Presentation on the topic linear function. Presentation “Linear function, its graph, properties”

Deputy Director for Water Resources Management,

math teacher

Municipal educational institution "Secondary school No. 65 named after. B.P.Agapitova UIPMEC"

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 intersect.

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 the intersection point

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

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 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 depends on the ordinate of the point of intersection with the axis Y .

b=4,(0,4)- dot

Y-Axis Intersections

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

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:

A) y=x- 4 And y=2x b) y=x-4 And y=x-0.5

V) y=-x-4 And y=x-0.5 G) y=2x And y=2x-3

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Participants: 8th grade correctional school(or 7th grade of secondary school).

Lesson time: 1 academic hour (35 minutes).

Lesson Objectives:

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

Lesson Objectives:

Carry out comparative analysis functions y=kx and y=kx+b;
Introduce students to the concept of " Linear function"and its schedule;

Equipment for the lesson:

Textbook Sh.A. Alimova “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 ( Appendix 1);
Cards with tasks for constructing a graph of a linear function ( appendix 2);
Card “Rectangular coordinate system” ( Appendix 3);
Cards for research work"Similarities and Differences" ( Appendix 4);
Card “Definition of a linear function” ( Appendix 5).

Lesson Plan:

Organizational moment– 2 min;
Updating knowledge – 5 min;
Explanation of new material – 15 min;
Problem solving – 10 min;
Summing up the lesson – 2 min;
Homework – 1 min.

Lesson progress

I. Organizational moment

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

II. Updating knowledge

Task 1: graph the function y=2x.

To complete the task, students with severe damage to the musculoskeletal system are given the “Rectangular Coordinate System” card.

If students do not cope with the task, analyze the task together with the students.

Job Analysis:

This function belongs to the function y=kx. What object is the graph of this function?
Through how many points can a straight line be drawn unambiguously?
This means that in order to construct a graph of the function y=2x, it is necessary to construct two points in the coordinate system that belong to this function. How to find the coordinates of a point that belongs to the graph of a function given by the formula?

After the analysis, students independently construct a graph.

Task 2: Let's consider the properties of the constructed function.

Is this function increasing or decreasing?
Name the values of x for which the function is positive.
Name the values of x for which the function is negative.

So, we repeated the plotting of the function y=kx and its properties. Today we will get acquainted with another type of function, which is related to the function y=kx. We will conduct a comparative analysis of the two functions to clarify their relationship. If someone is the first to see similarities and differences and draw conclusions, write them down on a card (give out a “Similarities and Differences” card).

III. Explanation of new material

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

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

Task 4: Orally complete 579 on page 140. Students take turns naming the function and giving a detailed answer to the question.

y=-x-2 – is a linear function. The coefficient before x is -2, the free term is -2.
y=2x2+3 – is not a linear function, since x is to the second power.
y=x/3- is a linear function, since the coefficient of x is 1/3, the free term is 0. Help from the teacher in case of difficulty: what number is the independent variable x multiplied by, if written x/3=x*1/3 ? What is the value of the free term if it is not in the record?
y=250 is a linear function, since the coefficient of x is 0, the free term is 250. Teacher help in case of difficulty: by what number can the independent variable x be multiplied if the product kx is missing?
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 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 studying the linear function.

Let us show that the graph of a linear function, just like 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 the independent one?

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 plot the found points in the coordinate system (click the mouse to display the coordinate system). We mark the points we found (click the mouse to plot the found points). Connect the constructed points (click the mouse to construct a straight line). It really turns out straight. If necessary, you can further select values of the independent variable to obtain a more accurate construction.

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

How many points are enough to construct so that a straight line can be unambiguously drawn through them?

This means that to build a graph of a linear function, it is enough to (click the mouse to display the algorithm):

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

Task 5: in the rectangular coordinate system constructed for task 1, construct a graph of the function: y=2x+5, y=2x+3, y=2x-4, y=2x-2, y=2x+1. Give students task cards (Appendix 3). Each student constructs one of the functions (at the teacher’s discretion). When constructing a graph, try to answer the questions on the “Similarities and Differences” card yourself.

Let's check the function graphs you have built (slide 4). First, students name their chosen points.

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

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

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

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

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

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

Last opportunity to fill out the “Similarities and Differences” cards yourself.

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

How did the similarities show up in their graphs? After receiving the answer, click the mouse.

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

Each of the constructed functions has k = 2, therefore the angles between the graphs and the Ox axis are equal, which means the 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 show up on their graphs? After receiving the answer, click the mouse to display 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 to each other?

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 axis (slide 5);
graphs of functions with identical values of coefficient k are parallel lines.

Let's look at other examples:

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

After summing up the comparison, fill out the “Similarities and Differences” cards. Provide individual assistance to students as needed.

IV. Problem solving

Task 6: construct a rectangular coordinate system with a unit segment equal to two notebook cells. In the coordinate system, construct the graphs of the functions specified in 581. Students with severe damage to the musculoskeletal system are given ready-made system coordinates

V. Summing up the lesson

What function did you get acquainted with today? After receiving the answer, click the mouse and say the definition of a linear function again.

Which object is the graph of a linear function? After receiving the answer, click the mouse and once again talk about the method of constructing a graph of a linear function.

How are the graphs of the functions y=kx+b and y=kx related to each other? After receiving the answer, click the mouse and once again talk about 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 graph of a linear function and to determine the values of the variables x and y from the graph, 589 (oral) – give a complete answer to the question (with explanation).

Thanks for the lesson(slide 7) !

Slide 1

Algebra lesson in 7th grade “Linear function and its graph” Prepared by Tatchin U.V. mathematics teacher MBOU secondary school No. 3, Surgut

Slide 2

Goal: developing the concept of “linear function”, the skill of constructing its graph using an algorithm. Objectives: Educational: - study the definition of a linear function, - introduce and study the algorithm for constructing a graph of a linear function, - practice the skill of recognizing a linear function using a given formula, graph, verbal description. Developmental: - develop visual memory, mathematically literate speech, accuracy, accuracy in construction, ability to analyze. Educational: - to cultivate a responsible attitude to academic work, accuracy, discipline, perseverance. - develop skills of self-control and mutual control

Slide 3

Lesson plan: I. Organizational moment II. Update background knowledge III. Studying new topic IV. Consolidation: oral exercises, graphing tasks V. Solving entertaining tasks VI. Summing up the lesson, recording homework VII. Reflection

Slide 4

I. Organizational moment Having solved the words horizontally, you will learn keyword 1. An exact set of instructions describing the order of actions of the performer to achieve the result of solving a 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. An angle whose degree measure 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 are equal to the corresponding values of the function 8. The road that we choose A L G O R I T M A B S C I S S A F U N C C I A D E C A R T T U P O Y A R G U M E N T GRAPH I C P R Y M A Y

Slide 5

1. An exact set of instructions describing the order of actions of the performer to achieve the result of solving a 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. An angle whose degree measure 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 are equal to the corresponding values of the function 8. The road that we choose A L G O R I T M A B S C I S S A F U N C C I A D E C A R T T U P O Y A R G U M E N T GRAPH I C P R Y M A Y

Slide 6

II. Updating basic knowledge Many real situations are described by mathematical models that are linear functions. Let's give an example. The tourist traveled 15 km by bus from point A to point B, and then continued moving 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 The mathematical model y = 15 + 4x is a linear function. A B C

Slide 7

III. Studying 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, you need to specify a specific x value and calculate the corresponding y value. Typically these results are presented in table form. 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

Slide 8

Algorithm for constructing a graph of a linear function 1) Create a table for a linear function (associate each value of the independent variable with the value of the dependent variable) 2) Construct points on the coordinate plane xOy 3) Draw a straight line through them - graph of a linear function Theorem Graph of a linear function y = k x + m is a straight line.

Slide 9

Let's consider the use of an algorithm for constructing a graph of a linear function Example 1 Construct a graph of a linear function y = 2x + 3 1) Make a table 2) Construct points (0;3) and (1;5) in the xOy coordinate plane 3) Draw a straight line through them

Slide 10

If the linear function y=k x+ m is considered not for all values of x, but only for values of x from a certain 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 on any non-negative value, but in practice a tourist cannot walk at a constant speed without sleep and rest for any amount of time. This means that it was necessary to make reasonable restrictions on x, say, a tourist walks no more than 6 hours. Now let’s write down a more accurate mathematical model: y = 15 + 4x, x 0; 6

Slide 11

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

Slide 12

Slide 13

We plot the function y = -2x + 1, (-3; 2) How does this example differ from the previous one?

Slide 14

Slide 15

IV. Reinforce the topic you've learned Select which function is a linear function

Slide 16

Slide 17

Slide 18

Complete the following task: A linear function is given by the formula y = -3x – 5. Find its value at x = 23, x = -5, x = 0

Slide 19

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

Slide 20

Find the value of the argument at which the linear function y = -2x + 2.4 takes the value equal to 20.4? Checking the solution When 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

Slide 21

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

Slide 22

Slide 23

Slide 24

Slide 25

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

Full name of educational institution:

Secondary municipal educational institution secondary school No. 3 of the village of Kochubeevskoye, Stavropol Territory

Subject area: mathematics

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

Age group: 7th grade

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

Slide 2 - updating of background knowledge: definition of a linear equation, orally select those that are linear from those proposed.

Slide 3 - definition of a linear function.

4 slide recognition of a linear function from those proposed.

5 slide - conclusion.

6 slides - ways to set a function.

Slide 7 I give an example and show.

Slide 8 - I give an example and show it.

9 slide task for students.

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

11 slide output.

Slide 12 - working with the graph of a linear function.

13 slide-Tasks for independent solution:build graphs of functions (do it in a notebook).

Slides 14-17 - showing the correct execution of the task.

Slides 18-27 are oral and written tasks. I don’t choose all tasks, but only those that are suitable for the level of readiness of the class.if there is time.

28 slide task for strong students.

29 slides - let's summarize.

30-31 slides - conclusions.

Slides 32-36 - historical background. (subject to time availability)

Slide 37 - Used literature

List of used literature and Internet resources:

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

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

3. Algebra 7th grade, edited by Makarychev Yu.N. 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, mathematics teacher, Municipal Educational Institution Secondary School No. 3, village. Kochubeevskoye, Stavropol Territory

Specify the 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 y = kx + b is called linear. The graph of a function of the form y = kx +b is a straight line. To construct a straight line, only two points are needed, since only one straight line passes through two points.

Find 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=0.00 5x; y=13 3 ,13 3 13 3 x; y= 3 - 1 0 , 01x ; y=2: x ; y = -0.004 9; y= x:6 2 .

y = kx + b – linear function x – argument (independent variable) y – function (dependent variable) k, b – numbers (coefficients) k ≠ 0

x X 1 X 2 X 3 y U 1 U 2 U 3

y = - 2x + 3 – linear function. The graph of a linear function is a straight line, to construct 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 is obtained by substituting the selected value of x into 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 ourselves

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

I option II option y=x-4 y =- x+4 Determine the relationship between the coefficients k and b and the location of the lines Plot a linear function

y=x-4 y=-x+4 I option II option 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 at x = 3 b) y  0 at x  3 If x  3, then the straight line is located above the x axis, which means the ordinates of the corresponding points of the straight line are positive c) y  0 at 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 (do it in a notebook) 1. y = 2x – 2 2. y = x + 2 3. y = 4 – x 4. y = 1 – 3x Please note: the points you choose to construct a straight line may be different, but the location of the graphs must 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? Explain the answer. 1 2 3 4 5 x y x y x y x y x y

The student made a mistake when graphing a function. In what 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 picture is the coefficient k negative? x

State 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 whose graph 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 about it!

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

Create an equation for a linear function using the following conditions:

let's sum it up

Write down your conclusions in your 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. *To construct a straight line, only two points are needed, since only one straight line passes through two points. *Coefficient k shows whether the straight line is increasing or decreasing. *Coefficient b shows at what point the straight line intersects the OY axis. *Condition of parallelism of two lines.

I wish you success!

Algebra - this word comes from the title of the work of Muhammad Al-Khorezmi “Aljabr and Al-Mukabala”, in which algebra was presented as an independent subject

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

Gottfried Leibniz was a German mathematician (1646 – 1716), who was the first to introduce the term “abscissa” in 1695, “ordinate” in 1684, and “coordinates” in 1692.

Rene Descartes - French philosopher and mathematician (1596 - 1650), who first introduced the concept of “function”

Used literature 1. Mordkovich A.G. and others. Algebra: textbook for 7th grade of general education institutions - M.: Prosveshchenie, 2010. 2. Zvavich L.I. and others. Didactic materials on algebra for grade 7 - M.: Prosveshchenie, 2010. 3. Algebra 7th grade, edited by Makarychev Yu.N. and others, Education, 2010. 4. Internet resources: www.symbolsbook.ru/Article.aspx %...id%3D222

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

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B k b>0b0 y=kx+b (y=2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y=kx I, III quarters Through the beginning of the coordinate K 0b0 y=kx+b (y=2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y=kx I, III quarters Through the beginning of the coordinate K"> 0b0 y=kx +b (y=2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y=kx I, III quarters Through the beginning of the coordinate K"> 0b0 y=kx+b (y =2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y=kx I, III quarters Through the beginning of the coordinate K" title="b k b>0b0 y=kx +b (y=2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y=kx I, III quarters Through the beginning of the coordinate K"> title="b k b>0b0 y=kx+b (y=2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y=kx I, III quarters Through the beginning of the coordinate K"> !}

B k b>0b0 y=kx+b (y=2x+1) I, III quarter. y=kx+b (y=2x-1) I, III quarter. y=kx I, III quarters Through the origin of the coordinate K 0b0 y=kx+b (y=2x+1) I, III quarter. y=kx+b (y=2x-1) I, III quarter. y=kx I, III quarters Through the origin of the coordinate K"> 0b0 y=kx+b (y=2x+1) I, III quarters y=kx+b (y=2x-1) I, III quarters y= kx I, III quarters Through the beginning of the coordinate K"> 0b0 y=kx+b (y=2x+1) I, III quarters. y=kx+b (y=2x-1) I, III quarter. y=kx I, III quarters Through the beginning of the coordinate K" title="b k b>0b0 y=kx+b (y=2x+1) I, III quarters y=kx+b (y=2x-1 ) I, III quarter y=kx I, III quarter Through the beginning of the coordinate K"> title="b k b>0b0 y=kx+b (y=2x+1) I, III quarter. y=kx+b (y=2x-1) I, III quarter. y=kx I, III quarters Through the beginning of the coordinate K"> !}

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