Linear function properties and its presentation graph. Linear function and its graph. Lesson summary and presentation. Lesson information card

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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Answers 1. a; b 2.a) 1; 3 b) 2; x y 1. a; c 2. a) 2; 4 b) 1; x y option 2 option

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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 coord 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 K-coord "> 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 origin of coord 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 origin of the K coordinate "title =" (! LANG: bkb> 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 origin of coord 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 origin of coord K"> !}

The presentation for the 7th grade on the topic "Linear function and its graph" refers to such a concept as "linear function". In the process of work, the students will need to convey the main idea that a linear function should contain the necessary conditions when constructing its graph.

slides 1-2 (Presentation topicand "Linear function and its graph", example)

The first slide shows the formula by which each linear formula is built. Accordingly, any function that takes the form of this formula will be linear. Students should learn this formula so that later they can plot a linear function from it.

slides 3-4 (examples)

In order for schoolchildren to more or less understand how to use this formula, it is necessary to analyze several examples that clearly show how exactly you need to obtain data from a specific problem, so that you can then substitute them instead of the variables of this formula. For this, the first example is given.

In the second examples, another task is given with different meanings so that students have the opportunity to consolidate the knowledge they have just acquired on this topic.

slides 5-6 (example, definition of a linear function)

The next slide shows the results of two examples, namely two equations of a linear function, compiled using the corresponding formula. Below it is disassembled into separate components. That is, here it is important to convey to schoolchildren that a linear function consists of two important elements, or rather the coefficients of a binomial. If we are guided by the formula, then they are the variables k and b.

Further, students should carefully analyze the definition of the linear function itself. In his formula, x is the independent variable, while k and b can be any number. In order for the linear function itself to exist, a certain condition must be met. It says that the number b should be equal, provided that the number k, on the contrary, should not be equal to zero.

slides 7-8 (examples)

For clarity, the next slide shows an example of building a graph, compiled using a formula in two ways. That is, when constructing, two conditions were taken into account: the first - the coefficient b is equal to the number 3, the second - the coefficient b is equal to zero. With the help of the presentation, it can be seen that these graphs differ only in the location of the straight line along the Y axis.

In the second example of plotting a graph of a linear function, students should understand the following: firstly, the graph with a coefficient k equal to zero passes through the origin, and secondly, the coefficient k is responsible, depending on its value, for the degree of inclination of the resulting graph along the Y axis.

slides 9-10 (example, linear function graph)

The next slide examines an example of a special graph, where the coefficient k is zero, and the function itself is equal to the value of the coefficient b.

So, having brought the above material to the students, the teacher must now explain that a graph built using a linear function is always a line, that is, a straight line.

Now we should analyze several examples of plotting in order to understand the dependence of the conditions for the values of the coefficients, as well as learn how to determine the coordinates of points on the graph.

slides 13-14 (examples)

In example number 4, 7th grade students must independently determine the coordinates of the graph in accordance with the condition.

The following example was created in order for schoolchildren to understand as much as possible how to build a graph of a linear function with a positive coefficient x, on which the location of a straight line on the X axis directly depends.

slides 15-16 (examples)

For the same reason, the presentation provides an example of plotting a graph with a negative value of the coefficient x.

The last example is a graph with a negative coefficient x. To complete it, students must determine the coordinates of the specified graph and build a graph based on these coordinates. This slide ends the presentation.

This material can be used both by teachers when conducting lessons according to the curriculum, and by students when studying the material on their own. The clarity of this presentation allows you to easily understand the training material on this topic.

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Answers 1. a; b 2.a) 1; 3 b) 2; x y 1. a; c 2. a) 2; 4 b) 1; x y option 2 option

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B k b> 0b0 K 0b0 K "> 0b0 K"> 0b0 K "title =" (! LANG: b k b> 0b0 K"> title="b k b> 0b0 K"> !}

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:

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

Slide 1

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

Slide 2

Purpose: 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

Slide 3

Lesson plan: I. Organizational moment II. Updating basic knowledge 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

Slide 4

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, in 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 are the corresponding values of function 8. The road we choose T G R A F I K P R Y M A Z

Slide 5

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 abscissa 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 F 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

Slide 6

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

Slide 7

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

Slide 8

Algorithm for constructing a graph of a linear function 1) Make a table for a linear function (put each value of the independent variable in correspondence with the value of the 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.

Slide 9

Consider the application of the algorithm for plotting the graph of a linear function Example 1 Plot a plot of a linear function y = 2x + 3 1) Draw up a table 2) Draw the points (0; 3) and (1; 5) in the coordinate plane xОy 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 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. Hence, 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

Slide 11

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) Draw up a table for the linear function y = -2x + 1 2) Build 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 plot 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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Slide 18

Perform the following task 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 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

Slide 21

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

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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) Check yourself: With the OY axis: (0; 3)