When k 0. How to find the slope of the equation

Linear function is a function of the form

x-argument (independent variable),

y- function (dependent variable),

k and b are some constant numbers

The graph of the linear function is straight.

enough to plot the graph. two points, because through two points you can draw a straight line, and moreover, only one.

If k˃0, then the graph is located in the 1st and 3rd coordinate quarters. If k˂0, then the graph is located in the 2nd and 4th coordinate quarters.

The number k is called the slope of the direct graph of the function y(x)=kx+b. If k˃0, then the angle of inclination of the straight line y(x)= kx+b to the positive direction Ox is acute; if k˂0, then this angle is obtuse.

The coefficient b shows the intersection point of the graph with the y-axis (0; b).

y(x)=k∙x-- special case typical function is called direct proportionality. The graph is a straight line passing through the origin, so one point is enough to build this graph.

Linear function graph

Where coefficient k = 3, hence

The graph of the function will increase and have sharp corner with the Ox axis because coefficient k has a plus sign.

OOF of a linear function

FRF of a linear function

Except the case where

Also a linear function of the form

It is a general function.

B) If k=0; b≠0,

In this case, the graph is a straight line parallel to the Ox axis and passing through the point (0;b).

C) If k≠0; b≠0, then the linear function has the form y(x)=k∙x+b.

Example 1 . Plot the function y(x)= -2x+5

Example 2 . Find the zeros of the function y=3x+1, y=0;

are zeros of the function.

Answer: or (;0)

Example 3 . Determine function value y=-x+3 for x=1 and x=-1

y(-1)=-(-1)+3=1+3=4

Answer: y_1=2; y_2=4.

Example 4 . Determine the coordinates of their intersection point or prove that the graphs do not intersect. Let the functions y 1 =10∙x-8 and y 2 =-3∙x+5 be given.

If the graphs of functions intersect, then the value of the functions at this point is equal to

Substitute x=1, then y 1 (1)=10∙1-8=2.

Comment. You can also substitute the obtained value of the argument into the function y 2 =-3∙x+5, then we will get the same answer y 2 (1)=-3∙1+5=2.

y=2 - ordinate of the intersection point.

(1;2) - the point of intersection of the graphs of the functions y \u003d 10x-8 and y \u003d -3x + 5.

Answer: (1;2)

Example 5 .

Construct graphs of functions y 1 (x)= x+3 and y 2 (x)= x-1.

It can be seen that the coefficient k=1 for both functions.

It follows from the above that if the coefficients of a linear function are equal, then their graphs in the coordinate system are parallel.

Example 6 .

Let's build two graphs of the function.

The first graph has the formula

The second graph has the formula

IN this case before us is a graph of two straight lines intersecting at the point (0; 4). This means that the coefficient b, which is responsible for the height of the rise of the graph above the x-axis, if x=0. So we can assume that the coefficient b of both graphs is 4.

Editors: Ageeva Lyubov Alexandrovna, Gavrilina Anna Viktorovna

Let's consider the problem. A motorcyclist leaving town A currently is located 20 km away. At what distance s (km) from A will the motorcyclist be after t hours if he moves at a speed of 40 km/h?

It is obvious that in t hours the motorcyclist will travel 50t km. Consequently, after t hours it will be at a distance of (20 + 50t) km from A, i.e. s = 50t + 20, where t ≥ 0.

Each value of t corresponds to a single value of s.

The formula s = 50t + 20, where t ≥ 0, defines a function.

Let's consider one more problem. For sending a telegram, a fee of 3 kopecks is charged for each word and an additional 10 kopecks. How many kopecks (u) should be paid for sending a telegram containing n words?

Since the sender must pay 3n kopecks for n words, the cost of sending a telegram in n words can be found by the formula u = 3n + 10, where n is any natural number.

In both problems considered, we encountered functions that are given by formulas of the form y \u003d kx + l, where k and l are some numbers, and x and y are variables.

A function that can be given by a formula of the form y = kx + l, where k and l are some numbers, is called linear.

Since the expression kx + l makes sense for any x, the domain of a linear function can be the set of all numbers or any of its subsets.

A special case of a linear function is the previously considered direct proportionality. Recall that for l \u003d 0 and k ≠ 0, the formula y \u003d kx + l takes the form y \u003d kx, and this formula, as you know, for k ≠ 0, direct proportionality is given.

Let us need to plot a linear function f given by the formula
y \u003d 0.5x + 2.

Let's get several corresponding values ​​of the variable y for some values ​​of x:

Let's note the points with the coordinates we received: (-6; -1), (-4; 0); (-2; 1), (0; 2), (2; 3), (4; 4); (6; 5), (8; 6).

It is obvious that the constructed points lie on some straight line. It does not yet follow from this that the graph of this function is a straight line.

To find out what form the graph of the considered function f has, let's compare it with the graph of direct proportionality x - y familiar to us, where x \u003d 0.5.

For any x, the value of the expression 0.5x + 2 is greater than the corresponding value of the expression 0.5x by 2 units. Therefore, the ordinate of each point of the graph of the function f is greater than the corresponding ordinate of the direct proportionality graph by 2 units.

Therefore, the graph of the considered function f can be obtained from the graph of direct proportionality by parallel translation by 2 units in the direction of the y-axis.

Since the graph of direct proportionality is a straight line, then the graph of the considered linear function f is also a straight line.

In general, the graph of a function given by a formula of the form y \u003d kx + l is a straight line.

We know that to construct a straight line, it is enough to determine the position of its two points.

Let, for example, you need to plot a function that is given by the formula
y \u003d 1.5x - 3.

Let's take two arbitrary values ​​of x, for example, x 1 = 0 and x 2 = 4. Calculate the corresponding values ​​of the function y 1 = -3, y 2 = 3, construct points A (-3; 0) and B (4; 3) and draw a line through these points. This straight line is the desired graph.

If the domain of the linear function is not represented by all mi numbers, then its graph will be a subset of points on a straight line (for example, a ray, a segment, a set of individual points).

The location of the graph of the function given by the formula y \u003d kx + l depends on the values ​​\u200b\u200bof l and k. In particular, the value of the angle of inclination of the graph of a linear function to the x-axis depends on the coefficient k. If k is positive number, then this angle is acute; if k is a negative number, then the angle is obtuse. The number k is called the slope of the line.

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>>Math: 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. Usually they put forward claims to the first two steps of the algorithm. Why, they say, solve the equation twice with respect to the variable y: first ax1 + bu + c = O, then axi + bu + c = O? Wouldn't it be 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. Consider first the equation 3x - 2y + 6 = 0 (see example 2 from § 28).

Giving x specific values, it is easy to calculate the corresponding y values. For example, for 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 can 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.

Similarly, the equation bx - 2y = 0 (see example 4 of § 28) could be converted to the form 2y = 16 -3x. then y = 2.5x; it is easy to find points (0; 0) and (2; 5) satisfying this equation.

Finally, the equation 3x + 2y - 16 = 0 from the same example can be converted to the form 2y = 16 -3x and then it is easy to find points (0; 0) and (2; 5) that satisfy it.

Let us now consider the indicated transformations into general view.

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

This particular form of the linear equation will be called a linear function.

Using equality (2), it is easy, by specifying a specific value of x, to calculate the corresponding value of y. 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.

Usually these results are presented in the form tables:

The y values ​​from the second row of the table are called the values ​​of the linear function y \u003d 2x + 3, respectively, at the points x \u003d 0, x \u003d 1, x \u003d -1, x \u003d -3.

In equation (1) the variables xnu are equal, but in equation (2) they are not: we assign specific values ​​to one of them - the variable x, while the value of the variable y depends on the chosen value of the variable x. Therefore, it is usually said 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 a linear function y = kx + mp. Thus, the following theorem is true.

Example 1 Construct a graph of a linear function y \u003d 2x + 3.

Solution. Let's make a table:

In the second situation, the independent variable x, denoting, as in the first situation, the number of days, can only take on the values ​​1, 2, 3, ..., 16. Indeed, if x \u003d 16, then using the formula y \u003d 500 - Z0x we find : y \u003d 500 - 30 16 \u003d 20. This means that already on the 17th day it will not be possible to take out 30 tons of coal from the warehouse, since only 20 tons will remain in the warehouse by this day and the process of coal export will have to be stopped. Therefore, the refined mathematical model of the second situation looks like this:

y \u003d 500 - ZOD:, where x \u003d 1, 2, 3, .... 16.

In the third situation, independent variable x can theoretically take on any non-negative value (e.g., x value = 0, x value = 2, x value = 3.5, etc.), but in practice a tourist cannot walk at a constant speed without sleeping and resting for as long as he wants . So we had to make reasonable limits on x, say 0< х < 6 (т. е. турист идет не более 6 ч).

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

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

If the linear function y \u003d kx + m should be considered not for all values ​​of x, but only for values ​​of x from some numerical interval X, then they write:

Example 2. Graph a linear function:

Solution, a) Make a table for the linear function y = 2x + 1

Let's build points (-3; 7) and (2; -3) on the xOy coordinate plane and draw a straight line through them. This is the graph of the equation y \u003d -2x: + 1. Next, select the segment connecting the constructed points (Fig. 38). This segment is the graph of the linear function y \u003d -2x + 1, where xe [-3, 2].

Usually they say this: we plotted a linear function y \u003d - 2x + 1 on the segment [- 3, 2].

b) How is this example different from the previous one? The linear function is the same (y \u003d -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 the interval on the coordinate line? Light circles (Fig. 39), we talked about this in § 26. Similarly, the 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 straight line y \u003d - 2x + 1 are taken that lie between the points marked with circles (Fig. 40). However, sometimes in such cases, not light circles are used, but arrows (Fig. 41). This is not fundamental, the main thing is to understand what is at stake.

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

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

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

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

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

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

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

Solution. Let's make a table for the linear function y \u003d -l, 5x + 3.5:

We construct points (1; 2) and (5; - 4) on the xOy coordinate plane and draw a straight line through them (Fig. 43-47). Let us single out on the constructed straight line the part corresponding to the values ​​of x 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 \u003d 2 (the linear function reaches this value at x \u003d 1), and y max. = - 4 (the linear function reaches this value at x = 5).

b) Using Figure 44, we conclude that this linear function has neither the largest nor the smallest values ​​in the 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) With the help of Figure 45 we conclude that y max. = 2 (as in the first case), and the smallest value the linear function does not (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. Plot a Linear Function

y \u003d 2x - 6. Using the graph, answer the following questions:

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

Solution. Let's make a table for the linear function y \u003d 2x-6:

Draw a straight line through the points (0; - 6) and (3; 0) - the graph of the function y \u003d 2x - 6 (Fig. 48).

a) y \u003d 0 at x \u003d 3. The graph intersects the x axis at the point x \u003d 3, this is the point with the ordinate y \u003d 0.
b) y > 0 for x > 3. Indeed, 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) at< 0 при х < 3. В самом деле если х < 3, то прямая расположена ниже оси х, значит, ординаты соответствующих точек прямой отрицательны. A

Note that in this example, we decided with the help of the graph:

a) equation 2x - 6 = 0 (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”, “hut”, “hut”. In mathematical language, the situation is about the same. Let's say equality with two variables y = kx + m, where k, m are specific numbers, can be called a linear function, can be called linear equation with two variables x and y (or with two unknowns x and y), you can call it a formula, you can call it a relationship between x and y, you can finally call it a relationship between x and y. It does not 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 a linear function shown in Figure 49, a. If we move along this graph from left to right, then the ordinates of the graph points increase all the time, we seem to “climb up the hill”. In such cases, mathematicians use the term increase and say this: if k>0, then the linear function y \u003d kx + m increases.

Consider the graph of a linear function shown in Figure 49, b. If we move along this graph from left to right, then the ordinates of the graph points decrease all the time, we seem to be “going down the 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 real life

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

Let's think about what real life situations are associated with such a concept as linear functions? Also, between what quantities or life situations perhaps establish a linear dependency?

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

Of course, the most common examples of linear dependence functions that we have come across are math lessons.

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

The calorie content of dairy products depends on fat content, and such a dependence, as a rule, is a linear function. So, for example, with an increase in the percentage of fat content in sour cream, 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 got a linear relationship.

To know the speed of sound propagation depending on temperature, it is possible to find out by applying 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 dependence, 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 height, and even proverbs in literature. And this list can be continued indefinitely.

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

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

Instruction

There are several ways to solve linear functions. Let's take a look at most of them. The most commonly used step-by-step substitution method. In one of the equations, it is necessary to express one variable in terms of another, and substitute it into another equation. And so on until only one variable remains in one of the equations. To solve it, you need to leave the variable on one side of the equal sign (it can be with a coefficient), and on the other side of the equal sign all the numeric data, not forgetting to change the sign of the number to the opposite when transferring. Having calculated one variable, substitute it into other expressions, continue the calculations according to the same algorithm.

For take an example linear functions, consisting of two equations:
2x+y-7=0;
x-y-2=0.
From the second equation it is convenient to express x:
x=y+2.
As you can see, when transferring from one part of the equality to another, the sign of and variables changed, as described above.
We substitute the resulting expression into the first equation, thus excluding the variable x from it:
2*(y+2)+y-7=0.
Expanding the brackets:
2y+4+y-7=0.
We compose variables and numbers, add them:
3y-3=0.
We transfer to the right side of the equation, change the sign:
3y=3.
We divide by the total coefficient, we get:
y=1.
Substitute the resulting value into the first expression:
x=y+2.
We get x=3.

Another way to solve similar ones is to term-by-term two equations to get a new one with one variable. The equation can be multiplied by a certain coefficient, the main thing is to multiply each term of the equation and not forget, and then add or subtract one equation from. This method saves a lot when finding a linear functions.

Let's take the already familiar system of equations with two variables:
2x+y-7=0;
x-y-2=0.
It is easy to see that the coefficient of the variable y is identical in the first and second equations and differs only in sign. This means that when adding these two equations term by term, we get a new one, but with one variable.
2x+x+y-y-7-2=0;
3x-9=0.
We transfer the numerical data to the right side of the equation, while changing the sign:
3x=9.
We find a common factor equal to the coefficient at x and divide both sides of the equation by it:
x=3.
The resulting one can be substituted into any of the equations of the system to calculate y:
x-y-2=0;
3-y-2=0;
-y+1=0;
-y=-1;
y=1.

You can also calculate data by plotting an accurate graph. To do this, you need to find the zeros functions. If one of the variables is equal to zero, then such a function is called homogeneous. By solving such equations, you will get two points necessary and sufficient to build a straight line - one of them will be located on the x-axis, the other on the y-axis.

We take any equation of the system and substitute the value x \u003d 0 there:
2*0+y-7=0;
We get y=7. Thus, the first point, let's call it A, will have coordinates A (0; 7).
In order to calculate a point lying on the x-axis, it is convenient to substitute the value y \u003d 0 into the second equation of the system:
x-0-2=0;
x=2.
The second point (B) will have coordinates B (2;0).
We mark the obtained points on the coordinate grid and draw a straight line through them. If you build it fairly accurately, other x and y values ​​can be computed directly from it.