Linear function and its graph presentation. Linear function and its graph Linear function its properties and graph presentation

Ladders and railings 12.08.2021

Ladders and railings

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Algebra lesson in grade 7 "Linear function and its graph" Prepared by U. V. Tatchin teacher of mathematics MBOU secondary school №3 city of Surgut

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Purpose: the formation of the concept of "linear function", the skill of constructing its graph according to the algorithm Objectives: Educational: - to study the definition of a linear function, - to introduce and study the algorithm for constructing a graph of a linear function, - to work out the skill of recognizing a linear function according to a given formula, graph, verbal description. Developing: - to develop visual memory, mathematically literate speech, accuracy, accuracy in construction, the ability to analyze. Educational: - to bring up a responsible attitude to educational work, accuracy, discipline, perseverance. - to form the skills of self-control and mutual control

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Lesson plan: I. Organizational moment II. Basic knowledge updating III. Study of a new topic IV. Reinforcement: oral exercises, tasks for building graphs V. Solving entertaining tasks VI. Summing up the lesson, recording homework VII. Reflection

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I. Organizational moment Having guessed the words horizontally, you will recognize the keyword 1. The exact set of instructions describing the order of the executor's actions to achieve the result of solving the problem in a finite time 2. One of the coordinates of a point 3. The dependence of one variable on another, at which each value of the argument corresponds to the only value of the dependent variable 4. The French mathematician, who introduced a rectangular coordinate system 5. Angle, the degree measure of which is greater than 900, but less than 1800 6. Independent variable 7. The set of all points of the coordinate plane, the abscissas of which are equal to the values of the argument, and the ordinates - the corresponding values of function 8. The road we choose T G R A F I K P R Y M A Z

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1. An exact set of instructions describing the order of the executor's actions to achieve the result of solving the problem in a finite time 2. One of the coordinates of a point 3. The dependence of one variable on another, in which each value of the argument corresponds to a single value of the dependent variable 4. The French mathematician who introduced the rectangular coordinate system 5. Angle, the degree measure of which is more than 900, but less than 1800 6. Independent variable 7. The set of all points of the coordinate plane, the abscissas of which are equal to the values of the argument, and the ordinates are equal to the corresponding values of the function 8. The road we choose ALG O R I T M A B S C I S S A V U N K C I D E K A R T T U P O J A R G U M E N T G R A F I K P R Y M A Z

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II. Basic knowledge actualization Many real situations are described by mathematical models that are linear functions. Let's give an example. The tourist traveled by bus 15 km from point A to point B, and then continued to move from point B in the same direction to point C, but on foot, at a speed of 4 km / h. At what distance from point A will the tourist be after 2 hours, after 4 hours, after 5 hours of walking? The mathematical model of the situation is the expression y = 15 + 4x, where x is the walking time in hours, y is the distance from A (in kilometers). Using this model, we answer the question of the problem: if x = 2, then y = 15 + 4 ∙ 2 = 23 if x = 4, then y = 15 + 4 ∙ 4 = 31 if x = 6, then y = 15 + 4 ∙ 6 = 39 Mathematical model y = 15 + 4x is a linear function. A B C

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III. Learning a new topic. An equation of the form y = k x + m, where k and m are numbers (coefficients) is called a linear function. To plot a linear function, it is necessary, by specifying a specific value for x, to calculate the corresponding value for y. Usually these results are presented in the form of a table. They say that x is the independent variable (or argument), y is the dependent variable. 2 1 1 2 x x x y y x

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Algorithm for constructing a graph of a linear function 1) Make a table for a linear function (for each value of an independent variable, assign the value of a dependent variable) 2) Plot points on the coordinate plane xOy 3) Draw a straight line through them - a graph of a linear function Theorem Graph of a linear function y = kx + m is a straight line.

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Consider the application of the algorithm for plotting a linear function graph Example 1 Plot a linear function y = 2x + 3 1) Create a table 2) Plot points (0; 3) and (1; 5) in the coordinate plane xОy 3) Draw a straight line through them

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If the linear function y = k x + m is considered not for all values of x, but only for values of x from some numerical set X, then they write: y = k x + m, where x X (is the sign of membership) Let's return to the problem In our situation, the independent the variable can take any non-negative value, but in practice a tourist cannot walk at a constant speed without sleep and rest for as long as he wants. This means that it was necessary to make reasonable restrictions on x, say, a tourist walks no more than 6 hours. Now we write down a more accurate mathematical model: y = 15 + 4x, x 0; 6

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Consider the following example Example 2 Build a graph of a linear function a) y = -2x + 1, -3; 2; b) y = -2x + 1, (-3; 2) 1) Let's compose a table for the linear function y = -2x + 1 2) Construct on the coordinate plane xOy points (-3; 7) and (2; -3) and let's draw a straight line through them. This is the graph of the equation y = -2x + 1. Next, select the segment connecting the constructed points. x -3 2 y 7 -3

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We are building a graph of the function y = -2x + 1, (-3; 2) What is the difference between this example and the previous one?

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IV. Consolidation of the studied topic Choose which function is a linear function

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Perform the following task Linear function is given by the formula y = -3x - 5. Find its value at x = 23, x = -5, x = 0

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Checking the solution If x = 23, then y = -3 23 - 5 = -69 - 5 = -74 If x = -5, then y = -3 (-5) - 5 = 15– 5 = 10 If x = 0 , then y = -3 0– 5 = 0 - 5 = -5

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Find the value of the argument that makes the linear function y = -2x + 2.4 equal to 20.4? Checking the solution For x = -9, the value of the function is 20.4 20.4 = - 2x + 2.4 2x = 2.4 - 20.4 2x = -18 x = -18: 2 x = -9

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Next task Without performing construction, answer the question: to which function graph does A (1; 0) belong?

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Name the coordinates of the points of intersection of the graph of this function with the coordinate axes With the OX axis: (-3; 0) Check yourself: With the OY axis: (0; 3)

Lesson objectives: to formulate the definition of a linear function, an idea of its graph; identify the role of the parameters b and k in the location of the graph of the linear function; form the ability to build a graph of a linear function; develop the ability to analyze, generalize, draw conclusions; develop logical thinking; development of skills of independent activity

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Participants: 8th grade of a special school (or 7th grade of a comprehensive school).

Lesson time: 1 academic hour (35 minutes).

Lesson objectives:

To consolidate knowledge and skills on the topic "Function y = kx";
Learn to build a graph of a linear function;
Develop the desire for independent research activities;
Continue to develop the ability to work with drawing tools (ruler).

Lesson Objectives:

Carry out a comparative analysis of the functions y = kx and y = kx + b;
To acquaint students with the concept of "Linear function" and its graph;

Equipment for the lesson:

The textbook by Sh.A. Alimov "Algebra 7";
Presentation on the topic "Linear function and its graph";
Computer;
Touch screen;
Cards with images of graphs of functions y = 2x and y = - 2x ( Annex 1);
Cards with tasks for plotting a linear function ( Appendix 2);
Card "Rectangular coordinate system" ( Appendix 3);
Cards for the research paper "Similarities and Differences" ( Appendix 4);
Card "Definition of a linear function" ( Appendix 5).

Lesson plan:

Organizational moment - 2 minutes;
Knowledge update - 5 min;
Explanation of new material - 15 min;
Problem solving - 10 minutes;
Summing up the lesson - 2 min;
Homework - 1 min.

During the classes

I. Organizational moment

Verification of compliance with the orthopedic regime of students; recording the date of the lesson, the topic of the lesson; familiarizing students with the goals and objectives of the lesson.

II. Knowledge update

Exercise 1: graph the function y = 2x.

To complete the assignment, students with a severe degree of damage to the musculoskeletal system should be given a card "Rectangular coordinate system".

If students are not doing well on the assignment, review the assignment with the students.

Analysis of the task:

This function refers to the function y = kx. Which object is the graph of this function?
How many points can you draw a straight line through?
This means that in order to plot the function y = 2x, it is necessary to plot two points in the coordinate system that belong to this function. How to find the coordinates of a point that belongs to the graph by a given function formula?

After the analysis, the students independently build the graph.

Assignment 2: Consider the properties of the constructed function.

Is this function increasing or decreasing?
What are the values of x at which the function is positive.
What are the values of x at which the function is negative.

So, we repeated the construction of the graph of the function y = kx and its properties. Today we will get acquainted with another kind of function, which is related to the function y = kx. We will carry out a comparative analysis of the two functions to clarify their relationship. If someone first sees the similarities and differences, draws conclusions, write them down on the card (issue a card "Similarities and differences").

III. Explanation of the new material

A linear function is a function of the form y = kx + b, where k and b are given numbers. (slide 2)

Assignment 3: Functions are written on the board. Name the coefficients k and b in the linear functions indicated on the board (Figure 1):

Assignment 4: Verbally complete 579 on page 140. Students take turns calling a function and giving a detailed answer to a question.

y = -x-2 - is a linear function. The coefficient in front of x is -2, the intercept is -2.
y = 2x2 + 3 - is not a linear function, since x is in the second power.
y = x / 3- is a linear function, since the coefficient in front of x is 1/3, the free term is 0. Teacher's help in case of difficulty: by what number is the independent variable x multiplied if x / 3 = x * 1/3 ? What is the free term if it is absent from the record?
y = 250 - is a linear function, since the coefficient in front of x is 0, the free term is 250. Teacher's help in case of difficulty: what number can be multiplied by the independent variable x if the product kx is absent?
y = 3 / x + 8 - is not a linear function, since division by x is performed, not multiplication. Teacher help in case of difficulty: When multiplying a fraction by a number, is this number multiplied by the numerator or denominator?
y = -x / 5 + 1 - is a linear function, since the coefficient in front of x is 1/5, the free term is 1. Teacher help in case of difficulty: When multiplying a fraction by a number, is this number multiplied by the numerator or denominator?

Let's continue our study of the linear function.

Let us show that the graph of a linear function, as well as the graph of the function y = kx, is a straight line. To do this, we define a linear function, for example, y = x + 1, in the form of a table for a certain number of points.

So, the function is given by the formula y = x + 1. What are the coefficient k and the free term b of this function? Which variable is independent?

We will take arbitrary values of the independent variable x located close to each other on the coordinate axis:

x	-2,5	-2	-1,5	-1	-0,5	0	0,5	1	1,5	2	2,5
y	-1,5	-1	-0,5	0	0,5	1	1,5	2	2,5	3	3,5

Let's build the found points in the coordinate system (click with the mouse to display the coordinate system). We mark the points found by us (click with the mouse to build the found points). Let's connect the constructed points (click with the mouse to draw a straight line). It really turns out to be a straight line. If necessary, you can further select the values of the independent variable to obtain a more accurate plot.

So, the graph of a linear function is a straight line (slide 3).

How many points is it enough to build so that a straight line can be drawn unambiguously through them?

So, to plot a graph of a linear function, it is enough (click with the mouse for the algorithm to appear):

choose two convenient values for the independent variable x;
find the value of the function from the selected values of x;
Mark the found points on the coordinate plane;
Draw a straight line through the constructed points.

Assignment 5: in the rectangular coordinate system built for task 1, plot the function: y = 2x + 5, y = 2x + 3, y = 2x-4, y = 2x-2, y = 2x + 1. Issue cards with assignments to students (Appendix 3). Each student builds one of the functions (at the discretion of the teacher). When building the graph, try to answer the questions on the "Similarities and Differences" card yourself.

Let's check the graphs of functions that you have built (slide 4). Students name their chosen points first.

We build a graph of the function y = 2x + 5 (click with the mouse): take convenient points (-2; 1) and (0; 5), draw a straight line through them (click with the mouse).

We build a graph of the function y = 2x + 3 (click with the mouse): take convenient points (0; 3) and (1; 5), draw a straight line through them (click with the mouse).

We build a graph of the function y = 2x + 1 (click with the mouse): take convenient points (0; 1) and (1; 3), draw a straight line through them (click with the mouse).

We build a graph of the function y = 2x-2 (click with the mouse): take convenient points (0; -2) and (1; 0), draw a straight line through them (click with the mouse).

We build a graph of the function y = 2x-4 (click with the mouse): take convenient points (0; -4) and (2; 0), draw a straight line through them (click with the mouse).

Earlier, you plotted the function y = 2x (click). Now each of you has built one more graph y = 2x + 5, y = 2x + 3, y = 2x-4, y = 2x-2, y = 2x + 1.

The last opportunity to independently fill in the cards "Similarities and differences".

What is common between the formulas of the linear functions you have constructed? After receiving the answer, click the mouse.

How did the similarity appear on their charts? After receiving the answer, click the mouse.

Why did it happen? What is the coefficient k responsible for?

Each of the constructed functions has k = 2, therefore, the angles between the graphs and the Ox axis are equal, which means that the straight lines are parallel (click the mouse).

What is the difference between the formulas of the constructed linear functions? After receiving the answer, click the mouse.

How did the difference appear on their graphs? After receiving the answer, click the mouse to show the coefficient b of each function and display it on the graph.

What do you think the free term b is responsible for?

What conclusion can you draw? How are the graphs of the functions y = kx and y = kx + b related?

the graph of the function y = kx + b is obtained by shifting the graph of the function y = kx by b units along the ordinate (slide 5);
the graphs of functions with the same values of the coefficient k are parallel straight lines.

Consider other examples:

The graphs of the functions y = -1 / 2x + 1 and y = -1 / 2x (click with the mouse) are parallel. One of the other is obtained by shifting one unit along the Oy axis.
The graphs of the functions y = 3x-5 and y = 3x (click with the mouse) are parallel. One from the other is obtained by shifting five units along the Oy axis.
The graphs of the functions y = -3 / 7x-3 and y = -3 / 7x (click with the mouse) are parallel. One of the other is obtained by shifting three units along the Oy axis.

After summing up the comparison, fill in the "Similarities and Differences" cards. Provide individual assistance to students if necessary.

IV. Solving problems

Assignment 6: Build a rectangular coordinate system with a unit line equal to two cells in the notebook. In the coordinate system, plot the graphs of the functions indicated in 581. For students with severe musculoskeletal disorders, issue a ready-made coordinate system.

V. Lesson summary

What function did you meet today? After receiving the answer, click the mouse and speak the definition of the linear function again.

Which object is a linear function graph? After receiving the answer, click the mouse and speak again the method of plotting the linear function.

How are the graphs of the functions y = kx + b and y = kx related? After receiving the answer, click the mouse and speak again the similarities and differences of the functions y = kx and y = kx + b.

Vi. Homework

Know the definition of a linear function, 582 - to plot a linear function and to determine the values of the variables x and y from the graph, 589 (orally) - give a complete answer to the question (with an explanation).

Thank you for the lesson(slide 7) !

Full name of the educational institution:

Municipal educational institution secondary school No. 3 of the village of Kochubeevskoe, Stavropol Territory

Subject area: mathematics

Lesson title: "Linear function, its schedule, properties ”.

Age group: Grade 7

Presentation title:"Linear function, its graph, properties."

Number of slides: 37

Environment (editor) in which the presentation was made: Power Point 2010

This presentation

1 slide - title

2 slide - basic knowledge actualization: definition of a linear equation, orally from the proposed to choose those that are linear.

3 slide - definition of a linear function.

4 slide recognition of the linear function of the proposed.

5 slide output.

6 slide ways to set the function.

7 slide I give an example, I show.

8 slide - I give an example, I show.

9 slide task for students.

10 slide - checking the correctness of the task. I draw the attention of the students to the relationship between the coefficients k and b and the location of the graphs.

11 slide output.

12 slide - work with a graph of a linear function.

13 Slide Tasks for self-solution:build graphs of functions (perform in a notebook).

Slides 14-17 - show the correct performance of the task.

18-27 slides - tasks of oral and written nature. I do not choose all assignments, but only those that are suitable for the level of preparedness of the classupon availability of time.

28 slide assignment for strong learners.

29 slides - to summarize.

30-31 slides - conclusions.

32-36 slides - historical background. (If there is time)

37 slide-Used literature

List of used literature and Internet resources:

1.Mordkovich A.G. and others. Algebra: a textbook for the 7th grade of educational institutions - M.: Education, 2010.

2.Zvavich L.I. and other Didactic materials on algebra for grade 7 - M.: Education, 2010.

3.Algebra grade 7, edited by Yu.N. Makarychev et al., Education, 2010

4. Internet resources:www.symbolsbook.ru/Article.aspx%...id%3D222

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Slide captions:

Linear function, its graph, properties. Kiryanova Marina Vladimirovna, teacher of mathematics, secondary school №3, p. Kochubeevskoe, Stavropol Territory

Specify linear equations: 1) 5y = x 2) 3y = 0 3) y 2 + 16x 2 = 0 4) + y = 4 5) x + y = 4 6) y = -x + 11 7) + 0.5x - 2 = 0 8) 25d - 2m + 1 = 0 9) y = 3 - 2x 5

A function of the form у = kx + b is called linear. The graph of a function of the form у = kx + b is a straight line. To draw a straight line, only two points are needed, since the only straight line passes through two points.

Find the equations of linear functions y = -x + 0.2; y = 1 2, 4x-5.7; y = - 9 x - 1 8; y = 5.04x; y = - 5.04x; y = 1 26.35 + 8.75x; y = x -0, 2; y = x: 8; y = 0.005x; y = 13 3, 13 3 13 3 x; y = 3 - 1.01x; y = 2: x; y = -0.004 9; y = x: 6 2.

y = kx + b - linear function х - argument (independent variable) y - function (dependent variable) k, b - numbers (coefficients) to ≠ 0

x X 1 X 2 X 3 y Y 1 Y 2 Y 3

y = - 2x + 3 is a linear function. The graph of a linear function is a straight line; to build a straight line, you need to have two points x - an independent variable, so we will choose its values ourselves; Y is a dependent variable, its value will be obtained as a result of substitution of the selected value of x in the function. We write the results in the table: x y 0 2 If x = 0, then y = - 2 · 0 + 3 = 3. 3 If x = 2, then y = -2 2 + 3 = - 4 + 3 = -1. - 1 Mark the points (0; 3) and (2; -1) on the coordinate plane and draw a straight line through them. x y 0 1 1 Y = - 2x + 3 3 2 - 1 we choose

Construct a graph of a linear function y = - 2 x +3 Let's compose a table: x y 03 1 1 Let's construct points (0; 3) and (1; 5) on the coordinate plane and draw a straight line x 1 0 1 3 y through them

Option I Option II y = x-4 y = - x + 4 Determine the relationship between the coefficients k and b and the location of the straight lines Construct a graph of a linear function

y = x-4 y = -x + 4 Option I Option II x y 1 2 0 -4 x 1 2 0 4 y

x 0 y y = kx + m (k> 0) x 0 y y = kx + m (k 0, then the linear function y = kx + b increases if k

Using the graph of the linear function y = 2x - 6, answer the questions: a) at what value of x will y = 0? b) at what values of x will y  0? c) at what values of x will y  0? 1 0 3 y 1 x -6 a) y = 0 for x = 3 b) y  0 for x  3 If x  3, then the line is located above the x-axis, which means that the ordinates of the corresponding points of the line are positive c) y  0 for x  3 If x  3, then the line is located below the x-axis, which means that the ordinates of the corresponding points of the line are negative

Tasks for independent solution: build graphs of functions (perform in a notebook) 1.y = 2x - 2 2.y = x + 2 3.y = 4 - x 4. y = 1 - 3x Please note: the points chosen by you to build a straight line may be different, but the location of the graphs must necessarily coincide

Answer to task 1

Answer to task 2

Answer to task 3

Answer to task 4

Which figure shows the graph of the linear function y = kx? The answer is to explain. 1 2 3 4 5 x y x y x y x y x y

The student made a mistake while plotting the function. Which picture? 1.y = x + 2 2.y = 1.5x 3.y = -x-1 x y 2 1 x y 3 1 x y 3 3

1 2 3 4 5 x y x y y x y x y In which figure is the coefficient k negative? x

What is the sign of the coefficient k for each of the linear functions:

In which figure is the free term b in the equation of a linear function negative? 1 2 3 4 5 x y x y x y x y x y

Select the linear function, the graph of which is shown in the figure y = x - 2 y = x + 2 y = 2 - x y = x - 1 y = - x + 1 y = - x - 1 y = 0.5x y = x + 2 y = 2x Well done! Think!

xy 1 2 0 1 2 3 -1 -2 -1 -2 xy 1 2 0 1 2 3 -1 -2 -1 -2 y = 2x y = 2x + 1 y = 2x- 1 y = -2x + 1 y = - 2x- 1 y = -2x

y = -0.5x + 2, y = -0.5x, y = -0.5x- 2 xy 1 2 0 1 2 3 -1 -2 -1 -2 3 4 5 6 -3 xy 1 2 0 2 3 -1 -2 -1 -2 3 4 5 6 -3 1 y = 0.5x + 2 y = 0.5x- 2 y = 0.5x y = -0.5x + 2 y = -0.5x y = -0 , 5x- 2

y = x + 1 y = x- 1, y = xy 1 2 0 1 2 3 -1 -2 -1 -2 3 4 5 6 -3 xy 1 2 0 1 2 3 -1 -2 -1 -2 3 4 5 6 -3 xy = -xy = -x + 3 y = -x- 3 y = x + 1 y = x- 1 y = x

Draw up the equation of a linear function according to the following conditions:

summarize

Write down the conclusions in a notebook We learned: * A function of the form y = kx + b is called linear. * The graph of a function of the form y = kx + b is a straight line. * Only two points are needed to draw a straight line, as there is only one straight line passing through two points. * The coefficient k shows the increase or decrease of the straight line. * Coefficient b shows at what point the line intersects the OY axis. * Condition of parallelism of two lines.

Wish you luck!

Algebra - this word comes from the title of the work of Muhammad Al-Khwarizmi "Al-djebr and Al-muqabala", in which algebra was presented as an independent subject

Robert Record is an English mathematician who in 1556. introduced an equal sign and explained his choice by the fact that nothing could be more equal than two parallel segments.

Gottfried Leibniz - German mathematician (1646 - 1716), who was the first to introduce the term "abscissa" - in 1695, "ordinate" - in 1684, "coordinates" - in 1692.

René Descartes - French philosopher and mathematician (1596-1650), who was the first to introduce the concept of "function"

References 1.Mordkovich A.G. and others. Algebra: a textbook for the 7th grade of educational institutions - M .: Education, 2010. 2.Zvavich L.I. and other Didactic materials on algebra for grade 7 - M .: Education, 2010. 3.Algebra grade 7, edited by Yu.N. Makarychev et al., Education, 2010 4. Internet resources: www.symbolsbook.ru/Article.aspx% ... id% 3D222

Deputy Director for OIA,

mathematic teacher

MOU "Secondary School No. 65 named after B.P.Agapitov UIPMETS "

the city of Magnitogorsk

y = kx + b

The graph of the equation y = kx + b is a straight line. When b = 0, the equation takes the form y = kx, its graph passes through the origin.

1.y = 3x-7 and y = -6x + 2

3 is not equal to –6, then the graphs overlap.

2. Solve the equation:

3x-7 = -6x + 2

1-abscissa of the intersection point.

3. Find the ordinate:

Y = 3x-7 = -6x + 2 = 3-7 = -4

-4-ordinate of intersection point

4. А (1; -4) coordinates of the intersection point.

The geometric meaning of the coefficient k

The angle of inclination of the straight line to the X axis depends on the values of k.

Y = 0.5x + 3

Y = 0.5x-3.3

As / k / increases, the angle of inclination to the X-axis of the straight lines increases.

k are equal to 0.5 and the angle of inclination to the X-axis is the same for straight lines

The coefficient k is called the slope

From value b the ordinate of the point of intersection with the axis depends Y .

b = 4, (0.4) - dot

Y-Intersections

b = -3, (0, -3) - Y-intercept

1. The functions are given by the formulas: Y = X-4, Y = 2x-3,

Y = -x-4, Y = 2x, Y = x-0.5 ... Find pairs of parallel lines. Answers:

but) y = x- 4 and y = 2x b) y = x-4 and y = x-0.5

in) y = -x-4 and y = x-0.5 G) y = 2x and y = 2x-3