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>>Mathematics: Linear function and its graph

Linear function and its graph


 The algorithm for constructing a graph of the equation ax + by + c = 0, which we formulated in § 28, for all its clarity and certainty, mathematicians do not really like. They usually make claims about the first two steps of the algorithm. Why, they say, solve the equation twice for the variable y: first ax1 + by + c = O, then ax1 + by + c = O? Isn’t it better to immediately express y from the equation ax + by + c = 0, then it will be easier to carry out calculations (and, most importantly, faster)? Let's check. Let's consider first the equation 3x - 2y + 6 = 0 (see example 2 from § 28).

By giving x specific values, it is easy to calculate the corresponding y values. For example, when x = 0 we get y = 3; at x = -2 we have y = 0; for x = 2 we have y = 6; for x = 4 we get: y = 9.

You see how easily and quickly the points (0; 3), (- 2; 0), (2; 6) and (4; 9) were found, which were highlighted in example 2 from § 28.

In the same way, the equation bx - 2y = 0 (see example 4 from § 28) could be transformed to the form 2y = 16 -3x. further y = 2.5x; it is not difficult to find points (0; 0) and (2; 5) satisfying this equation.

Finally, the equation 3x + 2y - 16 = 0 from the same example can be transformed to the form 2y = 16 -3x and then it is not difficult to find points (0; 0) and (2; 5) that satisfy it.

Let us now consider these transformations in general form.


Thus, linear equation (1) with two variables x and y can always be transformed to the form
y = kx + m,(2) where k,m are numbers (coefficients), and .

We will call this particular type of linear equation a linear function.

Using equality (2), it is easy to specify a specific x value and calculate the corresponding y value. Let, for example,

y = 2x + 3. Then:
if x = 0, then y = 3;
if x = 1, then y = 5;
if x = -1, then y = 1;
if x = 3, then y = 9, etc.

Typically these results are presented in the form tables:

The values ​​of y from the second row of the table are called the values ​​of the linear function y = 2x + 3, respectively, at the points x = 0, x = 1, x = -1, x = -3.

In equation (1) the variables hnu are equal, but in equation (2) they are not: we assign specific values ​​to one of them - variable x, while the value of variable y depends on the selected value of variable x. Therefore, we usually say that x is the independent variable (or argument), y is the dependent variable.

Note that a linear function is a special kind of linear equation with two variables. Equation graph y - kx + m, like any linear equation with two variables, is a straight line - it is also called the graph of the linear function y = kx + m. Thus, the following theorem is valid.


Example 1. Construct a graph of the linear function y = 2x + 3.

Solution. Let's make a table:

In the second situation, the independent variable x, which, as in the first situation, denotes the number of days, can only take the values ​​1, 2, 3, ..., 16. Indeed, if x = 16, then using the formula y = 500 - 30x we find : y = 500 - 30 16 = 20. This means that already on the 17th day it will not be possible to remove 30 tons of coal from the warehouse, since by this day only 20 tons will remain in the warehouse and the process of coal removal will have to be stopped. Therefore, the refined mathematical model of the second situation looks like this:

y = 500 - ZOD:, where x = 1, 2, 3, .... 16.

In the third situation, independent variable x can theoretically take on any non-negative value (for example, x value = 0, x value = 2, x value = 3.5, etc.), but practically a tourist cannot walk at a constant speed without sleep and rest for any amount of time . So we needed to make reasonable restrictions on x, say 0< х < 6 (т. е. турист идет не более 6 ч).

Recall that the geometric model of the non-strict double inequality 0< х < 6 служит отрезок (рис. 37). Значит, уточненная модель третьей ситуации выглядит так: у = 15 + 4х, где х принадлежит отрезку .

Let us agree to write instead of the phrase “x belongs to the set X” (read: “element x belongs to the set X”, e is the sign of membership). As you can see, our acquaintance with mathematical language is constantly ongoing.

If the linear function y = kx + m should be considered not for all values ​​of x, but only for values ​​of x from a certain numerical interval X, then they write:

Example 2. Graph a linear function:

Solution, a) Let's make a table for the linear function y = 2x + 1

Let's construct points (-3; 7) and (2; -3) on the xOy coordinate plane and draw a straight line through them. This is a graph of the equation y = -2x: + 1. Next, select a segment connecting the constructed points (Fig. 38). This segment is the graph of the linear function y = -2x+1, wherexe [-3, 2].

They usually say this: we have plotted a linear function y = - 2x + 1 on the segment [- 3, 2].

b) How does this example differ from the previous one? The linear function is the same (y = -2x + 1), which means that the same straight line serves as its graph. But - be careful! - this time x e (-3, 2), i.e. the values ​​x = -3 and x = 2 are not considered, they do not belong to the interval (- 3, 2). How did we mark the ends of an interval on a coordinate line? Light circles (Fig. 39), we talked about this in § 26. Similarly, points (- 3; 7) and B; - 3) will have to be marked on the drawing with light circles. This will remind us that only those points of the line y = - 2x + 1 are taken that lie between the points marked with circles (Fig. 40). However, sometimes in such cases they use arrows rather than light circles (Fig. 41). This is not fundamental, the main thing is to understand what is being said.


Example 3. Find the largest and smallest values ​​of a linear function on the segment.
Solution. Let's make a table for a linear function

Let's construct points (0; 4) and (6; 7) on the xOy coordinate plane and draw a straight line through them - a graph of the linear x function (Fig. 42).

We need to consider this linear function not as a whole, but on a segment, i.e. for x e.

The corresponding segment of the graph is highlighted in the drawing. We note that the largest ordinate of the points belonging to the selected part is equal to 7 - this is the largest value of the linear function on the segment. Usually the following notation is used: y max =7.

We note that the smallest ordinate of the points belonging to the part of the line highlighted in Figure 42 is equal to 4 - this is the smallest value of the linear function on the segment.
Usually the following notation is used: y name. = 4.

Example 4. Find y naib and y naim. for a linear function y = -1.5x + 3.5

a) on the segment; b) on the interval (1.5);
c) on a half-interval.

Solution. Let's make a table for the linear function y = -l.5x + 3.5:

Let's construct points (1; 2) and (5; - 4) on the xOy coordinate plane and draw a straight line through them (Fig. 43-47). Let us select on the constructed straight line the part corresponding to the x values ​​from the segment (Fig. 43), from the interval A, 5) (Fig. 44), from the half-interval (Fig. 47).

a) Using Figure 43, it is easy to conclude that y max = 2 (the linear function reaches this value at x = 1), and y min. = - 4 (the linear function reaches this value at x = 5).

b) Using Figure 44, we conclude: this linear function has neither the largest nor the smallest values ​​on a given interval. Why? The fact is that, unlike the previous case, both ends of the segment, in which the largest and smallest values ​​were reached, are excluded from consideration.

c) Using Figure 45, we conclude that y max. = 2 (as in the first case), and the linear function does not have a minimum value (as in the second case).

d) Using Figure 46, we conclude: y max = 3.5 (the linear function reaches this value at x = 0), and y max. does not exist.

e) Using Figure 47, we conclude: y max. = -1 (the linear function reaches this value at x = 3), and y max. does not exist.

Example 5. Graph a linear function

y = 2x - 6. Use the graph to answer the following questions:

a) at what value of x will y = 0?
b) for what values ​​of x will y > 0?
c) at what values ​​of x will y< 0?

Solution. Let's make a table for the linear function y = 2x-6:

Through the points (0; - 6) and (3; 0) we draw a straight line - the graph of the function y = 2x - 6 (Fig. 48).

a) y = 0 at x = 3. The graph intersects the x axis at the point x = 3, this is the point with ordinate y = 0.
b) y > 0 for x > 3. In fact, if x > 3, then the straight line is located above the x axis, which means that the ordinates of the corresponding points of the straight line are positive.

c) at< 0 при х < 3. В самом деле если х < 3, то прямая расположена ниже оси х, значит, ординаты соответствующих точек прямой отрицательны. A

Please note that in this example we used the graph to solve:

a) equation 2x - 6 = 0 (we got x = 3);
b) inequality 2x - 6 > 0 (we got x > 3);
c) inequality 2x - 6< 0 (получили х < 3).

Comment. In Russian, the same object is often called differently, for example: “house”, “building”, “structure”, “cottage”, “mansion”, “barrack”, “shack”, “hut”. In mathematical language the situation is approximately the same. Say, an equality with two variables y = kx + m, where k, m are specific numbers, can be called a linear function, can be called a linear equation with two variables x and y (or with two unknowns x and y), can be called a formula, can can be called a relationship connecting x and y, can finally be called a dependence between x and y. This doesn’t matter, the main thing is to understand that in all cases we are talking about the mathematical model y = kx + m

.

Consider the graph of the linear function shown in Figure 49, a. If we move along this graph from left to right, then the ordinates of the points on the graph are increasing all the time, as if we are “climbing up a hill.” In such cases, mathematicians use the term increase and say this: if k>0, then the linear function y = kx + m increases.

Consider the graph of the linear function shown in Figure 49, b. If we move along this graph from left to right, then the ordinates of the points on the graph are decreasing all the time, as if we are “going down a hill.” In such cases, mathematicians use the term decrease and say this: if k< О, то линейная функция у = kx + m убывает.

Linear function in life

Now let's summarize this topic. We have already become acquainted with such a concept as a linear function, we know its properties and learned how to build graphs. Also, you considered special cases of linear functions and learned what the relative position of graphs of linear functions depends on. But it turns out that in our everyday life we ​​also constantly intersect with this mathematical model.

Let us think about what real life situations are associated with such a concept as linear functions? And also, between what quantities or life situations is it possible to establish a linear relationship?

Many of you probably don’t quite understand why they need to study linear functions, because it’s unlikely to be useful in later life. But here you are deeply mistaken, because we encounter functions all the time and everywhere. Because even a regular monthly rent is also a function that depends on many variables. And these variables include square footage, number of residents, tariffs, electricity use, etc.

Of course, the most common examples of linear dependence functions that we have encountered are in mathematics lessons.

You and I solved problems where we found the distances traveled by cars, trains, or pedestrians at a certain speed. These are linear functions of movement time. But these examples are applicable not only in mathematics, they are present in our everyday life.

The calorie content of dairy products depends on the fat content, and such a dependence is usually a linear function. For example, when the percentage of fat in sour cream increases, the calorie content of the product also increases.



Now let's do the calculations and find the values ​​of k and b by solving the system of equations:


Now let's derive the dependency formula:

As a result, we obtained a linear relationship.

To know the speed of sound propagation depending on temperature, it is possible to find out by using the formula: v = 331 +0.6t, where v is the speed (in m/s), t is the temperature. If we draw a graph of this relationship, we will see that it will be linear, that is, it will represent a straight line.

And such practical uses of knowledge in the application of linear functional dependence can be listed for a long time. Starting from phone charges, hair length and growth, and even proverbs in literature. And this list goes on and on.

Calendar-thematic planning in mathematics, video in mathematics online, Mathematics at school download

A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions

LINEAR EQUATIONS AND INEQUALITIES I

§ 3 Linear functions and their graphs

Consider the equality

at = 2X + 1. (1)

Each letter value X this equality puts into correspondence a very specific meaning of the letter at . If, for example, x = 0, then at = 2 0 + 1 = 1; If X = 10, then at = 2 10 + 1 = 21; at X = - 1 / 2 we have y = 2 (- 1 / 2) + 1 = 0, etc. Let us turn to another equality:

at = X 2 (2)

Each value X this equality, like equality (1), associates a well-defined value at . If, for example, X = 2, then at = 4; at X = - 3 we get at = 9, etc. Equalities (1) and (2) connect two quantities X And at so that each value of one of them ( X ) is put into correspondence with a well-defined value of another quantity ( at ).

If each value of the quantity X corresponds to a very specific value at, then this value at called a function of X. Magnitude X this is called the function argument at.

Thus, formulas (1) and (2) define two different functions of the argument X .

Argument function X , having the form

y = ax + b , (3)

Where A And b - some given numbers are called linear. An example of a linear function can be any of the functions:

y = x + 2 (A = 1, b = 2);
at = - 10 (A = 0, b = - 10);
at = - 3X (A = - 3, b = 0);
at = 0 (a = b = 0).

As is known from the VIII grade course, function graph y = ax + b is a straight line. That is why this function is called linear.

Let us recall how to construct the graph of a linear function y = ax + b .

1. Graph of a function y = b . At a = 0 linear function y = ax + b looks like y = b . Its graph is a straight line parallel to the axis X and intersecting axis at at the ordinate point b . In Figure 1 you see a graph of the function y = 2 ( b > 0), and in Figure 2 is the graph of the function at = - 1 (b < 0).

If not only A , but also b equals zero, then the function y= ax+ b looks like at = 0. In this case, its graph coincides with the axis X (Fig. 3.)

2. Graph of a function y = ah . At b = 0 linear function y = ax + b looks like y = ah .

If A =/= 0, then its graph is a straight line passing through the origin and inclined to the axis X at an angle φ , whose tangent is equal to A (Fig. 4). To construct a straight line y = ah it is enough to find any one of its points different from the origin of coordinates. Assuming, for example, in the equality y = ah X = 1, we get at = A . Therefore, point M with coordinates (1; A ) lies on our straight line (Fig. 4). Now drawing a straight line through the origin and point M, we obtain the desired straight line y = ax .

In Figure 5, a straight line is drawn as an example at = 2X (A > 0), and in Figure 6 - straight y = - x (A < 0).

3. Graph of a function y = ax + b .

Let b > 0. Then the straight line y = ax + b y = ah on b units up. As an example, Figure 7 shows the construction of a straight line at = x / 2 + 3.

If b < 0, то прямая y = ax + b obtained by parallel shift of the line y = ah on - b units down. As an example, Figure 8 shows the construction of a straight line at = x / 2 - 3

Direct y = ax + b can be built in another way.

Any straight line is completely determined by its two points. Therefore, to plot a graph of the function y = ax + b It is enough to find any two of its points and then draw a straight line through them. Let us explain this using the example of the function at = - 2X + 3.

At X = 0 at = 3, and at X = 1 at = 1. Therefore, two points: M with coordinates (0; 3) and N with coordinates (1; 1) - lie on our line. By marking these points on the coordinate plane and connecting them with a straight line (Fig. 9), we obtain a graph of the function at = - 2X + 3.

Instead of points M and N, one could, of course, take the other two points. For example, as values X we could choose not 0 and 1, as above, but - 1 and 2.5. Then for at we would get the values ​​5 and - 2, respectively. Instead of points M and N, we would have points P with coordinates (- 1; 5) and Q with coordinates (2.5; - 2). These two points, as well as points M and N, completely define the desired line at = - 2X + 3.

Exercises

15. Construct function graphs on the same figure:

A) at = - 4; b) at = -2; V) at = 0; G) at = 2; d) at = 4.

Do these graphs intersect the coordinate axes? If they intersect, then indicate the coordinates of the intersection points.

16. Construct function graphs on the same figure:

A) at = x / 4 ; b) at = x / 2 ; V) at =X ; G) at = 2X ; d) at = 4X .

17. Construct function graphs on the same figure:

A) at = - x / 4 ; b) at = - x / 2 ; V) at = - X ; G) at = - 2X ; d) at = - 4X .

Construct graphs of these functions (No. 18-21) and determine the coordinates of the points of intersection of these graphs with the coordinate axes.

18. at = 3+ X . 20. at = - 4 - X .

19. at = 2X - 2. 21. at = 0,5(1 - 3X ).

22. Graph a function

at = 2x - 4;

using this graph, find out: a) at what values x y = 0;

b) at what values X values at negative and under what conditions - positive;

c) at what values X quantities X And at have the same signs;

d) at what values X quantities X And at have different signs.

23. Write the equations of the lines presented in Figures 10 and 11.

24. Which of the physical laws you know are described using linear functions?

25. How to graph a function at = - (ax + b ), if the function graph is given y = ax + b ?



  • What function is called linear?
  • What is the graph of a linear function?
  • What function is called direct proportionality?
  • In what case are the graphs of two linear functions parallel lines?
  • When do the graphs of two linear functions intersect?

  • In which figure does the graph of a linear function have a positive slope? Justify your answer.
  • Which figure shows a direct proportionality graph? Justify your answer.
  • In which figure does the graph of a linear function have a negative slope? Justify your answer.
  • Which function graph have we not studied? Justify your answer.

2. Who will write it down faster?

  • In a minute, make up the longest word related to the topic of our lesson from these letters

U, T, I, P, I, M, A, R, K, F, G, C, N, I, Ch, O


3. Find the error in the picture.


4. Find the correct answer.

  • What number is shown on the graph of the function given by the formula
  • y = O.5x + 3
  • y = - 4
  • y = 0.5x -3
  • x = - 4

  • Find the value of y corresponding to x=-14 if the linear function is given by the formula y=0.5x+5.

  • The linear function is given by the formula y=-4x+7. Find the value of x for which y=-13.
  • A. 1.5 B. –5 C. 5 D. -1.5

  • It is necessary to construct graphs of functions and select that part of it for the points of which the corresponding inequality is satisfied

  • y = x + 6, 4 ≤ x ≤ 6;
  • y = -x + 6, -6 ≤ x ≤-4;
  • y = - 1/3 x + 10, -6 ≤ x ≤ -3;
  • y = 1/3 x +10, 3 ≤ x ≤ 6;
  • y = -x + 14, 0 ≤ x ≤ 3;
  • y = x + 14, -3 ≤ x ≤ 0;
  • y = 9x – 18, 2 ≤ x ≤ 4;
  • y = - 9x – 18 -4 ≤ x ≤ -2;
  • y = 0, -2 ≤ x ≤ 2.

  • Tulip culture originated in Turkey.



  • 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 ringing laugh, and the golden bud opened.
  • The carefree children's laughter accomplished what no force could do.
  • Since then, it has become a custom to give tulips only to those who feel happiness.

  • Creative homework assignment:
  • draw a picture
  • using straight lines

Trainer on the topic

“Graphing a linear function using the displacement method”

https://pandia.ru/text/78/183/images/image001_208.gif" alt="*" width="13" height="13 src="> Schedule linear function is straight.

margin-top:0cm" type="disc"> up by “b” units if b > 0; down by “b” units if b< 0.

https://pandia.ru/text/78/183/images/image001_208.gif" alt="*" width="13" height="13 src="> Comment. Information that will be highlighted in the table (see below) bold italic , is an element of the solution, so it will need to be written when constructing each graph, changing the relevant data depending on the task.

Example 1. Graph the function y = 2x - 3

Solution to the task

Step 1 . y = 2x - 3 is a linear function, the graph is straight.

The graph of the function y = 2x - 3 can be obtained from the graph of the function y = 2x by shifting it along the op-amp axis by 3 units downward, therefore, you need to make a table to plot the function y = 2x.

y(0) = 2 0 = 0, then (0; 0) is the first point

y(1) = 2 1 = 2, then (1; 2) is the second point

Step 2. Draw a coordinate plane and mark the points found on it. Draw a straight line through these points, which will be the graph of the function y = 2x. It is better to construct this straight line with a dotted line, since when constructing using the displacement method, it is auxiliary.

Step 3. Shift the resulting graph down 3 units. This offset (shift) can be done in two ways:

1 way: take a ruler and use it to draw a straight line parallel to the one drawn by the dotted line, moving it down by 3 units;

Method 2: shift down by 3 units each point from the table from which the graph of the function y = 2x was constructed, and then draw a new straight line through these points

TTNO(SO)A7-05-2

© Gorina LV

Example 2. Graph the function y = 2 – x

Step-by-step comments and explanations

Solution to the task

Step 1. y = 2 - x is a linear function, the graph is a straight line.

The graph of the function y = 2 - x can be obtained from the graph of the function y = - x by shifting it along the op-amp axis by 2 units upward,

therefore, you need to create a table to plot the function y = - x.

y(0) = 0, then (0; 0) is the first point;

y(3) = - 3, then (3; - 3) is the second point.

Step 2. Draw a coordinate plane and mark the points found on it. Draw a straight line through these points, which will be the graph of the function y = - x. It is better to construct this straight line with a dotted line, since when constructing using the displacement method, it is auxiliary.

draw a graph of the linear function y=x+4. find a) the coordinates of the points of intersection of the graph with the coordinate axes b) the y value corresponding to the value

x, equal to -2;-1;1 c) THE VALUE TO WHICH CORRESPONDES TO Y, equal to 1;-2;7; d) find out whether the given linear function increases or decreases. Plot a graph of the linear function y=x+4. Find a) the coordinates of the points of intersection of the graph with the coordinate axes b) the y value corresponding to the x value equal to -2;-1;1 c) VALUE WHICH CORRESPONDES TO Y equal to 1;-2;7; d) find out whether a given linear function increases or decreases.

draw a graph of the linear function y = 2x+3 and use it to find a) the coordinates of the points of intersection of the graph with the coordinate axes b) the values ​​of the function at

x=-construct a graph of the linear function of step 1 and use it to find a) the coordinates of the points of intersection of the graph with the coordinate axes b) the values ​​of the function at x=-2;-1;2;B)2;-1;2;B) argument values ​​if y=-3;1;4

1. a) Find the coordinates of the points of intersection of the graph of the linear equation – 3x + 2y – 6 = 0 with the coordinate axes and construct its graph. b)

Does point K belong to the graph of this equation?

2. a) Transform the linear equation with two variables 2x + y – 1 = 0 into the form of a linear function and plot its graph.

b) Find the smallest and largest values ​​of this function on the segment [-1;2].

3. Find the coordinates of the intersection point of the lines y = 3 – x and y = 2x.

4. a) Define direct proportionality by a formula if it is known that its graph is parallel to the graph of the linear function y = 3x – 4.

5. At what value of p is the solution to the equation 5x + py – 3p = 0 a pair of numbers (1;1)?

1.Plot a graph of the linear function y=-2x.

a) the value of the function at x=-2;1;1.5.
b) the value of the argument when y = -4;1;2.
c) the largest and smallest values ​​of the function on the ray (- ;-2]
2.
a) define the linear function y=kx with a formula if it is known that its graph passes through point A(-4,-12)

HELP URGENTLY PEOPLE NEED.... 1. Graph the linear function y=-2x+1

Use the graph to find:
a) the smallest and largest values ​​of the function on the segment [-1; 2];
b) values ​​of the variable x for which y =0, y is less than 0.
2. Find the coordinates of the intersection point of the lines y = 3 -x and y =2x.
3. a) Find the coordinates of the intersection points of the graph of the linear equation
-3x+ 2 y - 6 = 0 with coordinate axes;
b) Determine whether the point belongs to the graph of this equation
K(1/3:,3,5)
4. a) Define the linear function y= kx by a formula if it is known that it
the graph is parallel to the straight line - 3x +y - 4 = 0.
b) Determine whether the given function is increasing or decreasing. Explain your answer.
_______________________________________________________________
5. At what value of p the solution to the equation 5x + py -3 p =0 is the pair
numbers (1;1) ?

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