Investigation of the graph of a function. Functions and their graphs X-axis and y-drawing

The basic elementary functions, their inherent properties and the corresponding graphs are one of the basics of mathematical knowledge, similar in importance to the multiplication table. Elementary functions are the basis, support for the study of all theoretical issues.

The article below provides key material on the topic of basic elementary functions. We will introduce terms, give them definitions; Let us study in detail each type of elementary functions and analyze their properties.

The following types of basic elementary functions are distinguished:

Definition 1

constant function (constant);
root of the nth degree;
power function;
exponential function;
logarithmic function;
trigonometric functions;
fraternal trigonometric functions.

A constant function is defined by the formula: y = C (C is some real number) and also has a name: constant. This function determines whether any real value of the independent variable x corresponds to the same value of the variable y – the value C .

The graph of a constant is a straight line that is parallel to the x-axis and passes through a point having coordinates (0, C). For clarity, we present graphs of constant functions y = 5 , y = - 2 , y = 3 , y = 3 (marked in black, red and blue in the drawing, respectively).

Definition 2

This elementary function is defined by the formula y = x n (n is a natural number greater than one).

Let's consider two variations of the function.

Root of the nth degree, n is an even number

For clarity, we indicate the drawing, which shows the graphs of such functions: y = x , y = x 4 and y = x 8 . These functions are color-coded: black, red and blue, respectively.

A similar view of the graphs of the function of an even degree for other values of the indicator.

Definition 3

Properties of the function root of the nth degree, n is an even number

the domain of definition is the set of all non-negative real numbers [ 0 , + ∞) ;
when x = 0 , the function y = x n has a value equal to zero;
this function is a function of general form (it is neither even nor odd);
range: [ 0 , + ∞) ;
this function y = x n with even exponents of the root increases over the entire domain of definition;
the function has a convexity with an upward direction over the entire domain of definition;
there are no inflection points;
there are no asymptotes;
the graph of the function for even n passes through the points (0 ; 0) and (1 ; 1) .

Root of the nth degree, n is an odd number

Such a function is defined on the entire set of real numbers. For clarity, consider the graphs of functions y = x 3 , y = x 5 and x 9 . In the drawing, they are indicated by colors: black, red and blue colors of the curves, respectively.

Other odd values of the exponent of the root of the function y = x n will give a graph of a similar form.

Definition 4

Properties of the function root of the nth degree, n is an odd number

the domain of definition is the set of all real numbers;
this function is odd;
the range of values is the set of all real numbers;
the function y = x n with odd exponents of the root increases over the entire domain of definition;
the function has concavity on the interval (- ∞ ; 0 ] and convexity on the interval [ 0 , + ∞) ;
the inflection point has coordinates (0 ; 0) ;
there are no asymptotes;
the graph of the function for odd n passes through the points (- 1 ; - 1) , (0 ; 0) and (1 ; 1) .

Power function

Definition 5

The power function is defined by the formula y = x a .

The type of graphs and properties of the function depend on the value of the exponent.

when a power function has an integer exponent a, then the form of the graph of the power function and its properties depend on whether the exponent is even or odd, and also what sign the exponent has. Let us consider all these special cases in more detail below;
the exponent can be fractional or irrational - depending on this, the type of graphs and the properties of the function also vary. We will analyze special cases by setting several conditions: 0< a < 1 ; a > 1 ; - 1 < a < 0 и a < - 1 ;
a power function can have a zero exponent, we will also analyze this case in more detail below.

Let's analyze the power function y = x a when a is an odd positive number, for example, a = 1 , 3 , 5 …

For clarity, we indicate the graphs of such power functions: y = x (black color of the graph), y = x 3 (blue color of the graph), y = x 5 (red color of the graph), y = x 7 (green graph). When a = 1 , we get a linear function y = x .

Definition 6

Properties of a power function when the exponent is an odd positive

the function is increasing for x ∈ (- ∞ ; + ∞) ;
the function is convex for x ∈ (- ∞ ; 0 ] and concave for x ∈ [ 0 ; + ∞) (excluding the linear function);
the inflection point has coordinates (0 ; 0) (excluding the linear function);
there are no asymptotes;
function passing points: (- 1 ; - 1) , (0 ; 0) , (1 ; 1) .

Let's analyze the power function y = x a when a is an even positive number, for example, a = 2 , 4 , 6 ...

For clarity, we indicate the graphs of such power functions: y \u003d x 2 (black color of the graph), y = x 4 (blue color of the graph), y = x 8 (red color of the graph). When a = 2, we get a quadratic function whose graph is a quadratic parabola.

Definition 7

Properties of a power function when the exponent is even positive:

domain of definition: x ∈ (- ∞ ; + ∞) ;
decreasing for x ∈ (- ∞ ; 0 ] ;
the function is concave for x ∈ (- ∞ ; + ∞) ;
there are no inflection points;
there are no asymptotes;
function passing points: (- 1 ; 1) , (0 ; 0) , (1 ; 1) .

The figure below shows examples of exponential function graphs y = x a when a is an odd negative number: y = x - 9 (black color of the chart); y = x - 5 (blue color of the graph); y = x - 3 (red color of the graph); y = x - 1 (green graph). When a \u003d - 1, we get an inverse proportionality, the graph of which is a hyperbola.

Definition 8

Power function properties when the exponent is odd negative:

When x \u003d 0, we get a discontinuity of the second kind, since lim x → 0 - 0 x a \u003d - ∞, lim x → 0 + 0 x a \u003d + ∞ for a \u003d - 1, - 3, - 5, .... Thus, the straight line x = 0 is a vertical asymptote;

range: y ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;
the function is odd because y (- x) = - y (x) ;
the function is decreasing for x ∈ - ∞ ; 0 ∪ (0 ; + ∞) ;
the function is convex for x ∈ (- ∞ ; 0) and concave for x ∈ (0 ; + ∞) ;
there are no inflection points;

k = lim x → ∞ x a x = 0 , b = lim x → ∞ (x a - k x) = 0 ⇒ y = k x + b = 0 when a = - 1 , - 3 , - 5 , . . . .

function passing points: (- 1 ; - 1) , (1 ; 1) .

The figure below shows examples of power function graphs y = x a when a is an even negative number: y = x - 8 (chart in black); y = x - 4 (blue color of the graph); y = x - 2 (red color of the graph).

Definition 9

Power function properties when the exponent is even negative:

domain of definition: x ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;

When x \u003d 0, we get a discontinuity of the second kind, since lim x → 0 - 0 x a \u003d + ∞, lim x → 0 + 0 x a \u003d + ∞ for a \u003d - 2, - 4, - 6, .... Thus, the straight line x = 0 is a vertical asymptote;

the function is even because y (- x) = y (x) ;
the function is increasing for x ∈ (- ∞ ; 0) and decreasing for x ∈ 0 ; +∞ ;
the function is concave for x ∈ (- ∞ ; 0) ∪ (0 ; + ∞) ;
there are no inflection points;
the horizontal asymptote is a straight line y = 0 because:

k = lim x → ∞ x a x = 0 , b = lim x → ∞ (x a - k x) = 0 ⇒ y = k x + b = 0 when a = - 2 , - 4 , - 6 , . . . .

function passing points: (- 1 ; 1) , (1 ; 1) .

From the very beginning, pay attention to the following aspect: in the case when a is a positive fraction with an odd denominator, some authors take the interval - ∞ as the domain of definition of this power function; + ∞ , stipulating that the exponent a is an irreducible fraction. At the moment, the authors of many educational publications on algebra and the beginnings of analysis DO NOT DEFINE power functions, where the exponent is a fraction with an odd denominator for negative values of the argument. Further, we will adhere to just such a position: we take the set [ 0 ; +∞) . Recommendation for students: find out the teacher's point of view at this point in order to avoid disagreements.

So let's take a look at the power function y = x a when the exponent is a rational or irrational number provided that 0< a < 1 .

Let us illustrate with graphs the power functions y = x a when a = 11 12 (chart in black); a = 5 7 (red color of the graph); a = 1 3 (blue color of the graph); a = 2 5 (green color of the graph).

Other values of the exponent a (assuming 0< a < 1) дадут аналогичный вид графика.

Definition 10

Power function properties at 0< a < 1:

range: y ∈ [ 0 ; +∞) ;
the function is increasing for x ∈ [ 0 ; +∞) ;
the function has convexity for x ∈ (0 ; + ∞) ;
there are no inflection points;
there are no asymptotes;

Let's analyze the power function y = x a when the exponent is a non-integer rational or irrational number provided that a > 1 .

We illustrate the graphs of the power function y = x a under given conditions on the example of such functions: y = x 5 4 , y = x 4 3 , y = x 7 3 , y = x 3 π (black, red, blue, green color of graphs, respectively).

Other values of the exponent a under the condition a > 1 will give a similar view of the graph.

Definition 11

Power function properties for a > 1:

domain of definition: x ∈ [ 0 ; +∞) ;
range: y ∈ [ 0 ; +∞) ;
this function is a function of general form (it is neither odd nor even);
the function is increasing for x ∈ [ 0 ; +∞) ;
the function is concave for x ∈ (0 ; + ∞) (when 1< a < 2) и выпуклость при x ∈ [ 0 ; + ∞) (когда a > 2);
there are no inflection points;
there are no asymptotes;
function passing points: (0 ; 0) , (1 ; 1) .

We draw your attention! When a is a negative fraction with an odd denominator, in the works of some authors there is a view that the domain of definition in this case is the interval - ∞; 0 ∪ (0 ; + ∞) with the proviso that the exponent a is an irreducible fraction. At the moment, the authors of educational materials on algebra and the beginnings of analysis DO NOT DEFINE power functions with an exponent in the form of a fraction with an odd denominator for negative values of the argument. Further, we adhere to just such a view: we take the set (0 ; + ∞) as the domain of power functions with fractional negative exponents. Suggestion for students: Clarify your teacher's vision at this point to avoid disagreement.

We continue the topic and analyze the power function y = x a provided: - 1< a < 0 .

Here is a drawing of graphs of the following functions: y = x - 5 6 , y = x - 2 3 , y = x - 1 2 2 , y = x - 1 7 (black, red, blue, green lines, respectively).

Definition 12

Power function properties at - 1< a < 0:

lim x → 0 + 0 x a = + ∞ when - 1< a < 0 , т.е. х = 0 – вертикальная асимптота;

range: y ∈ 0 ; +∞ ;
this function is a function of general form (it is neither odd nor even);
there are no inflection points;

The drawing below shows graphs of power functions y = x - 5 4 , y = x - 5 3 , y = x - 6 , y = x - 24 7 (black, red, blue, green colors of the curves, respectively).

Definition 13

Power function properties for a< - 1:

domain of definition: x ∈ 0 ; +∞ ;

lim x → 0 + 0 x a = + ∞ when a< - 1 , т.е. х = 0 – вертикальная асимптота;

range: y ∈ (0 ; + ∞) ;
this function is a function of general form (it is neither odd nor even);
the function is decreasing for x ∈ 0; +∞ ;
the function is concave for x ∈ 0; +∞ ;
there are no inflection points;
horizontal asymptote - straight line y = 0 ;
function passing point: (1 ; 1) .

When a \u003d 0 and x ≠ 0, we get the function y \u003d x 0 \u003d 1, which determines the line from which the point (0; 1) is excluded (we agreed that the expression 0 0 will not be given any value).

The exponential function has the form y = a x , where a > 0 and a ≠ 1 , and the graph of this function looks different based on the value of the base a . Let's consider special cases.

First, let's analyze the situation when the base of the exponential function has a value from zero to one (0< a < 1) . An illustrative example is the graphs of functions for a = 1 2 (blue color of the curve) and a = 5 6 (red color of the curve).

The graphs of the exponential function will have a similar form for other values of the base, provided that 0< a < 1 .

Definition 14

Properties of an exponential function when the base is less than one:

range: y ∈ (0 ; + ∞) ;
this function is a function of general form (it is neither odd nor even);
an exponential function whose base is less than one is decreasing over the entire domain of definition;
there are no inflection points;
the horizontal asymptote is the straight line y = 0 with the variable x tending to + ∞ ;

Now consider the case when the base of the exponential function is greater than one (a > 1).

Let's illustrate this special case with the graph of exponential functions y = 3 2 x (blue color of the curve) and y = e x (red color of the graph).

Other values of the base, greater than one, will give a similar view of the graph of the exponential function.

Definition 15

Properties of the exponential function when the base is greater than one:

the domain of definition is the entire set of real numbers;
range: y ∈ (0 ; + ∞) ;
this function is a function of general form (it is neither odd nor even);
an exponential function whose base is greater than one is increasing for x ∈ - ∞ ; +∞ ;
the function is concave for x ∈ - ∞ ; +∞ ;
there are no inflection points;
horizontal asymptote - straight line y = 0 with variable x tending to - ∞ ;
function passing point: (0 ; 1) .

The logarithmic function has the form y = log a (x) , where a > 0 , a ≠ 1 .

Such a function is defined only for positive values of the argument: for x ∈ 0 ; +∞ .

The graph of the logarithmic function has a different form, based on the value of the base a.

Consider first the situation when 0< a < 1 . Продемонстрируем этот частный случай графиком логарифмической функции при a = 1 2 (синий цвет кривой) и а = 5 6 (красный цвет кривой).

Other values of the base, not greater than one, will give a similar view of the graph.

Definition 16

Properties of a logarithmic function when the base is less than one:

domain of definition: x ∈ 0 ; +∞ . As x tends to zero from the right, the values of the function tend to + ∞;
range: y ∈ - ∞ ; +∞ ;
this function is a function of general form (it is neither odd nor even);
logarithmic
the function is concave for x ∈ 0; +∞ ;
there are no inflection points;
there are no asymptotes;

Now let's analyze a special case when the base of the logarithmic function is greater than one: a > 1 . In the drawing below, there are graphs of logarithmic functions y = log 3 2 x and y = ln x (blue and red colors of the graphs, respectively).

Other values of the base greater than one will give a similar view of the graph.

Definition 17

Properties of a logarithmic function when the base is greater than one:

domain of definition: x ∈ 0 ; +∞ . As x tends to zero from the right, the values of the function tend to - ∞;
range: y ∈ - ∞ ; + ∞ (the whole set of real numbers);
this function is a function of general form (it is neither odd nor even);
the logarithmic function is increasing for x ∈ 0; +∞ ;
the function has convexity for x ∈ 0; +∞ ;
there are no inflection points;
there are no asymptotes;
function passing point: (1 ; 0) .

Trigonometric functions are sine, cosine, tangent and cotangent. Let's analyze the properties of each of them and the corresponding graphs.

In general, all trigonometric functions are characterized by the property of periodicity, i.e. when the values of the functions are repeated for different values of the argument that differ from each other by the value of the period f (x + T) = f (x) (T is the period). Thus, the item "least positive period" is added to the list of properties of trigonometric functions. In addition, we will indicate such values of the argument for which the corresponding function vanishes.

Sine function: y = sin(x)

The graph of this function is called a sine wave.

Definition 18

Properties of the sine function:

domain of definition: the whole set of real numbers x ∈ - ∞ ; +∞ ;
the function vanishes when x = π k , where k ∈ Z (Z is the set of integers);
the function is increasing for x ∈ - π 2 + 2 π · k ; π 2 + 2 π k , k ∈ Z and decreasing for x ∈ π 2 + 2 π k ; 3 π 2 + 2 π k , k ∈ Z ;
the sine function has local maxima at the points π 2 + 2 π · k ; 1 and local minima at points - π 2 + 2 π · k ; - 1 , k ∈ Z ;
the sine function is concave when x ∈ - π + 2 π k; 2 π k , k ∈ Z and convex when x ∈ 2 π k ; π + 2 π k , k ∈ Z ;
there are no asymptotes.

cosine function: y=cos(x)

The graph of this function is called a cosine wave.

Definition 19

Properties of the cosine function:

domain of definition: x ∈ - ∞ ; +∞ ;
the smallest positive period: T \u003d 2 π;
range: y ∈ - 1 ; one ;
this function is even, since y (- x) = y (x) ;
the function is increasing for x ∈ - π + 2 π · k ; 2 π · k , k ∈ Z and decreasing for x ∈ 2 π · k ; π + 2 π k , k ∈ Z ;
the cosine function has local maxima at points 2 π · k ; 1 , k ∈ Z and local minima at the points π + 2 π · k ; - 1 , k ∈ z ;
the cosine function is concave when x ∈ π 2 + 2 π · k ; 3 π 2 + 2 π k , k ∈ Z and convex when x ∈ - π 2 + 2 π k ; π 2 + 2 π · k , k ∈ Z ;
inflection points have coordinates π 2 + π · k ; 0 , k ∈ Z
there are no asymptotes.

Tangent function: y = t g (x)

The graph of this function is called tangentoid.

Definition 20

Properties of the tangent function:

domain of definition: x ∈ - π 2 + π · k ; π 2 + π k , where k ∈ Z (Z is the set of integers);
The behavior of the tangent function on the boundary of the domain of definition lim x → π 2 + π · k + 0 t g (x) = - ∞ , lim x → π 2 + π · k - 0 t g (x) = + ∞ . Thus, the lines x = π 2 + π · k k ∈ Z are vertical asymptotes;
the function vanishes when x = π k for k ∈ Z (Z is the set of integers);
range: y ∈ - ∞ ; +∞ ;
this function is odd because y (- x) = - y (x) ;
the function is increasing at - π 2 + π · k ; π 2 + π k , k ∈ Z ;
the tangent function is concave for x ∈ [ π · k ; π 2 + π k) , k ∈ Z and convex for x ∈ (- π 2 + π k ; π k ] , k ∈ Z ;
inflection points have coordinates π k; 0 , k ∈ Z ;

Cotangent function: y = c t g (x)

The graph of this function is called the cotangentoid. .

Definition 21

Properties of the cotangent function:

domain of definition: x ∈ (π k ; π + π k) , where k ∈ Z (Z is the set of integers);

Behavior of the cotangent function on the boundary of the domain of definition lim x → π · k + 0 t g (x) = + ∞ , lim x → π · k - 0 t g (x) = - ∞ . Thus, the lines x = π k k ∈ Z are vertical asymptotes;

the smallest positive period: T \u003d π;
the function vanishes when x = π 2 + π k for k ∈ Z (Z is the set of integers);
range: y ∈ - ∞ ; +∞ ;
this function is odd because y (- x) = - y (x) ;
the function is decreasing for x ∈ π · k ; π + π k , k ∈ Z ;
the cotangent function is concave for x ∈ (π k ; π 2 + π k ] , k ∈ Z and convex for x ∈ [ - π 2 + π k ; π k) , k ∈ Z ;
inflection points have coordinates π 2 + π · k ; 0 , k ∈ Z ;
there are no oblique and horizontal asymptotes.

The inverse trigonometric functions are the arcsine, arccosine, arctangent, and arccotangent. Often, due to the presence of the prefix "arc" in the name, inverse trigonometric functions are called arc functions. .

Arcsine function: y = a r c sin (x)

Definition 22

Properties of the arcsine function:

this function is odd because y (- x) = - y (x) ;
the arcsine function is concave for x ∈ 0; 1 and convexity for x ∈ - 1 ; 0;
inflection points have coordinates (0 ; 0) , it is also the zero of the function;
there are no asymptotes.

Arccosine function: y = a r c cos (x)

Definition 23

Arccosine function properties:

domain of definition: x ∈ - 1 ; one ;
range: y ∈ 0 ; π;
this function is of general form (neither even nor odd);
the function is decreasing on the entire domain of definition;
the arccosine function is concave for x ∈ - 1 ; 0 and convexity for x ∈ 0 ; one ;
inflection points have coordinates 0 ; π2;
there are no asymptotes.

Arctangent function: y = a r c t g (x)

Definition 24

Arctangent function properties:

domain of definition: x ∈ - ∞ ; +∞ ;
range: y ∈ - π 2 ; π2;
this function is odd because y (- x) = - y (x) ;
the function is increasing over the entire domain of definition;
the arctangent function is concave for x ∈ (- ∞ ; 0 ] and convex for x ∈ [ 0 ; + ∞) ;
the inflection point has coordinates (0; 0), it is also the zero of the function;
horizontal asymptotes are straight lines y = - π 2 for x → - ∞ and y = π 2 for x → + ∞ (the asymptotes in the figure are green lines).

Arc cotangent function: y = a r c c t g (x)

Definition 25

Arc cotangent function properties:

domain of definition: x ∈ - ∞ ; +∞ ;
range: y ∈ (0 ; π) ;
this function is of a general type;
the function is decreasing on the entire domain of definition;
the arc cotangent function is concave for x ∈ [ 0 ; + ∞) and convexity for x ∈ (- ∞ ; 0 ] ;
the inflection point has coordinates 0 ; π2;
horizontal asymptotes are straight lines y = π at x → - ∞ (green line in the drawing) and y = 0 at x → + ∞.

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The length of the segment on the coordinate axis is found by the formula:

The length of the segment on the coordinate plane is sought by the formula:

To find the length of a segment in a three-dimensional coordinate system, the following formula is used:

The coordinates of the middle of the segment (for the coordinate axis only the first formula is used, for the coordinate plane - the first two formulas, for the three-dimensional coordinate system - all three formulas) are calculated by the formulas:

Function is a correspondence of the form y= f(x) between variables, due to which each considered value of some variable x(argument or independent variable) corresponds to a certain value of another variable, y(dependent variable, sometimes this value is simply called the value of the function). Note that the function assumes that one value of the argument X there can only be one value of the dependent variable at. However, the same value at can be obtained with various X.

Function scope are all values of the independent variable (function argument, usually X) for which the function is defined, i.e. its meaning exists. The domain of definition is indicated D(y). By and large, you are already familiar with this concept. The scope of a function is otherwise called the domain of valid values, or ODZ, which you have been able to find for a long time.

Function range are all possible values of the dependent variable of this function. Denoted E(at).

Function rises on the interval on which the larger value of the argument corresponds to the larger value of the function. Function Decreasing on the interval on which the larger value of the argument corresponds to the smaller value of the function.

Function intervals are the intervals of the independent variable at which the dependent variable retains its positive or negative sign.

Function zeros are those values of the argument for which the value of the function is equal to zero. At these points, the graph of the function intersects the abscissa axis (OX axis). Very often, the need to find the zeros of a function means simply solving the equation. Also, often the need to find intervals of constant sign means the need to simply solve the inequality.

Function y = f(x) are called even X

This means that for any opposite values of the argument, the values of the even function are equal. The graph of an even function is always symmetrical about the y-axis of the op-amp.

Function y = f(x) are called odd, if it is defined on a symmetric set and for any X from the domain of definition the equality is fulfilled:

This means that for any opposite values of the argument, the values of the odd function are also opposite. The graph of an odd function is always symmetrical about the origin.

The sum of the roots of even and odd functions (points of intersection of the abscissa axis OX) is always equal to zero, because for every positive root X has a negative root X.

It is important to note that some function does not have to be even or odd. There are many functions that are neither even nor odd. Such functions are called general functions, and none of the above equalities or properties hold for them.

Linear function is called a function that can be given by the formula:

The graph of a linear function is a straight line and in the general case looks like this (an example is given for the case when k> 0, in this case the function is increasing; for the case k < 0 функция будет убывающей, т.е. прямая будет наклонена в другую сторону - слева направо):

Graph of Quadratic Function (Parabola)

The graph of a parabola is given by a quadratic function:

A quadratic function, like any other function, intersects the OX axis at the points that are its roots: ( x one ; 0) and ( x 2; 0). If there are no roots, then the quadratic function does not intersect the OX axis, if there is one root, then at this point ( x 0; 0) the quadratic function only touches the OX axis, but does not intersect it. A quadratic function always intersects the OY axis at a point with coordinates: (0; c). The graph of a quadratic function (parabola) may look like this (the figure shows examples that far from exhaust all possible types of parabolas):

Wherein:

if the coefficient a> 0, in the function y = ax 2 + bx + c, then the branches of the parabola are directed upwards;
if a < 0, то ветви параболы направлены вниз.

Parabola vertex coordinates can be calculated using the following formulas. X tops (p- in the figures above) of a parabola (or the point at which the square trinomial reaches its maximum or minimum value):

Y tops (q- in the figures above) of a parabola or the maximum if the branches of the parabola are directed downwards ( a < 0), либо минимальное, если ветви параболы направлены вверх (a> 0), the value of the square trinomial:

Graphs of other functions

power function

Here are some examples of graphs of power functions:

Inversely proportional dependence call the function given by the formula:

Depending on the sign of the number k An inversely proportional graph can have two fundamental options:

Asymptote is the line to which the line of the graph of the function approaches infinitely close, but does not intersect. The asymptotes for the inverse proportionality graphs shown in the figure above are the coordinate axes, to which the graph of the function approaches infinitely close, but does not intersect them.

exponential function with base a call the function given by the formula:

a the graph of an exponential function can have two fundamental options (we will also give examples, see below):

logarithmic function call the function given by the formula:

Depending on whether the number is greater or less than one a The graph of a logarithmic function can have two fundamental options:

Function Graph y = |x| as follows:

Graphs of periodic (trigonometric) functions

Function at = f(x) is called periodical, if there exists such a non-zero number T, what f(x + T) = f(x), for anyone X out of function scope f(x). If the function f(x) is periodic with period T, then the function:

where: A, k, b are constant numbers, and k not equal to zero, also periodic with a period T 1 , which is determined by the formula:

Most examples of periodic functions are trigonometric functions. Here are the graphs of the main trigonometric functions. The following figure shows part of the graph of the function y= sin x(the whole graph continues indefinitely to the left and right), the graph of the function y= sin x called sinusoid:

Function Graph y= cos x called cosine wave. This graph is shown in the following figure. Since the graph of the sine, it continues indefinitely along the OX axis to the left and to the right:

Function Graph y=tg x called tangentoid. This graph is shown in the following figure. Like the graphs of other periodic functions, this graph repeats indefinitely along the OX axis to the left and to the right.

And finally, the graph of the function y=ctg x called cotangentoid. This graph is shown in the following figure. Like the graphs of other periodic and trigonometric functions, this graph repeats indefinitely along the OX axis to the left and to the right.

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How to successfully prepare for the CT in Physics and Mathematics?

In order to successfully prepare for the CT in Physics and Mathematics, among other things, three critical conditions must be met:

Study all the topics and complete all the tests and tasks given in the study materials on this site. To do this, you need nothing at all, namely: to devote three to four hours every day to preparing for the CT in physics and mathematics, studying theory and solving problems. The fact is that the DT is an exam where it is not enough just to know physics or mathematics, you also need to be able to quickly and without failures solve a large number of problems on various topics and varying complexity. The latter can only be learned by solving thousands of problems.
Learn all formulas and laws in physics, and formulas and methods in mathematics. In fact, it is also very simple to do this, there are only about 200 necessary formulas in physics, and even a little less in mathematics. In each of these subjects there are about a dozen standard methods for solving problems of a basic level of complexity, which can also be learned, and thus, completely automatically and without difficulty, solve most of the digital transformation at the right time. After that, you will only have to think about the most difficult tasks.
Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to solve both options. Again, on the DT, in addition to the ability to quickly and efficiently solve problems, and the knowledge of formulas and methods, it is also necessary to be able to properly plan time, distribute forces, and most importantly fill out the answer form correctly, without confusing either the numbers of answers and tasks, or your own surname. Also, during the RT, it is important to get used to the style of posing questions in tasks, which may seem very unusual to an unprepared person on the DT.

Successful, diligent and responsible implementation of these three points, as well as responsible study of the final training tests, will allow you to show an excellent result on the CT, the maximum of what you are capable of.

Found an error?

If you, as it seems to you, have found an error in the training materials, then please write about it by e-mail (). In the letter, indicate the subject (physics or mathematics), the name or number of the topic or test, the number of the task, or the place in the text (page) where, in your opinion, there is an error. Also describe what the alleged error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not a mistake.

Let's see how to explore a function using a graph. It turns out that looking at the graph, you can find out everything that interests us, namely:

function scope
function range
function zeros
periods of increase and decrease
high and low points
the largest and smallest value of the function on the interval.

Let's clarify the terminology:

Abscissa is the horizontal coordinate of the point.
Ordinate- vertical coordinate.
abscissa- the horizontal axis, most often called the axis.
Y-axis- vertical axis, or axis.

Argument is an independent variable on which the values of the function depend. Most often indicated.
In other words, we ourselves choose , substitute in the function formula and get .

Domain functions - the set of those (and only those) values of the argument for which the function exists.
Denoted: or .

In our figure, the domain of the function is a segment. It is on this segment that the graph of the function is drawn. Only here this function exists.

Function range is the set of values that the variable takes. In our figure, this is a segment - from the lowest to the highest value.

Function zeros- points where the value of the function is equal to zero, i.e. . In our figure, these are the points and .

Function values are positive where . In our figure, these are the intervals and .
Function values are negative where . We have this interval (or interval) from to.

The most important concepts - increasing and decreasing function on some set. As a set, you can take a segment, an interval, a union of intervals, or the entire number line.

Function increases

In other words, the more , the more , that is, the graph goes to the right and up.

Function decreasing on the set if for any and belonging to the set the inequality implies the inequality .

For a decreasing function, a larger value corresponds to a smaller value. The graph goes right and down.

In our figure, the function increases on the interval and decreases on the intervals and .

Let's define what is maximum and minimum points of the function.

Maximum point- this is an internal point of the domain of definition, such that the value of the function in it is greater than in all points sufficiently close to it.
In other words, the maximum point is such a point, the value of the function at which more than in neighboring ones. This is a local "hill" on the chart.

In our figure - the maximum point.

Low point- an internal point of the domain of definition, such that the value of the function in it is less than in all points sufficiently close to it.
That is, the minimum point is such that the value of the function in it is less than in neighboring ones. On the graph, this is a local “hole”.

In our figure - the minimum point.

The point is the boundary. It is not an interior point of the domain of definition and therefore does not fit the definition of a maximum point. After all, she has no neighbors on the left. In the same way, there can be no minimum point on our chart.

The maximum and minimum points are collectively called extremum points of the function. In our case, this is and .

But what if you need to find, for example, function minimum on the cut? In this case, the answer is: because function minimum is its value at the minimum point.

Similarly, the maximum of our function is . It is reached at the point .

We can say that the extrema of the function are equal to and .

Sometimes in tasks you need to find the largest and smallest values of the function on a given segment. They do not necessarily coincide with extremes.

In our case smallest function value on the interval is equal to and coincides with the minimum of the function. But its largest value on this segment is equal to . It is reached at the left end of the segment.

In any case, the largest and smallest values of a continuous function on a segment are achieved either at the extremum points or at the ends of the segment.

This methodological material is for reference only and covers a wide range of topics. The article provides an overview of the graphs of the main elementary functions and considers the most important issue - how to correctly and FAST build a graph. In the course of studying higher mathematics without knowing the graphs of the basic elementary functions, it will be difficult, so it is very important to remember what the graphs of a parabola, hyperbola, sine, cosine, etc. look like, to remember some function values. We will also talk about some properties of the main functions.

I do not pretend to completeness and scientific thoroughness of the materials, the emphasis will be placed, first of all, on practice - those things with which one has to face literally at every step, in any topic of higher mathematics. Charts for dummies? You can say so.

By popular demand from readers clickable table of contents:

In addition, there is an ultra-short abstract on the topic
– master 16 types of charts by studying SIX pages!

Seriously, six, even I myself was surprised. This abstract contains improved graphics and is available for a nominal fee, a demo version can be viewed. It is convenient to print the file so that the graphs are always at hand. Thanks for supporting the project!

And we start right away:

How to build coordinate axes correctly?

In practice, tests are almost always drawn up by students in separate notebooks, lined in a cage. Why do you need checkered markings? After all, the work, in principle, can be done on A4 sheets. And the cage is necessary just for the high-quality and accurate design of the drawings.

Any drawing of a function graph starts with coordinate axes.

Drawings are two-dimensional and three-dimensional.

Let us first consider the two-dimensional case Cartesian coordinate system:

1) We draw coordinate axes. The axis is called x-axis , and the axis y-axis . We always try to draw them neat and not crooked. The arrows should also not resemble Papa Carlo's beard.

2) We sign the axes with capital letters "x" and "y". Don't forget to sign the axes.

3) Set the scale along the axes: draw zero and two ones. When making a drawing, the most convenient and common scale is: 1 unit = 2 cells (drawing on the left) - stick to it if possible. However, from time to time it happens that the drawing does not fit on a notebook sheet - then we reduce the scale: 1 unit = 1 cell (drawing on the right). Rarely, but it happens that the scale of the drawing has to be reduced (or increased) even more

DO NOT scribble from a machine gun ... -5, -4, -3, -1, 0, 1, 2, 3, 4, 5, .... For the coordinate plane is not a monument to Descartes, and the student is not a dove. We put zero and two units along the axes. Sometimes instead of units, it is convenient to “detect” other values, for example, “two” on the abscissa axis and “three” on the ordinate axis - and this system (0, 2 and 3) will also uniquely set the coordinate grid.

It is better to estimate the estimated dimensions of the drawing BEFORE the drawing is drawn.. So, for example, if the task requires drawing a triangle with vertices , , , then it is quite clear that the popular scale 1 unit = 2 cells will not work. Why? Let's look at the point - here you have to measure fifteen centimeters down, and, obviously, the drawing will not fit (or barely fit) on a notebook sheet. Therefore, we immediately select a smaller scale 1 unit = 1 cell.

By the way, about centimeters and notebook cells. Is it true that there are 15 centimeters in 30 notebook cells? Measure in a notebook for interest 15 centimeters with a ruler. In the USSR, perhaps this was true ... It is interesting to note that if you measure these same centimeters horizontally and vertically, then the results (in cells) will be different! Strictly speaking, modern notebooks are not checkered, but rectangular. It may seem like nonsense, but drawing, for example, a circle with a compass in such situations is very inconvenient. To be honest, at such moments you begin to think about the correctness of Comrade Stalin, who was sent to camps for hack work in production, not to mention the domestic automotive industry, falling planes or exploding power plants.

Speaking of quality, or a brief recommendation on stationery. To date, most of the notebooks on sale, without saying bad words, are complete goblin. For the reason that they get wet, and not only from gel pens, but also from ballpoint pens! Save on paper. For the design of tests, I recommend using the notebooks of the Arkhangelsk Pulp and Paper Mill (18 sheets, cell) or Pyaterochka, although it is more expensive. It is advisable to choose a gel pen, even the cheapest Chinese gel refill is much better than a ballpoint pen, which either smears or tears paper. The only "competitive" ballpoint pen in my memory is the Erich Krause. She writes clearly, beautifully and stably - either with a full stem, or with an almost empty one.

Additionally: the vision of a rectangular coordinate system through the eyes of analytical geometry is covered in the article Linear (non) dependence of vectors. Vector basis, detailed information about coordinate quarters can be found in the second paragraph of the lesson Linear inequalities.

3D case

It's almost the same here.

1) We draw coordinate axes. Standard: applicate axis – directed upwards, axis – directed to the right, axis – downwards to the left strictly at an angle of 45 degrees.

2) We sign the axes.

3) Set the scale along the axes. Scale along the axis - two times smaller than the scale along the other axes. Also note that in the right drawing, I used a non-standard "serif" along the axis (this possibility has already been mentioned above). From my point of view, it’s more accurate, faster and more aesthetically pleasing - you don’t need to look for the middle of the cell under a microscope and “sculpt” the unit right up to the origin.

When doing a 3D drawing again - give priority to scale
1 unit = 2 cells (drawing on the left).

What are all these rules for? Rules are there to be broken. What am I going to do now. The fact is that the subsequent drawings of the article will be made by me in Excel, and the coordinate axes will look incorrect in terms of proper design. I could draw all the graphs by hand, but it’s really scary to draw them, as Excel is reluctant to draw them much more accurately.

Graphs and basic properties of elementary functions

The linear function is given by the equation . Linear function graph is direct. In order to construct a straight line, it is enough to know two points.

Example 1

Plot the function. Let's find two points. It is advantageous to choose zero as one of the points.

If , then

We take some other point, for example, 1.

If , then

When preparing tasks, the coordinates of points are usually summarized in a table:

And the values themselves are calculated orally or on a draft, calculator.

Two points are found, let's draw:

When drawing up a drawing, we always sign the graphics.

It will not be superfluous to recall special cases of a linear function:

Notice how I placed the captions, signatures should not be ambiguous when studying the drawing. In this case, it was highly undesirable to put a signature next to the point of intersection of the lines, or at the bottom right between the graphs.

1) A linear function of the form () is called direct proportionality. For instance, . The direct proportionality graph always passes through the origin. Thus, the construction of a straight line is simplified - it is enough to find only one point.

2) An equation of the form defines a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is built immediately, without finding any points. That is, the entry should be understood as follows: "y is always equal to -4, for any value of x."

3) An equation of the form defines a straight line parallel to the axis, in particular, the axis itself is given by the equation. The graph of the function is also built immediately. The entry should be understood as follows: "x is always, for any value of y, equal to 1."

Some will ask, well, why remember the 6th grade?! That's how it is, maybe so, only during the years of practice I met a good dozen students who were baffled by the task of constructing a graph like or .

Drawing a straight line is the most common action when making drawings.

The straight line is discussed in detail in the course of analytic geometry, and those who wish can refer to the article Equation of a straight line on a plane.

Quadratic function graph, cubic function graph, polynomial graph

Parabola. Graph of a quadratic function () is a parabola. Consider the famous case:

Let's recall some properties of the function.

So, the solution to our equation: - it is at this point that the vertex of the parabola is located. Why this is so can be learned from the theoretical article on the derivative and the lesson on the extrema of the function. In the meantime, we calculate the corresponding value of "y":

So the vertex is at the point

Now we find other points, while brazenly using the symmetry of the parabola. It should be noted that the function – is not even, but, nevertheless, no one canceled the symmetry of the parabola.

In what order to find the remaining points, I think it will be clear from the final table:

This construction algorithm can be figuratively called a "shuttle" or the "back and forth" principle with Anfisa Chekhova.

Let's make a drawing:

From the considered graphs, another useful feature comes to mind:

For a quadratic function () the following is true:

If , then the branches of the parabola are directed upwards.

If , then the branches of the parabola are directed downwards.

In-depth knowledge of the curve can be obtained in the lesson Hyperbola and parabola.

The cubic parabola is given by the function . Here is a drawing familiar from school:

We list the main properties of the function

Function Graph

It represents one of the branches of the parabola. Let's make a drawing:

The main properties of the function:

In this case, the axis is vertical asymptote for the hyperbola graph at .

It will be a BIG mistake if, when drawing up a drawing, by negligence, you allow the graph to intersect with the asymptote.

Also one-sided limits, tell us that a hyperbole not limited from above and not limited from below.

Let's explore the function at infinity: , that is, if we start to move along the axis to the left (or right) to infinity, then the “games” will be a slender step infinitely close approach zero, and, accordingly, the branches of the hyperbola infinitely close approach the axis.

So the axis is horizontal asymptote for the graph of the function, if "x" tends to plus or minus infinity.

The function is odd, which means that the hyperbola is symmetrical with respect to the origin. This fact is obvious from the drawing, in addition, it can be easily verified analytically: .

The graph of a function of the form () represents two branches of a hyperbola.

If , then the hyperbola is located in the first and third coordinate quadrants(see picture above).

If , then the hyperbola is located in the second and fourth coordinate quadrants.

It is not difficult to analyze the specified regularity of the place of residence of the hyperbola from the point of view of geometric transformations of graphs.

Example 3

Construct the right branch of the hyperbola

We use the pointwise construction method, while it is advantageous to select the values so that they divide completely:

Let's make a drawing:

It will not be difficult to construct the left branch of the hyperbola, here the oddness of the function will just help. Roughly speaking, in the pointwise construction table, mentally add a minus to each number, put the corresponding dots and draw the second branch.

Detailed geometric information about the considered line can be found in the article Hyperbola and parabola.

Graph of an exponential function

In this paragraph, I will immediately consider the exponential function, since in problems of higher mathematics in 95% of cases it is the exponent that occurs.

I remind you that - this is an irrational number: , this will be required when building a graph, which, in fact, I will build without ceremony. Three points is probably enough:

Let's leave the graph of the function alone for now, about it later.

The main properties of the function:

Fundamentally, the graphs of functions look the same, etc.

I must say that the second case is less common in practice, but it does occur, so I felt it necessary to include it in this article.

Graph of a logarithmic function

Consider a function with natural logarithm .
Let's do a line drawing:

If you forgot what a logarithm is, please refer to school textbooks.

The main properties of the function:

Domain:

Range of values: .

The function is not limited from above: , albeit slowly, but the branch of the logarithm goes up to infinity.
Let us examine the behavior of the function near zero on the right: . So the axis is vertical asymptote for the graph of the function with "x" tending to zero on the right.

Be sure to know and remember the typical value of the logarithm: .

Fundamentally, the plot of the logarithm at the base looks the same: , , (decimal logarithm to base 10), etc. At the same time, the larger the base, the flatter the chart will be.

We will not consider the case, something I don’t remember when the last time I built a graph with such a basis. Yes, and the logarithm seems to be a very rare guest in problems of higher mathematics.

In conclusion of the paragraph, I will say one more fact: Exponential Function and Logarithmic Functionare two mutually inverse functions. If you look closely at the graph of the logarithm, you can see that this is the same exponent, just it is located a little differently.

Graphs of trigonometric functions

How does trigonometric torment begin at school? Right. From the sine

Let's plot the function

This line is called sinusoid.

I remind you that “pi” is an irrational number:, and in trigonometry it dazzles in the eyes.

The main properties of the function:

This function is periodical with a period. What does it mean? Let's look at the cut. To the left and to the right of it, exactly the same piece of the graph repeats endlessly.

Domain: , that is, for any value of "x" there is a sine value.

Range of values: . The function is limited: , that is, all the “games” sit strictly in the segment .
This does not happen: or, more precisely, it happens, but these equations do not have a solution.