Let's consider the problem. A motorcyclist who left city A to present moment is located 20 km from it. At what distance s (km) from A will the motorcyclist be after t hours if he moves at a speed of 40 km/h?

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

Each t value corresponds to single meaning s.

The formula s = 50t + 20, where t ≥ 0, defines the 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 you pay for sending a telegram containing n words?

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

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

A function that can be specified 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 definition of a linear function can be the set of all numbers or any subset of it.

A special case of a linear function is the previously discussed direct proportionality. Recall that for l = 0 and k ≠ 0 the formula y = kx + l takes the form y = kx, and this formula, as is known, for k ≠ 0 specifies direct proportionality.

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

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

X	-6	-4	-2	0	2	4	6	8
y	-1	0	1	2	3	4	5	6

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

Obviously, the constructed points lie on a certain line. It does not follow from this that the graph of this function is a straight line.

To find out what form the graph of the function f under consideration looks like, let’s compare it with the familiar graph of direct proportionality x – y, where x = 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 on the graph of the function f is 2 units greater than the corresponding ordinate on the graph of direct proportionality.

Consequently, the graph of the function f in question can be obtained from the graph of direct proportionality by parallel translation by 2 units in the direction of the ordinate.

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

In general, the graph of a function given by a formula of the form y = 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 = 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; 0) in the coordinate plane. 3) and draw a straight line through these points. This straight line is the desired graph.

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

The location of the graph of the function specified by the formula y = kx + l depends on the values ​​of l and k. In particular, the angle of inclination of the graph of a linear function to the x-axis depends on the coefficient k. If k – 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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Learn to take derivatives of functions. The derivative characterizes the rate of change of a function at a certain point lying on the graph of this function. IN in this case The graph can be either a straight or curved line. That is, the derivative characterizes the rate of change of a function at a specific point in time. Remember general rules, by which derivatives are taken, and only then proceed to the next step.

• Read the article.
• How to take the simplest derivatives, for example, the derivative of an exponential equation, is described. The calculations presented in the following steps will be based on the methods described therein.

Learn to distinguish problems in which the slope must be calculated through the derivative of a function. Problems do not always ask you to find the slope or derivative of a function. For example, you may be asked to find the rate of change of a function at point A(x,y). You may also be asked to find the slope of the tangent at point A(x,y). In both cases it is necessary to take the derivative of the function.

Take the derivative of the function given to you. There is no need to build a graph here - you only need the equation of the function. In our example, take the derivative of the function. Take the derivative according to the methods outlined in the article mentioned above:

• Derivative:

Substitute the coordinates of the point given to you into the found derivative to calculate the slope. The derivative of a function is equal to the slope at a certain point. In other words, f"(x) is the slope of the function at any point (x,f(x)). In our example:

• Find the slope of the function f (x) = 2 x 2 + 6 x (\displaystyle f(x)=2x^(2)+6x) at point A(4,2).
• Derivative of a function:
  • f ′ (x) = 4 x + 6 (\displaystyle f"(x)=4x+6)
• Substitute the value of the “x” coordinate of this point:
  • f ′ (x) = 4 (4) + 6 (\displaystyle f"(x)=4(4)+6)
• Find the slope:
• Slope function f (x) = 2 x 2 + 6 x (\displaystyle f(x)=2x^(2)+6x) at point A(4,2) is equal to 22.

If possible, check your answer on a graph. Remember that the slope cannot be calculated at every point. Differential calculus examines complex functions and complex graphs, where the slope cannot be calculated at every point, and in some cases the points do not lie on the graphs at all. If possible, use a graphing calculator to check that the slope of the function you are given is correct. Otherwise, draw a tangent to the graph at the point given to you and think about whether the slope value you found matches what you see on the graph.

• The tangent will have the same slope as the graph of the function at a certain point. To draw a tangent at a given point, move left/right on the X axis (in our example, 22 values ​​to the right), and then up one on the Y axis. Mark the point, and then connect it to the point given to you. In our example, connect the points with coordinates (4,2) and (26,3).

1) Function domain and function range.

The domain of a function is the set of all valid valid argument values x(variable x), for which the function y = f(x) determined. The range of a function is the set of all real values y, which the function accepts.

In elementary mathematics, functions are studied only on the set of real numbers.

2) Function zeros.

Function zero is the value of the argument at which the value of the function is equal to zero.

3) Intervals of constant sign of a function.

Intervals of constant sign of a function are sets of argument values ​​on which the function values ​​are only positive or only negative.

4) Monotonicity of the function.

An increasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a larger value of the function.

A decreasing function (in a certain interval) is a function in which a larger value of the argument from this interval corresponds to a smaller value of the function.

5) Even (odd) function.

An even function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality f(-x) = f(x). Schedule even function symmetrical about the ordinate axis.

An odd function is a function whose domain of definition is symmetrical with respect to the origin and for any X from the domain of definition the equality is true f(-x) = - f(x). Schedule odd function symmetrical about the origin.

6) Limited and unlimited functions.

A function is called bounded if there is a positive number M such that |f(x)| ≤ M for all values ​​of x. If such a number does not exist, then the function is unlimited.

7) Periodicity of the function.

A function f(x) is periodic if there is a non-zero number T such that for any x from the domain of definition of the function the following holds: f(x+T) = f(x). This smallest number is called the period of the function. All trigonometric functions are periodic. (Trigonometric formulas).

19. Basic elementary functions, their properties and graphs. Application of functions in economics.

Basic elementary functions. Their properties and graphs

1. Linear function.

Linear function is called a function of the form , where x is a variable, a and b are real numbers.

Number A called the slope of the line, it is equal to the tangent of the angle of inclination of this line to the positive direction of the x-axis. The graph of a linear function is a straight line. It is defined by two points.

Properties of a Linear Function

1. Domain of definition - the set of all real numbers: D(y)=R

2. The set of values ​​is the set of all real numbers: E(y)=R

3. The function takes a zero value when or.

4. The function increases (decreases) over the entire domain of definition.

5. Linear function continuous over the entire domain of definition, differentiable and .

2. Quadratic function.

A function of the form, where x is a variable, coefficients a, b, c are real numbers, is called quadratic

Instructions

There are several ways to solve linear functions. Let's list the most of them. The most commonly used method is the 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 a 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 numerical data, not forgetting to change the sign of the number to the opposite one when transferring. Having calculated one variable, substitute it into other expressions and continue calculations using the same algorithm.

For example, let's take a linear system functions, consisting of two equations:
2x+y-7=0;
x-y-2=0.
It is convenient to express x from the second equation:
x=y+2.
As you can see, when transferring from one part of the equality to another, the sign of y and variables changed, as was 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 put together variables and numbers and add them up:
3у-3=0.
We move it to the right side of the equation and change the sign:
3y=3.
Divide by the total coefficient, we get:
y=1.
We substitute the resulting value into the first expression:
x=y+2.
We get x=3.

Another way to solve similar ones is to add two equations term by term to get a new one with one variable. The equation can be multiplied by a certain coefficient, the main thing is to multiply each member of the equation and not forget, and then add or subtract one equation from. This method is very economical 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 notice that the coefficient of the variable y is identical in the first and second equations and differs only in sign. This means that when we add 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, 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 result can be substituted into any of the system equations to calculate y:
x-y-2=0;
3-у-2=0;
-y+1=0;
-y=-1;
y=1.

You can also calculate data by creating an accurate graph. To do this you need to find zeros functions. If one of the variables is equal to zero, then such a function is called homogeneous. Having solved such equations, you will get two points necessary and sufficient to construct 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=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=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 plot it fairly accurately, other values ​​of x and y can be calculated directly from it.

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 appearance graphics. 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.