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Linear function formula. Linear function and its graph

A 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 a linear function is straight.

To create a graph it is enough 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 straight 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.

Coefficient b shows the point of intersection of the graph with the op-amp axis (0; b).

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

Graph of a Linear Function

Where coefficient k = 3, therefore

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

OOF linear function

OPF of a linear function

Except in the case where

Also a linear function of the form

Is a function of general form.

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).

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

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

Example 2 . Let's find the zeros of the function y=3x+1, y=0;

– zeros of the function.

Answer: or (;0)

Example 3 . Determine the value of the function 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 values ​​of the functions at this point are equal

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

Comment. You can also substitute the resulting value of the argument into the function y 2 =-3∙x+5, then we 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=10x-8 and y=-3x+5.

Answer: (1;2)

Example 5 .

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

You can notice that the coefficient k=1 for both functions.

From the above it follows that if the coefficients of a linear function are equal, then their graphs in the coordinate system are located 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, we have a graph of two 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 Ox axis, if x = 0. This means we can assume that the b coefficient of both graphs is equal to 4.

Editors: Ageeva Lyubov Aleksandrovna, Gavrilina Anna Viktorovna

>>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

As practice shows, tasks on the properties and graphs of a quadratic function cause serious difficulties. This is quite strange, because they study the quadratic function in the 8th grade, and then throughout the first quarter of the 9th grade they “torment” the properties of the parabola and build its graphs for various parameters.

This is due to the fact that when forcing students to construct parabolas, they practically do not devote time to “reading” the graphs, that is, they do not practice comprehending the information received from the picture. Apparently, it is assumed that, after constructing a dozen or two graphs, a smart student himself will discover and formulate the relationship between the coefficients in the formula and the appearance of the graph. In practice this does not work. For such a generalization, serious experience in mathematical mini-research is required, which most ninth-graders, of course, do not possess. Meanwhile, the State Inspectorate proposes to determine the signs of the coefficients using the schedule.

We will not demand the impossible from schoolchildren and will simply offer one of the algorithms for solving such problems.

So, a function of the form y = ax 2 + bx + c called quadratic, its graph is a parabola. As the name suggests, the main term is ax 2. That is A should not be equal to zero, the remaining coefficients ( b And With) can equal zero.

Let's see how the signs of its coefficients affect the appearance of a parabola.

The simplest dependence for the coefficient A. Most schoolchildren confidently answer: “if A> 0, then the branches of the parabola are directed upward, and if A < 0, - то вниз". Совершенно верно. Ниже приведен график квадратичной функции, у которой A > 0.

y = 0.5x 2 - 3x + 1

In this case A = 0,5

And now for A < 0:

y = - 0.5x2 - 3x + 1

In this case A = - 0,5

Impact of the coefficient With It's also pretty easy to follow. Let's imagine that we want to find the value of a function at a point X= 0. Substitute zero into the formula:

y = a 0 2 + b 0 + c = c. It turns out that y = c. That is With is the ordinate of the point of intersection of the parabola with the y-axis. Typically, this point is easy to find on the graph. And determine whether it lies above zero or below. That is With> 0 or With < 0.

With > 0:

y = x 2 + 4x + 3

With < 0

y = x 2 + 4x - 3

Accordingly, if With= 0, then the parabola will necessarily pass through the origin:

y = x 2 + 4x


More difficult with the parameter b. The point at which we will find it depends not only on b but also from A. This is the top of the parabola. Its abscissa (axis coordinate X) is found by the formula x in = - b/(2a). Thus, b = - 2ax in. That is, we proceed as follows: we find the vertex of the parabola on the graph, determine the sign of its abscissa, that is, we look to the right of zero ( x in> 0) or to the left ( x in < 0) она лежит.

However, that's not all. We also need to pay attention to the sign of the coefficient A. That is, look at where the branches of the parabola are directed. And only after that, according to the formula b = - 2ax in determine the sign b.

Let's look at an example:

The branches are directed upwards, which means A> 0, the parabola intersects the axis at below zero, that is With < 0, вершина параболы лежит правее нуля. Следовательно, x in> 0. So b = - 2ax in = -++ = -. b < 0. Окончательно имеем: A > 0, b < 0, With < 0.

“Critical points of a function” - Critical points. Among the critical points there are extremum points. A necessary condition for an extremum. Answer: 2. Definition. But, if f" (x0) = 0, then it is not necessary that point x0 will be an extremum point. Extremum points (repetition). Critical points of the function. Extremum points.

“Coordinate plane 6th grade” - Mathematics 6th grade. 1. X. 1. Find and write down the coordinates of points A, B, C, D: -6. Coordinate plane. O. -3. 7. U.

“Functions and their graphs” - Continuity. The largest and smallest value of a function. The concept of an inverse function. Linear. Logarithmic. Monotone. If k > 0, then the formed angle is acute, if k< 0, то угол тупой. В самой точке x = a функция может существовать, а может и не существовать. Х1, х2, х3 – нули функции у = f(x).

“Functions 9th grade” - Valid arithmetic operations on functions. [+] – addition, [-] – subtraction, [*] – multiplication, [:] – division. In such cases, we talk about graphically specifying the function. Formation of a class of elementary functions. Power function y=x0.5. Iovlev Maxim Nikolaevich, a 9th grade student at RMOU Raduzhskaya Secondary School.

“Lesson Tangent Equation” - 1. Clarify the concept of a tangent to the graph of a function. Leibniz considered the problem of drawing a tangent to an arbitrary curve. ALGORITHM FOR DEVELOPING AN EQUATION FOR A TANGENT TO THE GRAPH OF THE FUNCTION y=f(x). Lesson topic: Test: find the derivative of a function. Tangent equation. Fluxion. Grade 10. Decipher what Isaac Newton called the derivative function.

“Build a graph of a function” - The function y=3cosx is given. Graph of the function y=m*sin x. Graph the function. Contents: Given the function: y=sin (x+?/2). Stretching the graph y=cosx along the y axis. To continue click on l. Mouse button. Given the function y=cosx+1. Graph displacement y=sinx vertically. Given the function y=3sinx. Horizontal displacement of the graph y=cosx.

There are a total of 25 presentations in the topic

The concept of a numerical function. Methods for specifying a function. Properties of functions.

A numeric function is a function that acts from one numeric space (set) to another numeric space (set).

Three main ways to define a function: analytical, tabular and graphical.

1. Analytical.

The method of specifying a function using a formula is called analytical. This method is the main one in the mat. analysis, but in practice it is not convenient.

2. Tabular method of specifying a function.

A function can be specified using a table containing the argument values ​​and their corresponding function values.

3. Graphical method of specifying a function.

A function y=f(x) is said to be given graphically if its graph is constructed. This method of specifying a function makes it possible to determine the function values ​​only approximately, since constructing a graph and finding the function values ​​on it is associated with errors.

Properties of a function that must be taken into account when constructing its graph:

1) The domain of definition of the function.

Domain of the function, that is, those values ​​that the argument x of the function F =y (x) can take.

2) Intervals of increasing and decreasing functions.

The function is called increasing on the interval under consideration, if a larger value of the argument corresponds to a larger value of the function y(x). This means that if two arbitrary arguments x 1 and x 2 are taken from the interval under consideration, and x 1 > x 2, then y(x 1) > y(x 2).

The function is called decreasing on the interval under consideration, if a larger value of the argument corresponds to a smaller value of the function y(x). This means that if two arbitrary arguments x 1 and x 2 are taken from the interval under consideration, and x 1< х 2 , то у(х 1) < у(х 2).

3) Function zeros.

The points at which the function F = y (x) intersects the abscissa axis (they are obtained by solving the equation y(x) = 0) are called zeros of the function.

4) Even and odd functions.

The function is called even, if for all argument values ​​from the scope



y(-x) = y(x).

The graph of an even function is symmetrical about the ordinate.

The function is called odd, if for all values ​​of the argument from the domain of definition

y(-x) = -y(x).

The graph of an even function is symmetrical about the origin.

Many functions are neither even nor odd.

5) Periodicity of the function.

The function is called periodic, if there is a number P such that for all values ​​of the argument from the domain of definition

y(x + P) = y(x).


Linear function, its properties and graph.

A linear function is a function of the form y = kx + b, defined on the set of all real numbers.

k– slope (real number)

b– dummy term (real number)

x– independent variable.

· In the special case, if k = 0, we obtain a constant function y = b, the graph of which is a straight line parallel to the Ox axis passing through the point with coordinates (0; b).

· If b = 0, then we get the function y = kx, which is direct proportionality.

o The geometric meaning of the coefficient b is the length of the segment that the straight line cuts off along the Oy axis, counting from the origin.

o The geometric meaning of the coefficient k is the angle of inclination of the straight line to the positive direction of the Ox axis, calculated counterclockwise.

Properties of a linear function:

1) The domain of definition of a linear function is the entire real axis;

2) If k ≠ 0, then the range of values ​​of the linear function is the entire real axis.

If k = 0, then the range of values ​​of the linear function consists of the number b;

3) Evenness and oddness of a linear function depend on the values ​​of the coefficients k and b.

a) b ≠ 0, k = 0, therefore, y = b – even;

b) b = 0, k ≠ 0, therefore y = kx – odd;

c) b ≠ 0, k ≠ 0, therefore y = kx + b is a function of general form;

d) b = 0, k = 0, therefore y = 0 is both an even and an odd function.

4) A linear function does not have the property of periodicity;

5) Points of intersection with coordinate axes:

Ox: y = kx + b = 0, x = -b/k, therefore (-b/k; 0) is the point of intersection with the x-axis.

Oy: y = 0k + b = b, therefore (0; b) is the point of intersection with the ordinate.

Comment. If b = 0 and k = 0, then the function y = 0 vanishes for any value of the variable x. If b ≠ 0 and k = 0, then the function y = b does not vanish for any value of the variable x.

6) The intervals of constant sign depend on the coefficient k.

a) k > 0; kx + b > 0, kx > -b, x > -b/k.

y = kx + b – positive at x from (-b/k; +∞),

y = kx + b – negative for x from (-∞; -b/k).

b)k< 0; kx + b < 0, kx < -b, x < -b/k.

y = kx + b – positive at x from (-∞; -b/k),

y = kx + b – negative for x of (-b/k; +∞).

c) k = 0, b > 0; y = kx + b is positive throughout the entire domain of definition,

k = 0, b< 0; y = kx + b отрицательна на всей области определения.

7) The monotonicity intervals of a linear function depend on the coefficient k.

k > 0, therefore y = kx + b increases throughout the entire domain of definition,

k< 0, следовательно y = kx + b убывает на всей области определения.

11. Function y = ax 2 + bx + c, its properties and graph.

The function y = ax 2 + bx + c (a, b, c are constants, a ≠ 0) is called quadratic In the simplest case, y = ax 2 (b = c = 0) the graph is a curved line passing through the origin. The curve serving as a graph of the function y = ax 2 is a parabola. Every parabola has an axis of symmetry called the axis of the parabola. The point O of the intersection of a parabola with its axis is called the vertex of the parabola.
The graph can be constructed according to the following scheme: 1) Find the coordinates of the vertex of the parabola x 0 = -b/2a; y 0 = y(x 0). 2) We construct several more points that belong to the parabola; when constructing, we can use the symmetries of the parabola relative to the straight line x = -b/2a. 3) Connect the indicated points with a smooth line. Example. Graph the function b = x 2 + 2x - 3. Solutions. The graph of the function is a parabola, the branches of which are directed upward. The abscissa of the vertex of the parabola x 0 = 2/(2 ∙1) = -1, its ordinates y(-1) = (1) 2 + 2(-1) - 3 = -4. So, the vertex of the parabola is point (-1; -4). Let's compile a table of values ​​for several points that are located to the right of the axis of symmetry of the parabola - straight line x = -1.

Function properties.