Instructions

If the graph is a straight line passing through the origin of coordinates and forming an angle α with the OX axis (the angle of inclination of the straight line to the positive semi-axis OX). The function describing this line will have the form y = kx. The proportionality coefficient k is equal to tan α. If a straight line passes through the 2nd and 4th coordinate quarters, then k< 0, и является убывающей, если через 1-ю и 3-ю, то k >0 and the function increases. Let it represent a straight line, located in different ways relative to the coordinate axes. This is a linear function and has the form y = kx + b, where the variables x and y are to the first power, and k and b can be either positive or negative or equal to zero. The line is parallel to the line y = kx and cuts off at the axis |b| units. If the line is parallel to the abscissa axis, then k = 0, if the ordinate axis, then the equation has the form x = const.

A curve consisting of two branches located in different quarters and symmetrical relative to the origin of coordinates is a hyperbola. This graph is the inverse dependence of the variable y on x and is described by the equation y = k/x. Here k ≠ 0 is the proportionality coefficient. Moreover, if k > 0, the function decreases; if k< 0 - функция возрастает. Таким образом, областью определения функции является вся числовая прямая, кроме x = 0. Ветви приближаются к осям координат как к своим асимптотам. С уменьшением |k| ветки гиперболы все больше «вдавливаются» в координатные углы.

The quadratic function has the form y = ax2 + bx + c, where a, b and c are constant quantities and a  0. If the condition b = c = 0 is met, the function equation looks like y = ax2 ( simplest case), and its graph is a parabola passing through the origin. The graph of the function y = ax2 + bx + c has the same shape as the simplest case of the function, but its vertex (the point of intersection with the OY axis) does not lie at the origin.

A parabola is also the graph of a power function expressed by the equation y = xⁿ, if n is any even number. If n is any odd number, the graph of such a power function will look like a cubic parabola.
If n is any , the equation of the function takes the form. The graph of the function for odd n will be a hyperbola, and for even n their branches will be symmetrical with respect to the op axis.

Back in school years The functions are studied in detail and their graphs are constructed. But, unfortunately, they practically do not teach how to read the graph of a function and find its type from the presented drawing. It's actually quite simple if you remember the basic types of functions.

Instructions

If the presented graph is , which is through the origin of coordinates and with the OX axis the angle α (which is the angle of inclination of the straight line to the positive semi-axis), then the function describing such a straight line will be presented as y = kx. In this case, the proportionality coefficient k is equal to the tangent of the angle α.

If a given line passes through the second and fourth coordinate quarters, then k is equal to 0 and the function increases. Let the presented graph be a straight line located in any way relative to the coordinate axes. Then the function of such graphics will be linear, which is represented by the form y = kx + b, where the variables y and x are in the first, and b and k can take both negative and positive values or .

If the line is parallel to the line with the graph y = kx and cuts off b units on the ordinate axis, then the equation has the form x = const, if the graph is parallel to the abscissa axis, then k = 0.

A curved line that consists of two branches, symmetrical about the origin and located in different quarters, is a hyperbola. Such a graph shows the inverse dependence of the variable y on the variable x and is described by an equation of the form y = k/x, where k should not be equal to zero, since it is a coefficient of inverse proportionality. Moreover, if the value of k is greater than zero, the function decreases; if k less than zero– increases.

If the proposed graph is a parabola passing through the origin, its function, subject to the condition that b = c = 0, will have the form y = ax2. This is the simplest case of a quadratic function. The graph of a function of the form y = ax2 + bx + c will have the same form as the simplest case, however, the vertex (the point where the graph intersects the ordinate axis) will not be at the origin. In a quadratic function, represented by the form y = ax2 + bx + c, the values ​​of a, b and c are constant, while a is not equal to zero.

A parabola can also be the graph of a power function expressed by an equation of the form y = xⁿ only if n is any even number. If the value of n is an odd number, such a graph of a power function will be represented by a cubic parabola. If the variable n is any negative number, the function equation takes the form .

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The coordinate of absolutely any point on the plane is determined by its two quantities: along the abscissa axis and the ordinate axis. The collection of many such points represents the graph of the function. From it you can see how the Y value changes depending on the change in the X value. You can also determine in which section (interval) the function increases and in which it decreases.

Instructions

What can you say about a function if its graph is a straight line? See if this line passes through the coordinate origin point (that is, the one where the X and Y values ​​​​are equal to 0). If it passes, then such a function is described by the equation y = kx. It is easy to understand that the larger the value of k, the closer to the ordinate axis this straight line will be located. And the Y axis itself actually corresponds infinitely of great importance k.

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 such 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. The domain of definition is 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

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