Follow the same logic and remember the properties and possible actions with degrees.

Read the theory on the topic “” and everything will become extremely clear to you.

For example, here is an expression:

In this example, the degree is even, but what if it is odd? Again, apply the properties of exponents and factor everything:

Everything seems clear with this, but how to extract the root of a number to a power? Here, for example, is this:

Pretty simple, right? What if the degree is greater than two? We follow the same logic using the properties of degrees:

Well, is everything clear? Then solve the examples yourself:

And here are the answers:

Entering under the sign of the root

What haven’t we learned to do with roots! All that remains is to practice entering the number under the root sign!

It's really easy!

Let's say we have a number written down

What can we do with it? Well, of course, hide the three under the root, remembering that the three is the square root of!

Why do we need this? Yes, just to expand our capabilities when solving examples:

How do you like this property of roots? Does it make life much easier? For me, that's exactly right! Only We must remember that we can only enter positive numbers under the square root sign.

Solve this example yourself -
Did you manage? Let's see what you should get:

Well done! You managed to enter the number under the root sign! Let's move on to something equally important - let's look at how to compare numbers containing a square root!

Comparison of roots

Why do we need to learn to compare numbers that contain a square root?

Very simple. Often, in large and long expressions encountered in the exam, we receive an irrational answer (remember what this is? We already talked about this today!)

We need to place the received answers on the coordinate line, for example, to determine which interval is suitable for solving the equation. And here the problem arises: there is no calculator in the exam, and without it, how can you imagine which number is greater and which is less? That's it!

For example, determine which is greater: or?

You can’t tell right away. Well, let's use the disassembled property of entering a number under the root sign?

Then go ahead:

Well, obviously, the larger the number under the root sign, the larger the root itself!

Those. if, then, .

From this we firmly conclude that. And no one will convince us otherwise!

Extracting roots from large numbers

Before this, we entered a multiplier under the sign of the root, but how to remove it? You just need to factor it into factors and extract what you extract!

It was possible to take a different path and expand into other factors:

Not bad, right? Any of these approaches is correct, decide as you wish.

Factoring is very useful when solving such non-standard problems as this:

Let's not be afraid, but act! Let's decompose each factor under the root into separate factors:

Now try it yourself (without a calculator! It won’t be on the exam):

Is this the end? Let's not stop halfway!

That's all, it's not so scary, right?

Happened? Well done, that's right!

Now try this example:

But the example is a tough nut to crack, so you can’t immediately figure out how to approach it. But, of course, we can handle it.

Well, let's start factoring? Let us immediately note that you can divide a number by (remember the signs of divisibility):

Now, try it yourself (again, without a calculator!):

Well, did it work? Well done, that's right!

Let's sum it up

1. The square root (arithmetic square root) of a non-negative number is a non-negative number whose square is equal to.
   .
2. If we simply take the square root of something, we always get one non-negative result.
3. Properties of an arithmetic root:
4. When comparing square roots, it is necessary to remember that the larger the number under the root sign, the larger the root itself.

How's the square root? All clear?

We tried to explain to you without any fuss everything you need to know in the exam about the square root.

It's your turn. Write to us whether this topic is difficult for you or not.

Did you learn something new or was everything already clear?

Write in the comments and good luck on your exams!

Basic goals:

1) form an idea of ​​the feasibility of a generalized study of the dependencies of real quantities using the example of quantities related by the relation y=

2) to develop the ability to construct a graph y= and its properties;

3) repeat and consolidate the techniques of oral and written calculations, squaring, extracting square roots.

Equipment, demonstration material: handouts.

1. Algorithm:

2. Sample for completing the task in groups:

3. Sample for self-test of independent work:

4. Card for the reflection stage:

1) I understood how to graph the function y=.

2) I can list its properties using a graph.

3) I did not make mistakes in independent work.

4) I made mistakes in my independent work (list these mistakes and indicate their reason).

During the classes

1. Self-determination for educational activities

Purpose of the stage:

1) include students in educational activities;

2) determine the content of the lesson: we continue to work with real numbers.

Organization of the educational process at stage 1:

– What did we study in the last lesson? (We studied the set of real numbers, operations with them, built an algorithm to describe the properties of a function, repeated functions studied in 7th grade).

– Today we will continue to work with a set of real numbers, a function.

2. Updating knowledge and recording difficulties in activities

Purpose of the stage:

1) update educational content that is 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 mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;

3) record all repeated concepts and algorithms in the form of diagrams and symbols;

4) record an individual difficulty in activity, demonstrating at a personally significant level the insufficiency of existing knowledge.

Organization of the educational process at stage 2:

1. Let's remember how you can set dependencies between quantities? (Using text, formula, table, graph)

2. What is a function called? (A relationship between two quantities, where each value of one variable corresponds to a single value of another variable y = f(x)).

What is the name of x? (Independent variable - argument)

What is the name of y? (Dependent variable).

3. In 7th grade did we study 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 goals for activities

Purpose of the stage:

1) organize communicative interaction, during which the distinctive property of the task that caused difficulty in learning activities is identified and recorded;

2) agree on the purpose and topic of the lesson.

Organization of the educational process at stage 3:

-What's special about this task? (The dependence is given by the formula y = which we have not yet encountered.)

– What is the purpose of the lesson? (Get acquainted with the function y =, its properties and graph. Use the function in the table to determine the type of dependence, build a formula and graph.)

– Can you formulate the topic of the lesson? (Function y=, its properties and graph).

– Write the topic in your notebook.

4. Construction of a project for getting out of a difficulty

Purpose of the stage:

1) organize communicative interaction to build a new method of action that eliminates the cause of the identified difficulty;

2) fix a new method of action in a symbolic, verbal form and with the help of a standard.

Organization of the educational process at stage 4:

Work at this stage can be organized in groups, asking the groups to build a graph y =, then analyze the results. Groups can also be asked to describe the properties of a given function using an algorithm.

5. Primary consolidation in external speech

The purpose of the stage: to record the studied educational content in external speech.

Organization of the educational process at stage 5:

Construct a graph of y= - and describe its properties.

Properties y= - .

1.Domain of definition of a function.

2. Range of values ​​of the function.

3. y = 0, y> 0, y<0.

y =0 if x = 0.

y<0, если х(0;+)

4.Increasing, decreasing functions.

The function decreases as x.

Let's build a graph of y=.

Let's select its part on the segment. Note that we have = 1 for x = 1, and y max. =3 at x = 9.

Answer: at our name. = 1, y max. =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 based on comparing your solution with a standard for self-test.

Organization of the educational process at stage 6:

Students complete the task independently, conduct a self-test against the standard, analyze, and correct errors.

Let's build a graph of y=.

Using a graph, find the smallest and largest values ​​of the function on the segment.

7. Inclusion in the knowledge system and repetition

The purpose of the stage: to train the skills of using new content together with previously studied: 2) repeat the educational content that will be required in the next lessons.

Organization of the educational process at stage 7:

Solve the equation graphically: = x – 6.

One student is at the blackboard, the rest are in notebooks.

8. Reflection of activity

Purpose of the stage:

1) record new content learned in the lesson;

2) evaluate your own activities in the lesson;

3) thank classmates who helped get the result of the lesson;

4) record unresolved difficulties as directions for future educational activities;

5) discuss and write down your homework.

Organization of the educational process at stage 8:

- Guys, what was our goal today? (Study the function y=, its properties and graph).

– What knowledge helped us achieve our goal? (Ability to look for patterns, ability to read graphs.)

– Analyze your activities in class. (Cards with reflection)

Homework

paragraph 13 (before example 2) № 13.3, 13.4

Solve the equation graphically.

Municipal educational institution

secondary school No. 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:

Problems solved in the lesson:

teach students to work independently;

make assumptions and guesses;

be able to generalize the factors being studied.

Equipment: board, chalk, multimedia projector, handouts

Timing of the lesson.

Determining the topic of the lesson together with students -1 min.

Determining the goals and objectives of the lesson together with students -1 min.

Updating knowledge (frontal survey) –3 min.

Oral work -3 min.

Explanation of new material based on creating problem situations -7min.

Fizminutka –2 minutes.

Plotting a graph together with the class, drawing up the construction in notebooks and determining the properties of a function, working with a textbook -10 min.

Consolidating acquired knowledge and practicing graph transformation skills –9min .

Summing up the lesson, providing feedback -3 min.

Homework -1 min.

Total 40 minutes.

During the classes.

Determining the topic of the lesson together with students (1 min).

The topic of the lesson is determined by students using guiding questions:

function- work performed by an organ, the organism as a whole.

function- possibility, option, skill of a program or device.

function- duty, range of activities.

function character in a literary work.

function- type of subroutine in computer science

function in mathematics - the law of dependence of one quantity on another.

Determining the goals and objectives of the lesson together with students (1 min).

The teacher, with the help of students, formulates and pronounces the goals and objectives of this lesson.

Updating knowledge (frontal survey – 3 min).

Oral work – 3 min.

Frontal work.

(A and B belong, C does not)

Explanation of new material (based on creating problem situations – 7 min).

Problem situation: describe the properties of an unknown function.

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

Form No. 1

y=0, with x=?

The scope of the function.

Set of function values.

One of the team representatives answers each question, the rest of the teams vote “for” or “against” with signal cards and, if necessary, complement the answers of their classmates.

Together with the class, draw a conclusion about the domain of definition, the set of values, and the zeros of the function y=.

Problem situation : try to build a graph of an unknown function (there is a discussion in teams, searching for a solution).

The teacher recalls the algorithm for constructing function graphs. Students in teams try to depict the graph of the function y= on forms, then exchange forms with each other for self- and mutual testing.

Fizminutka (Clowning)

Constructing a graph together with the class with the design in notebooks – 10 min.

After a general discussion, the task of constructing a graph of the function y= is completed individually by each student in a notebook. At this time, the teacher provides differentiated assistance to students. After students complete the task, the graph of the function is shown on the board and students are asked to answer the following questions:

Conclusion: Together with the students, draw a conclusion about the properties of the function and read them from the textbook:

Consolidating acquired knowledge and practicing graph transformation skills – 9 min.

Students work on their card (according to the options), then change and check each other. Afterwards, graphs are shown on the board, and students evaluate their work by comparing it with the board.

Card No. 1

Card No. 2

Conclusion: about graph transformations

1) parallel transfer along the op-amp axis

2) shift along the OX axis.

9. Summing up the lesson, providing feedback – 3 min.

SLIDES – insert missing words

The domain of definition of this function, all numbers except ...(negative).

The graph of the function is located in... (I) quarters.

When the argument x = 0, the value... (functions) y = ... (0).

The greatest value of the function... (does not exist), smallest value - …(equals 0)

10. Homework (with comments – 1 min).

According to the textbook- §13

According to the problem book– No. 13.3, No. 74 (repetition of incomplete quadratic equations)

Lesson and presentation on the topic: "Power functions. Cubic root. Properties of the cubic root"

Additional materials
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Educational aids and simulators in the Integral online store for grade 9
Educational complex 1C: "Algebraic problems with parameters, grades 9–11" Software environment "1C: Mathematical Constructor 6.0"

Definition of a power function - cube root

Guys, we continue to study power functions. Today we will talk about the "Cubic root of x" function.
What is a cube root?
The number y is called a cube root of x (root of the third degree) if the equality $y^3=x$ holds.
Denoted as $\sqrt(x)$, where x is a radical number, 3 is an exponent.
$\sqrt(27)=3$; $3^3=$27.
$\sqrt((-8))=-2$; $(-2)^3=-8$.
As we can see, the cube root can also be extracted from negative numbers. It turns out that our root exists for all numbers.
The third root of a negative number is equal to a negative number. When raised to an odd power, the sign is preserved; the third power is odd.

Let's check the equality: $\sqrt((-x))$=-$\sqrt(x)$.
Let $\sqrt((-x))=a$ and $\sqrt(x)=b$. Let's raise both expressions to the third power. $–x=a^3$ and $x=b^3$. Then $a^3=-b^3$ or $a=-b$. Using the notation for roots we obtain the desired identity.

Properties of cubic roots

a) $\sqrt(a*b)=\sqrt(a)*\sqrt(6)$.
b) $\sqrt(\frac(a)(b))=\frac(\sqrt(a))(\sqrt(b))$.

Let's prove the second property. $(\sqrt(\frac(a)(b)))^3=\frac(\sqrt(a)^3)(\sqrt(b)^3)=\frac(a)(b)$.
We found that the number $\sqrt(\frac(a)(b))$ cubed is equal to $\frac(a)(b)$ and then equals $\sqrt(\frac(a)(b))$, which and needed to be proven.

Guys, let's build a graph of our function.
1) Domain of definition is the set of real numbers.
2) The function is odd, since $\sqrt((-x))$=-$\sqrt(x)$. Next, consider our function for $x≥0$, then display the graph relative to the origin.
3) The function increases when $x≥0$. For our function, a larger value of the argument corresponds to a larger value of the function, which means increase.
4) The function is not limited from above. In fact, from an arbitrarily large number we can calculate the third root, and we can move upward indefinitely, finding ever larger values ​​of the argument.
5) For $x≥0$ the smallest value is 0. This property is obvious.
Let's build a graph of the function by points at x≥0.

Let's construct our graph of the function over the entire domain of definition. Remember that our function is odd.

Function properties:
1) D(y)=(-∞;+∞).
2) Odd function.
3) Increases by (-∞;+∞).
4) Unlimited.
5) There is no minimum or maximum value.

7) E(y)= (-∞;+∞).
8) Convex downward by (-∞;0), convex upward by (0;+∞).

Examples of solving power functions

Examples
1. Solve the equation $\sqrt(x)=x$.
Solution. Let's construct two graphs on the same coordinate plane $y=\sqrt(x)$ and $y=x$.

As you can see, our graphs intersect at three points.
Answer: (-1;-1), (0;0), (1;1).

2. Construct a graph of the function. $y=\sqrt((x-2))-3$.
Solution. Our graph is obtained from the graph of the function $y=\sqrt(x)$, by parallel translation two units to the right and three units down.

3. Graph the function and read it. $\begin(cases)y=\sqrt(x), x≥-1\\y=-x-2, x≤-1 \end(cases)$.
Solution. Let's construct two graphs of functions on the same coordinate plane, taking into account our conditions. For $x≥-1$ we build a graph of the cubic root, for $x≤-1$ we build a graph of a linear function.
1) D(y)=(-∞;+∞).
2) The function is neither even nor odd.
3) Decreases by (-∞;-1), increases by (-1;+∞).
4) Unlimited from above, limited from below.
5) There is no greatest value. The smallest value is minus one.
6) The function is continuous on the entire number line.
7) E(y)= (-1;+∞).

Problems to solve independently

1. Solve the equation $\sqrt(x)=2-x$.
2. Construct a graph of the function $y=\sqrt((x+1))+1$.
3.Plot a graph of the function and read it. $\begin(cases)y=\sqrt(x), x≥1\\y=(x-1)^2+1, x≤1 \end(cases)$.