What is an analytical model of a linear function. Linear function. Detailed theory with examples (2019). Maintaining your privacy at the company level

Instruction

To find the coordinates of a point that belongs to a line, select it on the line and drop perpendicular lines on the coordinate axis. Determine what number the intersection point corresponds to, the intersection with the x-axis is the value of the abscissa, that is, x1, the intersection with the y-axis is the ordinate, y1.

Try to choose a point whose coordinates can be determined without fractional values, for convenience and accuracy of calculations. To build an equation, you need at least two points. Find the coordinates of another point belonging to this line (x2, y2).

Substitute the values of the coordinates into the equation of a straight line, which has the general form y=kx+b. You will get a system of two equations y1=kx1+b and y2=kx2+b. Solve this system, for example, in the following way.

Express b from the first equation and plug into the second, find k, plug into any equation and find b. For example, the solution of the system 1=2k+b and 3=5k+b will look like this: b=1-2k, 3=5k+(1-2k); 3k=2, k=1.5, b=1-2*1.5=-2. Thus, the equation of a straight line has the form y=1.5x-2.

Knowing two points belonging to the line, try to use the canonical equation of the line, it looks like this: (x - x1) / (x2 - x1) \u003d (y - y1) / (y2 - y1). Substitute the values (x1; y1) and (x2; y2), simplify. For example, points (2;3) and (-1;5) belong to the line (x-2)/(-1-2)=(y-3)/(5-3); -3(x-2)=2(y-3); -3x+6=2y-6; 2y=12-3x or y=6-1.5x.

To find the equation of a function that has a non-linear graph, proceed as follows. View all standard plots y=x^2, y=x^3, y=√x, y=sinx, y=cosx, y=tgx, etc. If one of them reminds you of your schedule, take it as a basis.

Draw a standard base function plot on the same coordinate axis and find it from your plot. If the graph is moved up or down by several units, then this number has been added to the function (for example, y=sinx+4). If the graph is moved to the right or left, then the number is added to the argument (for example, y \u003d sin (x + P / 2).

An elongated graph in height indicates that the argument function is multiplied by some number (for example, y=2sinx). If the graph, on the contrary, is reduced in height, then the number in front of the function is less than 1.

Compare the graph of the base function and your function in width. If it is narrower, then x is preceded by a number greater than 1, wide - a number less than 1 (for example, y=sin0.5x).

note

Perhaps the graph corresponds to the found equation only on a certain segment. In this case, indicate for which values of x the resulting equality holds.

A straight line is an algebraic line of the first order. In a Cartesian coordinate system on a plane, the equation of a straight line is given by an equation of the first degree.

You will need

Knowledge of analytical geometry. Basic knowledge of algebra.

Instruction

The equation is given by two on , which this line must pass. Compose the ratio of the coordinates of these points. Let the first point have coordinates (x1,y1), and the second (x2,y2), then the equation of the line will be written as follows: (x-x1)/(x2-x1) = (y-y1)(y2-y1).

We transform the obtained equation of a straight line and explicitly express y in terms of x. After this operation, the straight line equation will take the final form: y=(x-x1)/((x2-x1)*(y2-y1))+y1.

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Maslova Angelina

Research work in mathematics. Angelina compiled a computer model of a linear function, with the help of which she conducted the study.

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Municipal Autonomous Educational Institution Secondary School No. 8 of the City District of Bor, Nizhny Novgorod Region

Research work in computer science and mathematics

Completed by a student of grade 7A, Maslova Angelina

Supervisor: computer science teacher, Voronina Anna Alekseevna.

Bor city district - 2015

Introduction

Examining a Linear Function in Spreadsheets

Conclusion

Bibliography

Introduction

This year, in algebra lessons, we got acquainted with a linear function. We learned how to graph a linear function, determined how the function graph should behave depending on its coefficients. A little later, in a computer science lesson, we learned that these actions can be considered mathematical modeling. I decided to see if it was possible to explore a linear function using spreadsheets.

Goal of the work: explore linear function in spreadsheets

Research objectives:

find and study information about a linear function;
build a mathematical model of a linear function in a spreadsheet;
explore a linear function using the constructed model.

Object of study:math modeling.

Subject of study:mathematical model of a linear function.

Modeling as a method of knowledge

Man knows the world almost from his birth. To do this, a person uses models that can be very diverse.

Model is a new object that reflects some essential properties of a real object.

Real object models are used in a variety of situations:

When an object is very large (for example, the Earth - a model: a globe or a map) or, conversely, too small (a biological cell).
When the object is very complex in its structure (car - model: children's car).
When an object is dangerous to study (volcano).
When the object is very far away.

Modeling is the process of creating and studying a model.

We create and use models ourselves, sometimes without even thinking about it. For example, we take pictures of some event in our lives and then show them to our friends.

According to the type of information, all models can be divided into several groups:

verbal models. These models may exist orally or in writing. It can be just a verbal description of some subject or a poem, or maybe an article in a newspaper or an essay - all these are verbal models.
Graphic models. These are our drawings, photographs, diagrams and graphs.
iconic models. These are models written in some sign language: notes, mathematical, physical or chemical formulas.

Linear function and its properties

Linear functionis called a function of the form

The graph of a linear function is a straight line.

1 . To plot a function, we need the coordinates of two points belonging to the graph of the function. To find them, you need to take two x values, substitute them into the equation of the function, and calculate the corresponding y values from them.

For example, to graph the function, convenient to take and , then the ordinates of these points will be equal And .

We get points A(0;2) and B(3;3). Connect them and get the graph of the function:

2 . In the function equation y=kx+b, the coefficient k is responsible for the slope of the function graph:

The coefficient b is responsible for shifting the graph along the OY axis:

The figure below shows the graphs of the functions; ;

Note that in all these functions the coefficient greater than zero to the right . Moreover, the greater the value, the steeper the straight line goes.

In all functions- and we see that all graphs intersect the OY axis at the point (0; 3)

Now consider the graphs of the functions; ;

This time in all functions the coefficient less than zero , and all function graphs are skewed to the left . The coefficient b is the same, b=3, and the graphs, as in the previous case, cross the OY axis at the point (0;3)

Consider the function graphs; ;

Now in all equations of functions the coefficientsare equal. And we got three parallel lines.

But the coefficients b are different, and these graphs intersect the OY axis at different points:

Function Graph (b=3) crosses the OY axis at the point (0;3)

Function Graph (b=0) crosses the OY axis at the point (0;0) - the origin.

Function Graph (b=-2) crosses the OY axis at the point (0;-2)

So, if we know the signs of the coefficients k and b, then we can immediately imagine what the graph of the function looks like.

If k 0 , then the graph of the function looks like:

If k>0 and b>0 , then the graph of the function looks like:

If k>0 and b , then the graph of the function looks like:

If k, then the graph of the function looks like:

If k=0 , then the function turns into a functionand its graph looks like:

Ordinates of all points of the graph of the function equal

If b=0 , then the graph of the functionpasses through the origin:

4. Condition for parallelism of two lines:

Function Graph parallel to the graph of the function, If

5. The condition of perpendicularity of two lines:

Function Graph perpendicular to the graph of the function if or

6 . Intersection points of the graph of the functionwith coordinate axes.

with OY axis. The abscissa of any point belonging to the OY axis is equal to zero. Therefore, to find the point of intersection with the OY axis, you need to substitute zero instead of x in the equation of the function. We get y=b. That is, the point of intersection with the OY axis has coordinates (0;b).

With OX axis: The ordinate of any point belonging to the OX axis is zero. Therefore, to find the point of intersection with the OX axis, you need to substitute zero instead of y in the equation of the function. We get 0=kx+b. From here. That is, the point of intersection with the OX axis has coordinates (;0):

Examining a Linear Function in Spreadsheets

To explore a linear function in a spreadsheet environment, I compiled the following algorithm:

Build a mathematical model of the Linear function in a spreadsheet.
Fill in the trace table of argument and function values.
Plot a Linear Function using the Chart Wizard.
Explore the Linear function depending on the values of the coefficients.

To study the linear function, I used the Microsoft Office Excel 2007 program. To compile tables of argument and function values, I used formulas. I got the following table of values:

On such a mathematical model, one can easily follow the changes in the graph of a linear function by changing the values of the coefficients in the table.

Also, using spreadsheets, I decided to follow how the relative position of the graphs of two linear functions changes. By building a new mathematical model in the spreadsheet, I got the following result:

By changing the coefficients of two linear functions, I was clearly convinced of the validity of the studied information about the properties of linear functions.

Conclusion

The linear function in algebra is considered the simplest. But at the same time, it has many properties that are not immediately clear. Having built a mathematical model of a linear function in spreadsheets, and having studied it, the properties of a linear function have become more clear to me. I was able to clearly see how the graph changes when the coefficients of the function change.

I think that the mathematical model I have built will help seventh grade students to independently explore the linear function and understand it better.

Bibliography

Algebra textbook for grade 7.
Informatics textbook for grade 7
wikipedia.org

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Object of research: linear function. Subject of study: mathematical model of a linear function.

The purpose of the work: to explore a linear function in spreadsheets. Research objectives: to find and study information about a linear function; build a mathematical model of a linear function in a spreadsheet; explore a linear function using the constructed model.

A linear function is a function of the form y= k x+ b, where x is an argument, and k and b are some numbers (coefficients). The graph of a linear function is a straight line.

Consider a function y=kx+b such that k 0 , b=0 . View: y=kx In one coordinate system, we construct graphs of these functions: y=3x y=x y=-7x We build each graph with the corresponding color x 0 1 y 0 3 x 0 1 y 0 1 x 0 1 y 0 7

The graph of a linear function of the form y \u003d k x passes through the origin. y=x y=3x y=-7x y x

Conclusion: The graph of a linear function of the form y = kx + b intersects the O Y axis at the point (0; b).

Consider the function y=kx+b , where k=0. View: y=b In one coordinate system, build graphs of functions: y=4 y=-3 y=0 We build each graph with the appropriate color

The graph of a linear function of the form y = b runs parallel to the OX axis and intersects the O Y axis at the point (0; b). y=4 y=-3 y=0 y x

In one coordinate system, build graphs of functions: Y=2x Y=2x+ 3 Y=2x-4 We build each graph with the appropriate color x 0 1 y 0 2 x 0 1 y 3 5 x 0 1 y -4 -2

Graphs of linear functions of the form y=kx+b are parallel if the coefficients at x are the same. y \u003d 2x + 3 y \u003d 2x y \u003d 2x-4 y x

In one coordinate system, we construct graphs of functions: y=3x+4 Y= - 2x+4 We build graphs with the appropriate color x 0 1 y 4 7 x 0 1 y 4 2

Graphs of two linear functions of the form y=kx+b intersect if the coefficients at x are different. y x

In one coordinate system, we construct graphs of functions: y=0, 5x-2 y=-2x-4 y= 4 x-1 y=- 0, 2 5 x- 3 x 0 4 y x 0 -2 y -4 0 x 0 4 y -2 0 x 0 1 y -1 3 x 0 - 4 y -3 -2

y=0, 5x-2 y=-2x-4 y= 4 x-1 y=- 0, 2 5 x- 1" .

Therefore, the coefficient k is called the slope of the straight line - the graph of the function y \u003d kx + b. If k 0 , then the angle of inclination of the graph to the O X axis is acute. The function is increasing. y x y x

Spreadsheet

Linear equations Algebraic condition Geometrical derivation 1 * to 2 = -1 Lines are parallel Lines coincide Lines are perpendicular Lines intersect

The mathematical model I have built will help seventh grade students to independently explore the linear function and understand it better.

Class: 7

The function occupies one of the leading places in the school algebra course and has numerous applications in other sciences. At the beginning of the study, in order to motivate, update the issue, I inform you that not a single phenomenon, not a single process in nature can be studied, no machine can be designed, and then operate without a complete mathematical description. One tool for this is a function. Its study begins in the 7th grade, as a rule, children do not delve into the definition. Particularly hard-to-reach concepts are such as domain of definition and domain of value. Using the known connections between the quantities in the problems of movement, the costs are shifting them into the language of the function, keeping the connection with its definition. Thus, in students the concept of function is formed at a conscious level. At the same stage, painstaking work is carried out on new concepts: domain of definition, domain of value, argument, value of a function. I use advanced learning: I introduce the notation D(y), E(y), introduce the concept of the zero of a function (analytically and graphically), when solving exercises with areas of constant sign. The earlier and more often students encounter difficult concepts, the better they are realized at the level of long-term memory. When studying a linear function, it is advisable to show the connection with the solution of linear equations and systems, and later with the solution of linear inequalities and their systems. At the lecture, students receive a large block (module) of new information, so at the end of the lecture, the material is "wrung out" and a summary is drawn up that students should know. Practical skills are developed in the process of performing exercises using various methods based on individual and independent work.

1. Some information about the linear function.

Linear function is very common in practice. The rod length is a linear function of temperature. The length of rails, bridges is also a linear function of temperature. The distance traveled by a pedestrian, train, car at a constant speed is a linear function of the time of movement.

A linear function describes a number of physical dependencies and laws. Let's consider some of them.

1) l \u003d l o (1 + at) - linear expansion of solids.

2) v \u003d v o (1 + bt) - volumetric expansion of solids.

3) p=p o (1+at) - the dependence of the resistivity of solid conductors on temperature.

4) v \u003d v o + at - the speed of uniformly accelerated movement.

5) x= x o + vt is the coordinate of uniform motion.

Task 1. Define a linear function from tabular data:

X	1	3
at	-1	3

Solution. y \u003d kx + b, the problem is reduced to solving the system of equations: 1 \u003d k 1 + b and 3 \u003d k 3 + b

Answer: y \u003d 2x - 3.

Problem 2. Moving uniformly and rectilinearly, the body passed 14 m in the first 8s, and 12 m in another 4s. Compose an equation of motion based on these data.

Solution. According to the condition of the problem, we have two equations: 14 \u003d x o +8 v o and 26 \u003d x o +12 v o, solving the system of equations, we get v \u003d 3, x o \u003d -10.

Answer: x = -10 + 3t.

Problem 3. A car leaving the city moving at a speed of 80 km/h. After 1.5 hours, a motorcycle drove after him, the speed of which was 100 km/h. How long will it take for the bike to overtake him? How far from the city will this happen?

Answer: 7.5 hours, 600 km.

Task 4. The distance between two points at the initial moment is 300m. The points move towards each other with speeds of 1.5 m/s and 3.5 m/s. When will they meet? Where will it happen?

Answer: 60 s, 90 m.

Task 5. A copper ruler at 0 ° C has a length of 1 m. Find the increase in its length with an increase in its temperature by 35 o, by 1000 o C (the melting point of copper is 1083 o C)

Answer: 0.6mm.

2. Direct proportionality.

Many laws of physics are expressed through direct proportionality. In most cases, a model is used to write these laws.

in some cases -

Let's take a few examples.

1. S \u003d v t (v - const)

2. v = a t (a - const, a - acceleration).

3. F \u003d kx (Hooke's law: F - force, k - stiffness (const), x - elongation).

4. E = F/q (E is the strength at a given point of the electric field, E is const, F is the force acting on the charge, q is the magnitude of the charge).

As a mathematical model of direct proportionality, one can use the similarity of triangles or the proportionality of segments (Thales' theorem).

Task 1. The train passed a traffic light in 5 seconds, and past a platform 150 m long, in 15 seconds. What is the length of the train and its speed?

Solution. Let x be the length of the train, x+150 be the total length of the train and the platform. In this problem, the speed is constant, and the time is proportional to the length.

We have a proportion: (x + 150): 15 = x: 5.

Where x = 75, v = 15.

Answer. 75 m, 15 m/s.

Problem 2. The boat went downstream 90 km in some time. In the same time, he would have passed 70 km against the current. How far will the raft travel in this time?

Answer. 10 km.

Task 3. What was the initial temperature of the air if, when heated by 3 degrees, its volume increased by 1% of the original.

Answer. 300 K (Kelvin) or 27 0 C.

Lecture on the topic "Linear function".

Algebra, 7th grade

1. Consider examples of tasks using well-known formulas:

S = v t (path formula), (1)

C \u003d c c (cost formula). (2)

Problem 1. The car, having driven away from point A at a distance of 20 km, continued its journey at a speed of 62 km/h. How far from point A will the car be after t hours? Compose an expression for the problem, denoting the distance S, find it at t = 1h, 2.5h, 4h.

1) Using formula (1), we find the path traveled by a car at a speed of 62 km/h in time t, S 1 = 62t;
2) Then from point A in t hours the car will be at a distance S = S 1 + 20 or S = 62t + 20, find the value of S:

at t = 1, S = 62*1 + 20, S = 82;
at t = 2.5, S = 62 * 2.5 + 20, S = 175;
at t = 4, S = 62*4+ 20, S = 268.

We note that when finding S, only the value of t and S changes, i.e. t and S are variables, and S depends on t, each value of t corresponds to a single value of S. Denoting the variable S for Y, and t for x, we get a formula for solving this problem:

Y= 62x + 20. (3)

Problem 2. A textbook was bought in a store for 150 rubles and 15 notebooks for n rubles each. How much did you pay for the purchase? Make an expression for the problem, denoting the cost C, find it for n = 5,8,16.

1) Using formula (2), we find the cost of notebooks С 1 = 15n;
2) Then the cost of the entire purchase is С= С1 +150 or С= 15n+150, we find the value of C:

at n = 5, C = 15 5 + 150, C = 225;
at n = 8, C = 15 8 + 150, C = 270;
at n = 16, C = 15 16+ 150, C = 390.

Similarly, we notice that C and n are variables, for each value of n there corresponds a single value of C. Denoting the variable C for Y, and n for x, we get the formula for solving Problem 2:

Y= 15x + 150. (4)

Comparing formulas (3) and (4), we make sure that the variable Y is found through the variable x according to one algorithm. We considered only two different problems that describe the phenomena around us every day. In fact, there are many processes that change according to the obtained laws, so such a relationship between variables deserves to be studied.

Problem solutions show that the values of the variable x are chosen arbitrarily, satisfying the conditions of the problems (positive in problem 1 and natural in problem 2), i.e. x is an independent variable (it is called an argument), and Y is a dependent variable and there is a one-to-one correspondence between them , and by definition such a dependence is a function. Therefore, denoting the coefficient at x by the letter k, and the free term by the letter b, we obtain the formula

Y= kx + b.

Definition.View function y= kx + b, where k, b are some numbers, x is an argument, y is the value of the function, is called a linear function.

To study the properties of a linear function, we introduce definitions.

Definition 1. The set of admissible values of an independent variable is called the domain of definition of the function (admissible - this means those numerical values x for which y is calculated) and is denoted by D (y).

Definition 2. The set of values of the dependent variable is called the range of the function (these are the numerical values that y takes) and is denoted by E(y).

Definition 3. The graph of a function is a set of points of the coordinate plane, the coordinates of which turn the formula into a true equality.

Definition 4. The coefficient k at x is called the slope.

Consider the properties of a linear function.

1. D(y) - all numbers (multiplication is defined on the set of all numbers).
2. E(y) - all numbers.
3. If y \u003d 0, then x \u003d -b / k, the point (-b / k; 0) - the point of intersection with the Ox axis, is called the zero of the function.
4. If x= 0, then y= b, the point (0; b) is the point of intersection with the Oy axis.
5. Find out in which line the linear function will line up the points on the coordinate plane, i.e. which is the graph of the function. To do this, consider the functions

1) y= 2x + 3, 2) y= -3x - 2.

For each function we will make a table of values. Let's set arbitrary values for the variable x, and calculate the corresponding values for the variable Y.

X	-1,5	-2	0	1	2
Y	0	-1	3	5	7

Having built the resulting pairs (x; y) on the coordinate plane and connecting them for each function separately (we took the values of x with a step of 1, if you reduce the step, then the points will line up more often, and if the step is close to zero, then the points will merge into a solid line ), we notice that the points line up in a straight line in case 1) and in case 2). Due to the fact that the functions are chosen arbitrarily (build your own graphs y= 0.5x - 4, y= x + 5), we conclude that that the graph of a linear function is a straight line. Using the property of a straight line: a single straight line passes through two points, it is enough to take two points to construct a straight line.

6. It is known from geometry that lines can either intersect or be parallel. We investigate the relative position of the graphs of several functions.

1) y= -x + 5, y= -x + 3, y= -x - 4; 2) y= 2x + 2, y= x + 2, y= -0.5x + 2.

Let's build groups of graphs 1) and 2) and draw conclusions.

Graphs of functions 1) are located in parallel, examining the formulas, we notice that all functions have the same coefficients at x.

Function graphs 2) intersect at one point (0;2). Examining the formulas, we notice that the coefficients are different, and the number b = 2.

In addition, it is easy to see that the lines given by linear functions with k › 0 form an acute angle with the positive direction of the Ox axis, and an obtuse angle with k ‹ 0. Therefore, the coefficient k is called the slope coefficient.

7. Consider special cases of a linear function, depending on the coefficients.

1) If b=0, then the function takes the form y= kx, then k = y/x (the ratio shows how many times it differs or what part is y from x).

A function of the form Y= kx is called direct proportionality. This function has all the properties of a linear function, its feature is that when x=0 y=0. The graph of direct proportionality passes through the origin point (0; 0).

2) If k = 0, then the function takes the form y = b, which means that for any values of x, the function takes the same value.

A function of the form y = b is called a constant. The graph of the function is a straight line passing through the point (0;b) parallel to the Ox axis, with b=0 the graph of the constant function coincides with the abscissa axis.

Abstract

1. Definition A function of the form Y= kx + b, where k, b are some numbers, x is an argument, Y is the value of the function, is called a linear function.

D(y) - all numbers.

E(y) - all numbers.

The graph of a linear function is a straight line passing through the point (0;b).

2. If b=0, then the function takes the form y= kx, called direct proportionality. The direct proportionality graph passes through the origin.

3. If k = 0, then the function takes the form y= b, is called a constant. The graph of the constant function passes through the point (0;b), parallel to the x-axis.

4. Mutual arrangement of graphs of linear functions.

The functions y= k 1 x + b 1 and y= k 2 x + b 2 are given.

If k 1 = k 2, then the graphs are parallel;

If k 1 and k 2 are not equal, then the graphs intersect.

5. See examples of graphs of linear functions above.

Literature.

Textbook Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov and others. "Algebra, 8".
Didactic materials on algebra for grade 8 / V.I. Zhokhov, Yu.N. Makarychev, N.G. Mindyuk. - M .: Education, 2006. - 144 p.
Supplement to the newspaper September 1 "Mathematics", 2001, No. 2, No. 4.

Summarize and systematize knowledge on the topic “Linear function”:

consolidate the ability to read and build graphs of functions given by the formulas y = kx + b, y = kx;
consolidate the ability to determine the relative position of graphs of linear functions;
develop skills in working with graphs of linear functions.

Develop ability to analyze, compare, draw conclusions. Development of cognitive interest in mathematics, competent oral mathematical speech, accuracy and accuracy in construction.

Upbringing attentiveness, independence in work, ability to work in pairs.

Equipment: ruler, pencil, task cards, colored pencils.

Type of lesson: a lesson to consolidate the studied material.

Lesson plan:

Organizing time.
oral work. Mathematical dictation with self-examination and self-assessment. Historical excursion.
Training exercises.
Independent work.
Summary of the lesson.
Homework.

During the classes

1. Communication of the purpose of the lesson.

The purpose of the lesson is to generalize and systematize knowledge on the topic “Linear function”.

2. Let's start by testing your theoretical knowledge.

- Define the function. What is an independent variable? Dependent variable?

- Define the graph of a function.

– Formulate the definition of a linear function.

What is the graph of a linear function?

How to plot a linear function?

- Formulate the definition of direct proportionality. What is a graph? How to build a graph? How is the graph of the function y = kx located in the coordinate plane for k > 0 and for k< 0?

Mathematical dictation with self-examination and self-assessment.

Look at the pictures and answer the questions.

1) The graph of which function is superfluous?

2) Which figure shows a graph of direct proportionality?

3) In which figure does the graph of a linear function have a negative slope?

4) Determine the sign of the number b. (Write the answer as an inequality)

Checking work. Evaluation.

Work in pairs.

Decipher the name of the mathematician who first used the term function. To do this, in the boxes, enter the letter corresponding to the graph of the given function. In the remaining square, enter the letter C. Complete the drawing with a graph of the function corresponding to this letter.

Picture 1

Figure 2

Figure 3

Gottfried Wilhelm Leibniz, 1646-1716, German philosopher, mathematician, physicist and linguist. He and the English scientist I. Newton created (independently of each other) the foundations of an important branch of mathematics - mathematical analysis. Leibniz introduced many concepts and symbols used in mathematics today.

3. 1. Given the functions given by the formulas: y = x-5; y=0.5x; y = – 2x; y=4.

Name the functions. Indicate the graphs of which of these functions will pass through the point M (8; 4). Schematically show what the drawing will be like if it depicts graphs of functions passing through point M.

2. The graph of direct proportionality passes through point C (2; 1). Write a formula for direct proportionality. At what value of m will the graph pass through point B (-4;m).

3. Plot the function given by the formula y=1/2X. How can you get a graph of the function given by the formula y=1/2X – 4 and y = 1/2X+3 from the graph of this function. Analyze the resulting graphs.

4. Functions are given by formulas:

1) y \u003d 4x + 9 and y \u003d 6x-5;
2) y=1/2x-3 and y=0.5x+2;
3) y \u003d x and y \u003d -5x + 2.4;
4) y= 3x+6 and y= -2.5x+6.

What is the relative position of the function graphs? Without constructing, find the coordinates of the intersection point of the first pair of graphs. (Self test)

4. Independent work in pairs. (perform on ml. paper). Intersubject communication.

It is necessary to build graphs of functions and select that part of it, for the points of which the corresponding inequality is true:

y \u003d x + 6, 4 < X < 6;
y \u003d -x + 6, -6 < X < -4;
y \u003d - 1/3 x + 10, -6 < X < -3;
y \u003d 1/3 x +10, 3 < X < 6;
y \u003d -x + 14, 0 < X < 3;
y \u003d x + 14, -3 < X < 0;
y \u003d 9x - 18, 2 < X < 4;
y \u003d - 9x - 18 -4 < X < -2;
y = 0, -2 < X < 2.

What drawing did you get? ( Tulip.)

A little about tulips:

About 120 species of tulips are known, distributed mainly in Central, Eastern and Southern Asia and Southern Europe. Botanists believe that the culture of tulips originated in Turkey in the 12th century. The plant gained world fame far from its homeland, in Holland, rightfully called the Land of Tulips.

Here is the legend of the tulip. Happiness was contained in the golden bud of a yellow tulip. No one could reach this happiness, because there was no such force that could open its bud. But one day a woman with a child was walking through the meadow. The boy escaped from his mother's arms, ran up to the flower with a sonorous laugh, and the golden bud opened. Carefree childish laughter did what no power could do. Since then, it has become customary to give tulips only to those who experience happiness.

Creative homework. Create a drawing in a rectangular coordinate system, consisting of segments and make its analytical model.

6. Independent work. Differentiated task (in two versions)

I option:

Draw schematic diagrams of functions:

II option:

Draw schematically the graphs of functions for which the conditions are met:

7. Summary of the lesson

Analysis of the work done. Grading.

y \u003d x + 6,	4 < X < 6;
y \u003d -x + 6,	-6 < X < -4;
y \u003d - 1/3 x + 10,	-6 < X < -3;
y \u003d 1/3 x +10,	3 < X < 6;
y \u003d -x + 14,	0 < X < 3;
y \u003d x + 14,	-3 < X < 0;
y \u003d 9x - 18,	2 < X < 4;
y \u003d - 9x - 18	-4 < X < -2;
y = 0,	-2 < X < 2.