Graph of the function y a x l m. Lesson 1. How to construct a graph of the function y = f(x-l), if the graph of the function y = f(x) is known. Parallel transfer of function graphs. Teacher's opening speech

Y = x 2yx 1 O y = (x-4) 2 y = (x+3) 2 by 4 y = x 2 by 3 y = x 2

Graph the function y = f(x) Graph the function y = f(x-l): l units to the right if l > 0 to – l units to the left if l "> 0 to – l units to the left if l "> 0 to – l units to the left if l " title="Graph the function y = f(x) Graph functions y = f(x-l): l units to the right, if l >0 – l units to the left, if l"> title="Construct a graph of the function y = f(x) Construct a graph of the function y = f(x-l): l units to the right, if l >0 - l units to the left, if l"> !}

Construct a graph of the function y = f(x) Construct a graph of the function y = f(x-l): l units to the right, if l >0 - l units to the left, if l 0 to – l units to the left if l "> 0 to – l units to the left if l "> 0 to – l units to the left if l " title="Graph the function y = f(x) Graph functions y = f(x-l): l units to the right, if l >0 – l units to the left, if l"> title="Construct a graph of the function y = f(x) Construct a graph of the function y = f(x-l): l units to the right, if l >0 - l units to the left, if l"> !}

Write the equation of the parabola y = (x + l) 2 shown in the figure x 0 y y = (x – 2) 2 ANSWER: -3

Write the equation of the parabola y = (x + l) 2 shown in the figure x 0 y y = (x + 3) 2 ANSWER: -3

Write the equation of the parabola y = (x + l) 2 shown in the figure x 0 y y = (x – 4) 2 ANSWER: -3

Lesson “How to graph the function y =f(x+ l)+ m, if the graph of the function y = is knownf(x).

8A class. Teacher Bobunova V.V. Municipal educational institution secondary school No. 1, Pugachev, Saratov region

Basic tutorial

The purpose of the lesson : repeat the rules for constructing graphs of functions y=(x+l)and y=f(x)+m, if the graph of the function y= is knownf(x); consider the rule for constructing a graph of a function y= f(x+ l)+ m, if the graph of the function y = is knownf(x); develop the ability to build graphs of various functions.

Tasks:

educational:

teach students to build a graph of the function y =f(x+l)+m, if the graph of the function y =f(x) is known; teach how to use these methods when performing exercises; improve the ability to build graphs of functions y=f(x)+m and y=(x+l), if the graph of the function y=f(x) is known;

R educational:

develop students’ ICT competence while completing independent tasks using electronic educational resources; develop the ability to justify your decision; develop the ability to analyze, compare, generalize and systematize;

V educational:

develop the ability to conduct individual and group discussions;

formation of responsibility of everyone for the final results of work in pairs, ethical behavior.

Lesson type - presentation of new material.

Teaching methods: illustrative-verbal (illustrative-verbal and partially search).

Forms of work - individual(front, work in pairs)

Equipment : Computer, multimedia projector, screen, multimedia presentation for the lesson, handouts.

During the classes.

1. Organizational moment , checking homework. The teacher scans the homework of one of the students, shows it to the class, and the students check their work.
2. Individual work .
Four students are given cards for individual work at the board.

Card 1
Construct graphs of these functions:
, , .

3. Updating knowledge. Working with function graphs. Write the equation of the graph of the function shown in the figure (slides 1-5).When checking the task, remember the already learned rules for constructing graphs of functions y= f(x+ l) and y=f(x)+m f(x) .

4. Explanation of new material.

Class assignment: on one coordinate plane, construct dashed line graphs of the following functions:y=x 2 , y=(x-2) 2 , y=x 2 -3.
Then students are asked to independently construct a solid line graph of the function y = (x-2) 2 -3. There is a discussion about constructing this graph and students are asked to formulate a rule for constructing a graph of a function y=f(x+l)+m , if the graph of the function is knownf(x) .
To plot a function y= f(x+ l)+ m, if the graph of the function is known y=f(x) , you need a graph of the function y= f(x) move along the axis x on / l/ units to the right ifl or left if l>0 , and then move the resulting graph along the axis y by /m/ units up if m>0 , down if m.

Class assignment. To what point will the vertex of the parabola move, given by the equation:

1.y=(x+1)²-2

2. y = (x-7)²-4

3.y=4(x-2)²+8

4. y=0.5(x-3.5)²+6

Question for the class: “Is it necessary to build three graphs forplotting the function y =f(x+ l)+ m? »
After the discussion, the conclusion is drawn: “In fact, the graph of the function y = (x - 2) 2 - 3 is the same parabola that served as the graph of the function y = x 2 ,
only the vertex of the parabola has moved from the origin to point (2; -3). Therefore, to construct it, you need to move the coordinate system to point (2; -3), and in the new coordinate system, construct a graph of the function y=x 2 .

5. Consolidation of new material.

Frontal work with full pronunciation of the rules of construction. Graph the function y = 0.5(x-5) 2 -7

Independent work (in pairs).

1. Graph the function y=2(x+3) 2 +1.

2. Construct a graph of the function y=√x+6+4.

3. No. 21.16(c)

Additional task.

4. Solve the equation graphically -3=x, using the graph in Exercise No. 21.16(c).

5. Solve the system of equations graphically

VI . Lesson summary

Guys, let's summarize the lesson. What did we repeat today, consolidate, learn something new in the lesson?(Students tell the main points of the lesson) What did you find most difficult when creating graphs?

You showed good knowledge. Well done! Ratings...

VII .Homework. clause 12, No. 21.7; 21.16(a);21.20(b). Additional task: plot the function y=x 2 -4x+6. This is a creative task to construct a graph of a quadratic function based on existing knowledge on transformations of graphs of functions.

Literature.

Mordkovich A. G. Algebra. 8th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich, P. V. Semenov. - 12th ed., erased. - M.: Mnemosyne, 2010. Problem book for students of general education institutions / [A. G. Mordkovich, L. A. Alexandrova, T. N. Mishustina and others |; Ed. A. G. Mordkovich. - 12th ed., rev. - M.: Mnemosyne, 2010.

Municipal state educational institution

"Malinovskaya secondary school of Zavyalovsky district" named after Hero of Russia Vitaly Wolf

Lesson topic : “How to plot graphs of functions y=f( x+ l) and y=f( x)+ l, if the graph of the function y = is knownf( x)" (lesson duration 45 minutes)

Item: algebra

Class: 8

The purpose of the lesson : studying and primary awareness of new educational material, understanding the connections and relationships in the objects of study, creating conditions for conscious and confident mastery of the skills of using the construction algorithmgraphs of functions using movement along the coordinate axes.

Lesson objectives: developing skills in constructing graphs of functions using movement along coordinate axes.

Educational objectives of the lesson (formation of cognitive UUD):

introduce students to the algorithm for constructing graphs of functions using movement along the coordinate axes

train the ability to use the derived algorithm;

organize student activities to acquire the necessary skills and abilities;

repeat and consolidate material about graphs of simple functions;

Developmental objectives of the lesson: (formation of regulatory UUD):

develop students’ abilities to analyze, draw conclusions, determine relationships and logical sequence of thoughts;

develop the ability to listen and correct the speech of your comrades;

train the ability to reflect on one’s own activities and the activities of one’s friends.

Educational objectives of the lesson (formation of communicative and personal UUD):

promote the development of students’ cognitive interest in the subject;

instill in students the skills of organizing independent work;

Lesson type Lesson for the initial presentation of new knowledge.

Forms of student work: Frontal, in pairs, group, individual

Equipment: computer, interactive whiteboard, handouts for practical work, self-assessment sheets.

8th grade algebra lesson project

Technological lesson map

Lesson stage

(in accordance with the structure of educational activities)

Teacher activities

Planned student activities

Developed (formed) educational activities

subject

universal

Organizational

Greeting students; teacher checking the class's readiness for the lesson; organization of attention; instructions on how to use the self-assessment sheet.

Familiarization with the self-assessment sheet, clarification of evaluation criteria.

Get ready for work.

L: the ability to highlight the moral aspect of behavior

R : the ability to reflect on one’s own activities and the activities of comrades.

P : conscious and voluntary construction of a speech utterance.

Motivational

Updating knowledge

Frontal work.

Offers to answer the questions posed, repeat the material on the topic “Graphs of elementary functions. Graph of a quadratic function." Prepare students to study a new topic. After checking the results, students are asked to make assumptions about the topic of the lesson and the purpose of the lesson.

Answer questions

Make a guess about the topic of the lesson.

They monitor the correctness of answers and information, and develop their own attitude towards the material studied.

Write down the topic of the lesson in your notebook.

Repeat the definition of a quadratic function, its graph, and how to construct a graph of a quadratic function.

L: development of motives for educational activities.

R: goal setting.

P: independently identify and formulate a cognitive goal. Highlight essential information, put forward hypotheses and update personal life experience

Planning actions to achieve the goal.

Frontal work.

Finds out from students what qualities they need to achieve the goal, how to achieve the goal, what will we do for this?

Offers to do practical work.

They list personality traits: perseverance, willpower, discipline.

List the actions that need to be taken to complete the assigned tasks. They outline a work plan and by what means they will carry out the plan.

The ability to apply previously acquired knowledge to learn new things.

L: acceptance of the social role of the student, meaning formation.

R: drawing up a plan and sequence of actions, predicting the result and level of mastery of the material.

TO: the ability to listen to the interlocutor, complement and clarify the opinions expressed.

P: the ability to consciously construct a speech utterance.

Implementation of what was planned (learning new material).

Work in groups.

Offers practical work in groups.

Do practical work. They formulate a rule, work from the textbook, compare it with their formulation, come up with and analyze their own examples. They speak out their assumptions, listen to their classmates’ options, draw conclusions,

apply the acquired knowledge in practice.

The ability to understand and formulate an algorithm for constructing graphs of quadratic functions, and to apply these algorithms when constructing graphs of elementary functions.

L: independence and critical thinking; development of cooperation skills.

R: Monitoring the correctness of answers to information in the textbook, developing students’ own attitude to the material studied. Correction.

P: Search and highlight the necessary information.

TO: Listen to the interlocutor, construct statements that are understandable to the interlocutor. Semantic reading

Primary comprehension and consolidation of knowledge

Organizes work on compiling an algorithm for constructing graphs of elementary functions.

Complete the task of constructing graphs using movement along the coordinate axes.

Ability to apply the rule of constructing graphs using movement along coordinate axes.

L:meaning formation.

R : train the ability to reflect on one’s own activities and the activities of one’s comrades.

TO: the ability to listen and engage in dialogue, participate in collective discussion of problems, integrate into a peer group and build productive interaction, cultivate responsibility and accuracy.

P: ability to use the derived algorithm;

Reinforcing the material learned

Organizes the development of skills in constructing graphs of a quadratic function using movement along the coordinate axes in notebooks and on the board.

Offers to solve independent work followed by self-test. (On the interactive whiteboard). Organizes the reproduction and correction of students’ basic knowledge

They complete the task, compare it with the solution on the board, and evaluate their solution.

Perform independent work and self-assessment.

Apply an algorithm for constructing graphs using movement along coordinate axes.

L: respect for the mistakes of classmates, independence and critical thinking.

R: carry out self-monitoring of the process of completing a task, evaluate the proposed solutions. Correction.

P: compare and summarize facts, build logical reasoning, use demonstrative mathematical speech.

TO: listen to the interlocutor, construct statements that are understandable to the interlocutor.

"Creative application of knowledge."

Work in groups.

Offers to find the most convenient way to graph a function by applying the transformations learned in the lesson together.

They work in groups, look for different ways to construct graphs, mutually monitor the process of completing the task, evaluate the proposed options for statements, and choose the most accurate one.

Apply the graphing algorithm using the obtained methods in combination

L: acceptance of the social role of the student; independence and critical thinking; development of motives for educational activities, development of cooperation skills.

R: accept and implement a learning task

P: compare and analyze the results of the proposed task, justify your opinion

TO: listen to the interlocutor, coordinate efforts to solve the educational task, negotiate and come to a common opinion in joint activities, construct statements that are understandable to the interlocutor.

Homework

Explains homework. Provides a selection of multi-level tasks using a textbook and additional sources of information:

Plan their actions in accordance with self-esteem. They independently choose the level to complete their homework.

Working at home with text.

Know the algorithm for constructing graphs of a quadratic function, using a shift along the x and y axes, and be able to apply it when performing practical tasks.

L.acceptance of the social role of the student

R. Self-assessment is carried out adequately.

P. Update acquired knowledge in accordance with the level of assimilation

Reflection

Organizes a discussion of achievements by asking prepared questions.

Offers self-assessment of achievements using the proposed algorithm.

Participate in a conversation to discuss achievements, answering questions prepared in advance.

They draw conclusions and self-assess their achievements using the proposed algorithm.

L: independence and critical thinking;

R: accept and save the educational goal and task, carry out final and step-by-step control based on the result, plan future activities

P: analyze the degree of assimilation of new material

TO: listen to classmates, voice their opinions.

During the classes:

1. Organizational stage

Teacher's opening remarks:

Hello guys, sit down. I'm glad to meet you. I see you are in a good mood, and I wish everyone in the lesson to rise one more step higher in knowledge.

The living connection between generations is not interrupted for a moment; every day we learn the experience accumulated by our ancestors. The ancient Greeks, based on observations and practical experience, drew conclusions, expressed assumptions and hypotheses, and then at meetings of scientists - symposiums, they tried to substantiate and prove these hypotheses. At that time, the statement arose: “Truth is born in a dispute.” Today's lesson will also be like a small symposium. We will express our assumption on the issue, try to prove it, and if we succeed, then we will see how it can be applied when solving problems. And as the epigraph of our lesson, I want to offer the words of Pythagoras:

Instructions on how to use the self-assessment sheet: You have self-assessment sheets on your desks. Sign them. During lesson, try to evaluate yourself and one of your classmates, according to the criteria indicated in the self-assessment sheet.(Annex 1)

Children's statements.

Students are ready to start work and have an idea of how to work with the self-assessment sheet.

2. Updating knowledge .

What basic functions have we learned?

What are the graphs of these functions? Match each graph with a function.

What is the name of the function y=x 2 ?

What is the graph of a function?

Tell us the algorithm for constructing a graph of a function?

Graph the Functiony=x 2

y=x 2 , y=k/x, y=kx+v, y=kx, y=√x, y=|x|

line, parabola, hyperbola

quadratic

parabola

at point (0;0)

make a table of the corresponding values of the argument and function.

Mark on the coordinate plane the points whose coordinates are indicated in the table.

build a graph by connecting the dots

Complete the task (one person at the board, the rest in the notebook)

3. Staging (summing up)

Are functions quadratic (opens entry) and why?

Divide them into 4 groups. Explain the principle.

y=x 2 +2, y=x 2 +4, y=x 2 -1, y=x 2 -3, y=(x-1) 2 , y=(x-2) 2 y=(x+1) 2 y=(x+4) 2

Try to formulate the topic of our lesson? Right.

Open your notebooks and write down the topic of the lesson "»

What will we do in class? So, what goal will you define for yourself in this lesson?

If you clearly understand what you will have to do during the lesson, put 2 points on the self-assessment sheet; if you doubt something, put 1 point; if you don’t understand the purpose and objectives of the lesson - 0. Rate the classmate written on your sheet. if he took part in determining the topic, or the purpose and objectives of the lesson - 1 point, if not - 0 points.

Yes, they are, because the variable is to the second power

Break down and explain

1st group:y= x 2 +2, y= x 2 +4

Group 2:y= x 2 -1, y= x 2 -3

Group 3:y=( x-1) 2 , y=( x-2) 2

Group 4:y=( x+1) 2 y=( x+4) 2

formulate the theme "Plotting graphs of elementary functions».

Let's get acquainted with the charting algorithmelementaryfunctions.

Write down the topic of the lesson.

4. Planning actions to achieve the goal.

Front work

Guys, what is easier to deal with a problem alone or together? What qualities would a friend have that you would want to work with to solve a problem? How to achieve the goal, what will we do for this?

We continue to evaluate ourselves and our friend according to the criteria specified in the self-assessment sheet.

Together.

Children list: smart, kind, resourceful.

Can be found in a textbook or on the Internet.

Evaluate themselves and one classmate.

5. Search

Plot graphs of these functions in groups. Write down the general form of the functions whose graphs you plotted. Draw a conclusion.

Let's listen to the first group.

How did you build?

Where is the vertex of the parabola?

y= x 2 +a?

Let's listen to the second group.

What functions did you plot?

How did you build?

What pattern did you see?

Where is the vertex of the parabola?

How to graph a function y= x 2 - huh?

Let's listen to the third group.

What functions did you plot?

How did you build?

What pattern did you see?

Where is the vertex of the parabola?

How to graph a function y=(x-c) 2 (c>0)?

Let's listen to the fourth group.

What functions did you plot?

How did you build?

What pattern did you see?

Where is the vertex of the parabola?

How to graph a function y=(x+c) 2 (c>0)?

Work in groups of four, speak the rule to each other, offer your functions other than quadratic to another pair, check the correctness of the solution. Enter points on the self-assessment sheet.

What did we do in class?

State the rules again.

Which function graph did we use and what did we do with it?

What should we add to the formulation of the topic of the lesson? (finishes)

Each group works with its functions according to general instructions in groups and draws conclusions.

Group 1 posts graphs

y= x 2 +a, (a>0)

By points

All points on the graph of a functiony= x 2 moved up along the O axisyon 2; by 4, i.e. on a unit segments.

At point (0;a).

y= x 2 move up along the O axisyon a unit segments.

Group 2 hangs up graphs

We have constructed graphs of functions of the formy= x 2 - a, (a>0)

By points

All points on the graph of a functiony= x 2 moved down along the O axisyby 1; by 3, i.e. on a unit segments.

At point (0;-a).

You need all the points on the graph of the functiony= x 2 move down along the O axisyon a unit segments.

Group 3 hangs up charts

We have constructed graphs of functions of the formy=( x-V) 2 (in>0)

By points

All points on the graph of a functiony= x 2 shifted to the right along the Ox axis by 1; by 2, i.e. on in unit segments.

At point (at;0).

You need all the points on the graph of the functiony= x 2 shift to the right along the Ox axis in unit segments.

Group 4 hangs up charts

We have constructed graphs of functions of the formy=( x+c) 2 (in>0)

By points

All points on the graph of a functiony= x 2 shifted to the left along the Ox axis by 1; by 2, i.e. on in unit segments.

At point (-at;0).

You need all the points on the graph of the functiony= x 2 shift to the left along the Ox axis in unit segments.

They work in groups of four, compare formulations, construct functions proposed by their comrades, and evaluate.

Plotting graphs of quadratic functions?

Listed by support

Graph of a function y= x 2 we shifted along the coordinate axes.

Construction using movement along coordinate axes.

6. Consolidation of acquired knowledge.

Work according to the textbook: complete No. 19.11 (a, b), No. 20.11 (a, b) at the board and in notebooks.

Oral frontal work

№ 19.3 (a, b), No. 19.5 (a, b), No. 19.7 (a, b), No. 20.1 (a, b), No. 20.2 (a, b), No. 20.4 (a, b)

Students come to the board in order, solve examples, pronounce the rule.

They work orally and conduct self-assessment.

7. “Creative application of knowledge”

Work in groups. (slide 7)

Graph the Function

Do the task in groups.

Conduct self-assessment.

8. Homework. (Slide 8)

You can write down your homework by choosing at least two numbers:

P.19, 20 learn algorithms.

№ 19.2, No. 19.9 (c, d), No. 20.2, No. 20.5 (c, d).

Select and write down homework.

Evaluate your choice of homework.

9. Reflection

During the entire lesson you filled out a self-assessment sheet, count the number of points and give yourself a grade for the lesson and please rate your classmate verbally. Let's listen to your friend's assessment, and the rest compare their assessment with that of their classmate. Try to explain your assessment. What goal did we set at the beginning of the lesson? Have you achieved your goal?

Parable: A sage was walking, and three people met him, carrying carts with stones for construction under the hot sun. The sage stopped and asked each one a question. The first one asked, “What have you been doing all day? And he answered with a grin that he had been carrying stones all day. The sage asked the second, “What did you do all day?” and he replied, “And I did my job conscientiously.” And the third smiled, his face lit up with joy and pleasure, “And I took part in the construction of the temple.”
– Guys, let’s try to evaluate each of your work for the lesson.
-Who carried the stones? (Pick up the yellow tokens)
– Who worked conscientiously? (Pick up the blue tokens)
-Who built the temple? (Pick up the red tokens)

Self-esteem. Give a classmate's assessment.

Using signal cards, they show the degree of mastery of the material.

Appendix 1 Self-assessment sheet

p/p

Student activity

Self-assessment criteria

Self-esteem

Classmate evaluation criteria

Classmate's rating (F.I.)

Formulation of the lesson topic, lesson goals and objectives

I myself was able to determine the topic, purpose and objectives of the lesson - 2 points.

I was only able to determine the topic of the lesson 1 point.

I could not determine the topic, purpose and objectives of the lesson - 0 points.

Participated in determining the topic of the lesson, the purpose of the lesson, or the objectives of the lesson - 1 point.

Did not participate in determining the topic of the lesson, the purpose of the lesson, or the objectives of the lesson 0 points.

What will I do to achieve the goal.

I myself determined how to achieve the goal of the lesson 1 point.

I could not determine how to achieve the lesson goal - 0 points.

Participated in planning actions to achieve the lesson goal - 1 point.

Did not participate in planning actions to achieve the lesson goal 0 points.

Doing practical work in a group

Participated in group work – 1 point.

Did not participate in the work of the group - 0 point.

Work in pairs to reinforce the rules.

Checking the correctness of tasks

Participated in pair work – 1 point.

Did not participate in the work of the pair - 0 point.

Not Evaluated

Execution No. 19.11 (a, b), No. 20.11 (a, b)

I made all the examples myself - 2 points.

Did more than half myself – 1 point

Did less than half myself - 0 points.

Completed the task at the board 1 point.

Didn't complete the task on the board 0 points.

Oral work

For each correctly completed task -1 point

Not Evaluated

Completing a creative task (group work)

Found a convenient way to solve 1 point.

I didn’t find a convenient way to solve 0 points.

Choosing homework

4 points – selected all tasks;

3 points - chose 3 tasks out of 4,

2 points – only 2 numbers were chosen.

Not Evaluated

Rate yourself:

if you scored 9-11 points - “5”

6 – 8 points – “4”

3 – 5 points – “3”

This video lesson will discuss the issue of graphical representation of the function y = f(x + l), provided that the graph of the function y = f(x) is known in advance.

For completeness of understanding, explanations will be accompanied by a visual supplement. To do this, we will construct graphs of the functions y = x 2 and y = (x + 3) 2 in the same coordinate system. The first of the functions has already been discussed in our video lessons earlier, and we know that its graph is a parabola. For the function y = (x + 3) 2, substituting the values of the argument x, we calculate the coordinates of the points, from which we build a graph. By connecting the points of a smooth curve, we see that the graph is a parabola. You will notice that this graph has the same appearance as in the case of y = x 2, but in this case it is moved to the left by three units along the x-axis. Accordingly, there is also a displacement of the vertex of the parabola to the position (- 3; 0), and not at the origin of coordinates, as we see for the parabola of equality y = x 2. The axis of symmetry is also shifted, and corresponds to the line at position x = - 3, and not x = 0, as we can observe in the case of the graph of the equation y = x 2.

When we depict, as the video demonstrates, graphs of the functions y = x 2 and y = (x - 2) 2 in one coordinate grid, you can notice that the second graph is similar to the first with the only peculiarity that there is a shift along the x-axis to the right by 2 positions. You can see what this looks like in person in the video provided.

After viewing this example, it becomes clear that graphically solving functions of this type occurs using the same algorithm.

Another example that our video offers is the equality y = -2 (x - 4) 2. Its graph is also a parabola of the form y = - 2x 2, which has undergone a shift, that is, a parallel translation along the x-axis to the right by four units. This video will introduce you to the chart itself.

Based on the above, the following conclusions can be drawn:

1) In order to draw a graph of a function like y = f(x + l), if l is a positive number specified by the condition, it is necessary to move the equality graph along the x axis to the left by l scale units;

2) In order to construct a graph of the function y = f(x - l), where the number l is a given positive number, you simply need to shift the graph of the function y = f(x) along the x axis by l scale units to the right.

That is, if the sign of the number l is positive, then we shift it in the direction of decreasing values along the abscissa axis, and if it is negative, then in the direction of increasing it.

Example 1. Using the knowledge gained in the video, it is necessary to plot the function y = - 3 / (x+5)

To solve this problem, we first construct a hyperbola for the equality y = -3/x, after which we shift the resulting graph along the x-axis to the left by 5 scale units. As a result, we got the required graph - this is a hyperbola with asymptotes x = -5 and y = 0. You saw the graph itself when watching the proposed video.

The next example is as follows: it is necessary to construct a graph of the function y = |x+2|. The essence of solving this problem is the same algorithm as in the previous case. First, we build a graph of the function y = |x|, and then shift it by two scale units to the left.

In addition, it should be said that when plotting a function of the form y = f(x + l), if l is any number different from zero, that is, both positive and negative. When solving function problems, we calculated the coordinates of points, which we used to construct graphs, without paying attention to the sign next to a certain number l, which was present in our functions, but simply noted the shift of the graph to one degree or another. However, it should be noted that the direction of the shift was still determined by the sign of the number l: in the case when the value of the number l was positive, the graph shifted to the left, and in the case when the number l was less than zero, the graph shifted to the right.

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Slide 1

Slide 2

x y 2 1 1 0 6 -2 3 Oral work for repetition 1) [-1;3] 2) 3) [-2;6] 4)

Slide 3

x y 3 1 1 0 6 -2 3 Oral work for repetition 1) [-1;3] 2) 3) [-2;6] 4) Find the range of values of the function

Slide 4

x y 4 1 1 0 6 -2 3 Oral repetition work 1) 1 2) 1;1 3) 1;4 4) 4 Find the zeros of the function

Slide 5

One of the figures shows a graph of a function increasing over the interval. Please indicate this pic. Oral work for repetition

Slide 6

Oral work for repetition

Slide 7

F(-1)

Slide 8

The domain of definition of a function... The domain of values of a function... Zeros of a function... Positive and negative values of a function... Monotonicity of a function... The largest and smallest value of a function... Continuity... Boundedness... Convexity... Oral work for repetition

Slide 9

How to construct a graph of the function y=f(x+l)+m from the graph of the function y=f(x)

Slide 10

10 m >0 m

Slide 11

The graph of the function y=a(x+l)2 is a parabola, which can be obtained from the graph of the function y = ax2 using parallel translation along the x-axis by l units to the left if l> 0 l

Slide 12

x y 12 X=5 y=4 1 1 0 5 4 5 units 4 units

Slide 13

In class No. 21.5 (oral) No. 21.12-21.13 (v, g) No. 21.10 (g)

Slide 14

Practical work

You can choose to build 2 graphs: No. 21.8 (a); No. 21.9 (a); No. 21.11 (c); No. 21.11 (g).

Slide 15

Converting Function Graphs

Homework §21. No. 21.11 (a, b) No. 21.12-19.13 (a, b)

Slide 16

Literature

Drawings for oral work from the textbook by S.A. Telyakovsky “Algebra. Textbook for 9th grade of general education institutions.” M.: Enlightenment. 2003

Slide 17

View all slides

Abstract

Municipal budgetary educational institution gymnasium No. 1, Lebedyan, Lipetsk region

(plan is for 2 hours)

Mathematic teacher

Gladunets Irina Vladimirovna

ANNOTATION

INTRODUCTION

LESSON PLAN

The purpose of the lesson:

Lesson objectives:

Educational:

Lesson type: learning new material.

Lesson type: combined.

Forms of student work: frontal, collective.

Interdisciplinary connections: physics.

Intrasubject connections:

LESSON STRUCTURE

Lesson stage	Presentation slide no.	Teacher activities	Student activity	(per minute)

Organizing time		Greets students.
Updating knowledge		Find the domain of the function Find the domain of the function Find the zeros of the function Find the zeros of the function One of the figures shows a graph of a function decreasing over the interval. Please indicate this pic. The figure shows a graph of the function y = f(x). From the given statements, choose the correct one List properties of a function	Answer teacher questions

Learning new material
Physical education minute			Doing exercises
		No. 21.5 (oral) No. 21.12-21.13 (c, d)
Practical work
Control of practical work
Homework		Sets a homework assignment.	Write down the task in your diary.
Lesson summary			Answer the teacher's questions.
Reflection			Summarize the lesson.

DURING THE CLASSES

The graph of the function y = ax2 + m is a parabola that can be obtained from the graph of the function y = ax2 by parallel translation along the x axis by m units up if m > 0, or by - m units down if m< 0.

The graph of the function y=a(x+l)2 is a parabola, which can be obtained from the graph of the function y = ax2 using parallel translation along the x axis by l units to the left if l>0, or by – l units to the right if l<0

Municipal budgetary educational institution gymnasium No. 1, Lebedyan, Lipetsk region

Development of an algebra lesson in 8th grade on the topic

How to construct a graph of the function y=f(x+l)+m from the graph of the function y=f(x)

(plan is for 2 hours)

Mathematic teacher

Gladunets Irina Vladimirovna

ANNOTATION

This lesson may be interesting because during the lesson of studying new material, practical work of a teaching nature is immediately carried out in order to consolidate what has been learned. Moreover, the work is carried out collectively (in groups). �The lesson helps to promote the development of cognitive activity of learning, develop students' attention and create the need to acquire knowledge, cultivate self-control skills, habits of reflection, achieve a change in the student's role in the educational process from a passive observer to an active researcher

INTRODUCTION

The relevance of this development lies in the fact that a modern lesson should not only be boring and interesting, but also display modern techniques and resources. In this case, independent practice of the studied material in the course of collective work, computer support, visibility, mutual assistance and mutual control of students are used, which means that the lesson ensures communication and scientific development of students in the lesson, which meets modern educational requirements. This lesson allows you to develop students’ logical thinking; develop the ability to generalize and draw conclusions; develop cognitive interest and communication skills when working with a partner. The lesson also helps to promote the formation of a responsible attitude towards learning; cultivate a culture of educational work, skills for economical use of educational time; cultivate the will and perseverance to achieve final results.

The lesson is designed for children of different levels of development; the main emphasis in the lesson methodology is on the collective method of work. This lesson is designed so that it meets the requirements for a modern lesson on the development of independence in learning and the development of communicative qualities.

LESSON PLAN

The purpose of the lesson:

study the algorithm for constructing a graph of the function y=f(x+l)+m from the graph of the function y=f(x) and consolidate the learned material during independent educational work.

Lesson objectives:

Educational:

consolidate the skill of plotting graphs of various functions;

consolidate the skill of shifting the graph of the function y=f(x) along the Ox and Oy axes

carry out verification of knowledge on this topic during practical (collective) work.

Educational:

promote the development of cognitive learning activities through the use of information and communication technologies in the classroom;

develop students’ logical thinking and attention; create the need to acquire knowledge.

Educational:

develop self-control skills and habits of reflection;

to achieve a change in the role of the student in the educational process from a passive observer to an active researcher.

Lesson type: learning new material.

Lesson type: combined.

Forms of student work: frontal, collective.

Material and technical equipment: computer, media projector, screen.

Formation of key competencies: ability to build graphs of previously studied functions and shift them along the Ox, Oy axes.

Interdisciplinary connections: physics.

Intrasubject connections: functions: linear, a special case of a quadratic function, inverse proportionality, y=√x.

LESSON STRUCTURE

Lesson stage	Presentation slide no.	Teacher activities (indicating actions with ESM, for example, demonstration)	Student activity	(per minute)

Organizing time		Greets students.	Greetings from the teachers. Write down the number
Updating knowledge		Asks students questions to review: Find the domain of the function Find the domain of the function Find the zeros of the function Find the zeros of the function One of the figures shows a graph of a function decreasing over the interval. Please indicate this pic. The figure shows a graph of the function y = f(x). From the given statements, choose the correct one List properties of a function	Answer teacher questions
Formulation of the topic and purpose of the lesson		Formulates the topic of the lesson and its purpose and objectives.	Write down the topic of the lesson in your notebook.
Learning new material		Demonstrates a presentation. Working with the graph of the function y=1/2x2. Transforming the graph by shifting it to the right by 5 units. scale and up by 4 units.	They watch the presentation, answer the teacher’s questions, summarize the material, and draw conclusions. Build graphs and their displacement in notebooks.
Physical education minute		Recites exercises in poetic form.	Doing exercises
Consolidation of acquired knowledge, skills and abilities		Offers to solve problems from the textbook No. 21.5 (oral) No. 21.12-21.13 (c, d)	Do it in a notebook and on the board.
Practical work		Offers to complete a practical task by dividing the class into groups of 4 (3) people.	They do practical work in a notebook, then make a report on a double piece of paper, giving marks to each other, according to the activity of group members and their participation in the work and the report.
Control of practical work		Checks reports in groups and assigns grades to students.	Submit a report on practical work to the teacher
Homework		Sets a homework assignment.	Write down the task in your diary.
Lesson summary		Asks students questions about the algorithm for constructing graphs of functions and their movement along the coordinate axes.	Answer the teacher's questions.
Reflection		Conducts psychological testing for reflection	Summarize the lesson.

DURING THE CLASSES

Repetition of previously studied material

Self-test (slides 2-8)

2. Updating knowledge (slides 9-11)

The topic of our lesson: “How to construct a graph of the function y=f(x+l)+m from the graph of the function y=f(x). We need to develop the skill of constructing a graph of the function y=f(x+l)+m by shifting the graph y=f(x) (original) along the coordinate axes, or by “linking” the graph of the original function to a new coordinate system. Then we will consolidate the acquired knowledge through practical collective training work.

Let's remember how we built graphs of the functions y=f(x)+m and y=f(x+l).

Learning new material (slide 12)

Working with the graph of the function y=1/2x2. Transforming the graph by shifting it to the right by 5 units. scale and up by 4 units.

Reinforcing what has been learned (slide 13)

No. 21.5 (oral), No. 21.12-21.13 (v, g), No. 21.10 (g)

Practical (educational) work (team) (slide 14)

You can choose to build 2 graphs: No. 21.8 (a); 21.9(a); 21.11 (c); 21.11 (g).

Students in the class are divided into groups of 4 people so that the group includes students of different learning levels. While discussing how to graph functions, everyone works in their own notebook. Then the students transfer the results of their collective work onto a double sheet of paper. Record all group members and rate each group member according to activity and participation in the work. Then the sheets of paper are submitted to the teacher for checking.

While working, students can ask the teacher for help.

Grades may be posted in a journal (at the discretion of the teacher).

Homework: §21, No. 21.11 (a, b), No. 21.12-19.13 (a, b) (slide 15)

Drawings for oral repetition work from the textbook by S.A. Telyakovsky “Algebra. Textbook for 9th grade of general education institutions.” M.: Enlightenment. 2003