Function graphs and their properties Table. Function graphics study

13.08.2020

The main elementary functions inherent in the properties and the corresponding graphs are one of the Azov. mathematical knowledgesimilar to the degree of importance with the multiplication table. Elementary functions are a base based for studying all theoretical issues.

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

The following types of basic elementary functions are distinguished:

Definition 1.

constant function (constant);
root n-degree;
power function;
exponential function;
logarithmic function;
trigonometric functions;
brother trigonometric functions.

A constant function is determined by the formula: Y \u003d C (C is a certain number) and also has a name: constant. This function determines the correspondence to any valid value of an independent variable x of the same variable y value c.

The schedule of the constant is straight, which is parallel to the abscissa axis and passes through a point having coordinates (0, c). For clarity, we give graphs of permanent functions y \u003d 5, y \u003d - 2, y \u003d 3, y \u003d 3 (in the drawing designated with black, red and blue colors, respectively).

Definition 2.

This elementary function is determined by the formula y \u003d x n (n - the natural number of more units).

Consider two variations of the function.

Root N -Y degree, n - even number

For clarity, point out the drawing, which shows graphs of such functions: y \u003d x, y \u003d x 4 and y \u003d x 8. These functions are marked with color: black, red and blue, respectively.

Similar view of the functions of an even degree function at different values \u200b\u200bof the indicator.

Definition 3.

Properties function root N-esh, n - even number

the definition area is the set of all non-negative valid numbers [0, + ∞);
when x \u003d 0, function y \u003d x n has a value equal to zero;
this function function general form (no or even odd);
value area: [0, + ∞);
this function y \u003d x n at even indicators of the root increases throughout the definition area;
the function has convexity towards upwards on the entire definition area;
there are no inflection points;
asymptotes are absent;
the graph of the function at even N passes through the points (0; 0) and (1; 1).

Root n -i degree, n is an odd number

This function is defined on the entire set of valid numbers. For clarity, consider graphs of functions y \u003d x 3, y \u003d x 5 and x 9. In the drawing, they are indicated by flowers: black, red and blue colors of curves, respectively.

Other odd values \u200b\u200bof the root rate of the function y \u003d x n will give a graph of a similar species.

Definition 4.

Properties function N-es degree root, N - odd number

the definition area is the set of all valid numbers;
this feature is odd;
the range of values \u200b\u200bis the set of all valid numbers;
the function y \u003d x n with odd root indicators increases throughout the definition area;
the function has a concave on the interval (- ∞; 0] and the convexity at the interval [0, + ∞);
the inflection point has coordinates (0; 0);
asymptotes are absent;
the graph of the function with odd n passes through the points (- 1; - 1), (0; 0) and (1; 1).

Power function

Definition 5.

The power function is determined by the formula y \u003d x a.

The view of the graphs and the properties of the function depend on the value of the indicator.

when a power function has a whole indicator A, the type of graph of the power function and its properties depend on the even or odd indicator, as well as the sign of the degree. Consider all these special cases in more detail below;
an indicator of a degree can be fractional or irrational - depending on this, the view of the graphs and the properties of the function varies. We will analyze special cases by setting several conditions: 0< a < 1 ; a > 1 ; - 1 < a < 0 и a < - 1 ;
the power function may have a zero indicator, this case is also read below.

We will analyze the power function y \u003d x a when a is an odd positive number, for example, a \u003d 1, 3, 5 ...

For clarity, we indicate graphics of such power functions: Y \u003d X (black graphics), y \u003d x 3 (blue color graphic), y \u003d x 5 (red graphics), y \u003d x 7 (green graphics). When a \u003d 1, we obtain a linear function y \u003d x.

Definition 6.

The properties of the power function, when the indicator of the degree is an odd positive

the function is increasing with x ∈ (- ∞; + ∞);
the function has a bulge at x ∈ (- ∞; 0] and a concave at x ∈ [0; + ∞) (excluding linear function);
the inflection point has coordinates (0; 0) (excluding linear function);
asymptotes are absent;
passage points of function: (- 1; - 1), (0; 0), (1; 1).

We will analyze the power function y \u003d x a when a is an even positive number, for example, a \u003d 2, 4, 6 ...

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

Definition 7.

The properties of the power function, when the degree indicator is even positive:

the definition area: x ∈ (- ∞; + ∞);
decreasing at x ∈ (- ∞; 0];
the function has a concave at x ∈ (- ∞; + ∞);
logging points are missing;
asymptotes are absent;
passage points of function: (- 1; 1), (0; 0), (1; 1).

Figure below shows examples of graphs of power function. y \u003d x a when a is an odd negative number: y \u003d x - 9 (black graphics); y \u003d x - 5 (blue color graphic); y \u003d x - 3 (red graphics); Y \u003d X - 1 (green graphics). When a \u003d - 1, we obtain inverse proportionality, whose graph is a hyperbole.

Definition 8.

The properties of the power function, when the degree indicator is an odd negative:

When x \u003d 0, we obtain the rupture of the second kind, since Lim X → 0 - 0 x A \u003d - ∞, Lim X → 0 + 0 x a \u003d + ∞ at a \u003d - 1, - 3, - 5, .... Thus, the straight line x \u003d 0 is the vertical asymptota;

the range of values: y ∈ (- ∞; 0) ∪ (0; + ∞);
the function is odd, since y (- x) \u003d - y (x);
the function is decreasing at x ∈ - ∞; 0 ∪ (0; + ∞);
the function has a bulge at x ∈ (- ∞; 0) and a concave at x ∈ (0; + ∞);
points of inflection are absent;

k \u003d lim x → ∞ x a x \u003d 0, b \u003d lim x → ∞ (x a - k x) \u003d 0 ⇒ y \u003d k x + b \u003d 0, when A \u003d - 1, - 3, - 5 ,. . . .

passage points of function: (- 1; - 1), (1; 1).

Figure below shows examples of graphs of the power function Y \u003d X A, when A is an even negative number: y \u003d x - 8 (black color graph); y \u003d x - 4 (blue color graph); Y \u003d X - 2 (red graphics).

Definition 9.

The properties of the power function, when the indicator of the degree is even negative:

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

When x \u003d 0, we obtain the rupture of the second kind, since Lim X → 0 - 0 x A \u003d + ∞, Lim X → 0 x 0 x a \u003d + ∞ at a \u003d - 2, - 4, - 6, .... Thus, the straight line x \u003d 0 is the vertical asymptota;

the function is even, since y (- x) \u003d y (x);
the function is increasing with x ∈ (- ∞; 0) and decreasing with x ∈ 0; + ∞;
the function has a concave at x ∈ (- ∞; 0) ∪ (0; + ∞);
points of inflection are absent;
horizontal asymptotta - straight y \u003d 0, because:

k \u003d lim x → ∞ x a x \u003d 0, b \u003d lim x → ∞ (x a - k x) \u003d 0 ⇒ y \u003d k x + b \u003d 0, when A \u003d - 2, - 4, - 6 ,. . . .

passage points of function: (- 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 are taken as the area of \u200b\u200bdetermination of this power function interval - ∞; + ∞, stinging at the same time that the indicator A is an unstable fraction. At the moment, the authors of many educational publications on algebra and the principle of analysis do not define power functions, where the indicator is a fraction with an odd denominator with negative values \u200b\u200bof the argument. Next, we will permit this position: take the area of \u200b\u200bdetermination of power functions with fractional positive indicators of the degree set [0; + ∞). Recommendation for students: Find out the teacher's opinion at this point to avoid disagreements.

So, we will analyze the power function y \u003d x a when the degree rate is rational or irrational number provided that 0< a < 1 .

We illustrate the graphs power functions y \u003d x a when a \u003d 11 12 (black graphics); a \u003d 5 7 (red graphics); a \u003d 1 3 (blue graphics color); A \u003d 2 5 (Green graphics).

Other values \u200b\u200bof the indicator of degree A (provided 0< a < 1) дадут аналогичный вид графика.

Definition 10.

The properties of the power function at 0< a < 1:

the range of values: y ∈ [0; + ∞);
the function is increasing with x ∈ [0; + ∞);
the function has a bulge at x ∈ (0; + ∞);
points of inflection are absent;
asymptotes are absent;

We will analyze the power function y \u003d x a, when the degree rate is a non-target rational or irrational number, provided that A\u003e 1.

We illustrate the graphs power function y \u003d x a under the specified conditions on the example of such functions: Y \u003d x 5 4, y \u003d x 4 3, y \u003d x 7 3, y \u003d x 3 π (black, red, blue, green graphics, respectively).

Other values \u200b\u200bof the indicator of the degree and under the condition A\u003e 1 will give a similar type of graphics.

Definition 11.

The properties of the power function at a\u003e 1:

definition area: x ∈ [0; + ∞);
the range of values: y ∈ [0; + ∞);
this function is a common form function (there is neither odd, nor even);
the function is increasing with x ∈ [0; + ∞);
the function has a concave at x ∈ (0; + ∞) (when 1< a < 2) и выпуклость при x ∈ [ 0 ; + ∞) (когда a > 2);
points of inflection are absent;
asymptotes are absent;
passage points of function: (0; 0), (1; 1).

We pay your attention! When a is a negative fraction with an odd denominator, in the works of some authors there is a look that the definition area in this case is the interval - ∞; 0 ∪ (0; + ∞) with the reservation, which is an indicator of degree A is an unstable fraction. At the moment, the authors of educational materials on algebra and the principle of analysis do not define power functions with an indicator in the form of a fraction with an odd denominator with negative values \u200b\u200bof the argument. Next, we adhere to such a look: Take the area of \u200b\u200bthe determination of power functions with fractional negative indicators set (0; + ∞). Recommendation for students: Specify the vision of your teacher at this point to avoid disagreements.

We continue the topic and disassemble the power function y \u003d x a provided: - 1< a < 0 .

We give the drawing of graphs Next functions: y \u003d x - 5 6, y \u003d x - 2 3, y \u003d x - 1 2 2, y \u003d x - 1 7 (black, red, blue, green lines, respectively).

Definition 12.

The properties of the power function at - 1< a < 0:

lim X → 0 + 0 x a \u003d + ∞, when - 1< a < 0 , т.е. х = 0 – вертикальная асимптота;

value area: y ∈ 0; + ∞;
this function is a common form function (there is neither odd, nor even);
points of inflection are absent;

The drawing below shows the graphs of the power functions y \u003d x - 5 4, y \u003d x - 5 3, y \u003d x - 6, y \u003d x - 24 7 (black, red, blue, green colors of curves, respectively).

Definition 13.

The properties of the power function at a< - 1:

definition area: x ∈ 0; + ∞;

lim X → 0 + 0 x a \u003d + ∞ when a< - 1 , т.е. х = 0 – вертикальная асимптота;

the range of values: y ∈ (0; + ∞);
this function is a common form function (there is neither odd, nor even);
the function is decreasing at x ∈ 0; + ∞;
the function has a concave at x ∈ 0; + ∞;
points of inflection are absent;
horizontal asymptotta - straight y \u003d 0;
the function of the function: (1; 1).

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

The indicative function has the form y \u003d a x, where a\u003e 0 and a ≠ 1, and the graph of this function looks different, based on the base value a. Consider private cases.

First we will analyze the situation when the base of the indicative function matters from zero to one (0< a < 1) . Visual example will serve graphs of functions at a \u003d 1 2 (blue color curve) and a \u003d 5 6 (red curve).

The same species will have graphs of an indicative function at other base values \u200b\u200bprovided 0< a < 1 .

Definition 14.

Properties of the indicative function when the base is less than one:

the range of values: y ∈ (0; + ∞);
this function is a common form function (there is neither odd, nor even);
the indicative function in which the base is less than the unit is descending throughout the definition area;
points of inflection are absent;
horizontal asymptotta - straight y \u003d 0 with a variable X, striving to + ∞;

Now consider the case when the base of the indicative function is greater than the unit (A 1).

We illustrate this particular case by graph of the indicative functions y \u003d 3 2 x (blue color curve) and y \u003d e x (red graphics).

Other base values, large units, give a similar type of graph of the indicative function.

Definition 15.

Properties of the indicative function when the base is greater than the unit:

the definition area is all many valid numbers;
the range of values: y ∈ (0; + ∞);
this function is a common form function (there is neither odd, nor even);
the indicative function in which the base is greater than the unit is increasing at x ∈ - ∞; + ∞;
the function has a concave at x ∈ - ∞; + ∞;
points of inflection are absent;
horizontal asymptotta - straight y \u003d 0 with a variable X, striving to - ∞;
function point: (0; 1).

The logarithmic function has the form y \u003d log a (x), where a\u003e 0, a ≠ 1.

This function is defined only with positive values \u200b\u200bof the argument: at x ∈ 0; + ∞.

The graph of the logarithmic function has different lookBased 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 base values, not large units, will give a similar type of graphics.

Definition 16.

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

definition area: x ∈ 0; + ∞. When x tends to zero to the right, the values \u200b\u200bof the function tend to + ∞;
the range of values: y ∈ - ∞; + ∞;
this function is a common form function (there is neither odd, nor even);
logarithmic
the function has a concave at x ∈ 0; + ∞;
points of inflection are absent;
asymptotes are absent;

Now we will analyze a special case when the basis of the logarithmic function is greater than: A 1 . In the drawing below, the logarithmic functions y \u003d log 3 2 x and y \u003d ln x (blue and red graphs, respectively).

Other base values \u200b\u200bgreater than the unit will give a similar type of graph.

Definition 17.

Properties of logarithmic function, when the base is greater than:

definition area: x ∈ 0; + ∞. When x tends to zero to the right, the values \u200b\u200bof the function tend to - ∞;
the range of values: y ∈ - ∞; + ∞ (all many valid numbers);
this function is a common form function (there is neither odd, nor even);
the logarithmic function is increasing at x ∈ 0; + ∞;
the function has a bulge at x ∈ 0; + ∞;
points of inflection are absent;
asymptotes are absent;
function point: (1; 0).

Trigonometric functions - It is sinus, cosine, tangent and catangenes. We will analyze the properties of each of them and the corresponding graphs.

In general, for all trigonometric functions, the property of the frequency is characteristic, i.e. When the values \u200b\u200bof the functions are repeated at different values \u200b\u200bof the argument, differing from each other by the period F (x + T) \u003d f (x) (T - period). Thus, in the list of properties of trigonometric functions, the item "The smallest positive period" is added. In addition, we will specify such values \u200b\u200bof the argument in which the corresponding function adds to zero.

Sinus function: Y \u003d sin (x)

The graph of this feature is called sinusoid.

Definition 18.

Sinus function properties:

definition area: all sets of valid numbers x ∈ - ∞; + ∞;
the function refers to zero when x \u003d π · k, where k ∈ Z (z is a set of integers);
the function is increasing at x ∈ - π 2 + 2 π · k; π 2 + 2 π · k, k ∈ Z and decreasing with x ∈ π 2 + 2 π · k; 3 π 2 + 2 π · k, k ∈ Z;
the sinus 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 function of the sine 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;
asymptotes are absent.

Cosine function: Y \u003d COS (x)

The graph of this feature is called a cosineida.

Definition 19.

Cosine function properties:

the definition area: x ∈ - ∞; + ∞;
the smallest positive period: T \u003d 2 π;
the range of values: y ∈ - 1; one ;
this function is even, since y (- x) \u003d y (x);
the function is increasing at x ∈ - π + 2 π · k; 2 π · k, k ∈ Z and decreasing with 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 π + 2 π · k points; - 1, k ∈ Z;
the function of the cosine 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;
points of inflection have coordinates π 2 + π · k; 0, k ∈ Z
asymptotes are absent.

Tangent function: Y \u003d T G (x)

The graph of this feature is called tangentSoid.

Definition 20.

Properties of the Tangent function:

the definition area: x ∈ - π 2 + π · k; π 2 + π · k, where k ∈ Z (Z is a set of integers);
The behavior of the function of the tangent at the boundary of the definition area of \u200b\u200bLim X → π 2 + π · k + 0 t g (x) \u003d - ∞, lim x → π 2 + π · k - 0 t g (x) \u003d + ∞. Thus, straight x \u003d π 2 + π · k k ∈ Z are vertical asymptotes;
the function refers to zero when X \u003d π · k for k ∈ Z (z is a plurality of integers);
the range of values: y ∈ - ∞; + ∞;
this function is an odd, since y (- x) \u003d - y (x);
the function is increasing at - π 2 + π · k; π 2 + π · k, k ∈ Z;
the Tangent function is concave at x ∈ [π · k; π 2 + π · k), k ∈ Z and convex at x ∈ (- π 2 + π · k; π · k], k ∈ Z;
points of inflection have coordinates π · k; 0, k ∈ Z;

Cotangent function: Y \u003d C T G (x)

The schedule of this feature is called Kotangensoid .

Definition 21.

Properties of the Cotangent function:

the definition area: x ∈ (π · k; π + π · k), where k ∈ Z (z is a plurality of integers);

The behavior of the Cotangent function at the boundary of the limit area of \u200b\u200bLim X → π · K + 0 T G (x) \u003d + ∞, Lim X → π · k - 0 T g (x) \u003d - ∞. Thus, straight x \u003d π · k k ∈ Z are vertical asymptotes;

the smallest positive period: T \u003d π;
the function refers to zero when x \u003d π 2 + π · k at k ∈ Z (z is the set of integers);
the range of values: y ∈ - ∞; + ∞;
this function is an odd, since y (- x) \u003d - y (x);
the function is decreasing at x ∈ π · k; π + π · k, k ∈ Z;
the Cotangent function is concave at x ∈ (π · k; π 2 + π · k], k ∈ Z and convex at x ∈ [- π 2 + π · k; π · k), k ∈ Z;
points of inflection have coordinates π 2 + π · k; 0, k ∈ Z;
inclined and horizontal asymptotes are absent.

Inverse trigonometric functions are Arksinus, Arkkosinus, Arctangen and Arkotangent. Often, due to the presence of the prefix "Ark" in the title, inverse trigonometric functions are called arcfunctions .

Arxinus function: y \u003d a r c sin (x)

Definition 22.

Arksinus function properties:

this function is an odd, since y (- x) \u003d - y (x);
the Arksinus function has a concave at x ∈ 0; 1 and the convexity at x ∈ - 1; 0;
points of inflection have coordinates (0; 0), it is also zero functions;
asymptotes are absent.

Arkkosinus function: y \u003d a r c cos (x)

Definition 23.

Properties of the Arkkosinus function:

definition area: x ∈ - 1; one ;
value area: y ∈ 0; π;
this function is a common form (neither even or odd);
the function is decreasing throughout the field of definition;
the function of the arcsinus has a concave at x ∈ - 1; 0 and convexity at x ∈ 0; one ;
points of inflection have coordinates 0; π 2;
asymptotes are absent.

Arctangent function: Y \u003d A R C T G (x)

Definition 24.

Properties of the ARCTANGENS function:

the definition area: x ∈ - ∞; + ∞;
the range of values: y ∈ - π 2; π 2;
this function is an odd, since y (- x) \u003d - y (x);
the function is increasing throughout the field of definition;
the Arctangent function has a concave at x ∈ (- ∞; 0] and the convexity at x ∈ [0; + ∞);
the inflection point has coordinates (0; 0), it is also zero functions;
horizontal asymptotes - straight y \u003d - π 2 at x → - ∞ and y \u003d π 2 at x → + ∞ (in the image of asymptotes are green lines).

Arkkothangent function: y \u003d a r c c t g (x)

Definition 25.

Properties of the function Arkkothangence:

the definition area: x ∈ - ∞; + ∞;
area of \u200b\u200bvalues: y ∈ (0; π);
this function is a common type;
the function is decreasing throughout the field of definition;
the Arccothange function has a concave at x ∈ [0; + ∞) and convexity at x ∈ (- ∞; 0];
the inflection point has coordinates 0; π 2;
horizontal asymptotes - straight y \u003d π with x → - ∞ (on the drawing - the line of green) and y \u003d 0 at x → + ∞.

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Build a function

We bring to your attention a service for leaving the schedules of functions online, all rights to which companies belong Desmos.. To enter functions, use the left column. You can enter manually either using a virtual keyboard at the bottom of the window. To increase the window with a schedule, you can hide both the left column and the virtual keyboard.

Advantages of building schedules online

Visual display of entered functions
Building very complex graphs
Construction of graphs specified implicitly (for example, ellipse X ^ 2/9 + Y ^ 2/16 \u003d 1)
Ability to save graphs and get a link on them that becomes available to all on the Internet.
Scale Management, Line Color
Ability to build graphs by points, use of constants
Building simultaneously several graphs of functions
Construction of graphs in the polar coordinate system (use R and θ (\\eta))

With us are easy to build graphs of varying complexity. Building is made instantly. The service is in demand for finding points of intersection of functions, for the image of the graphs to further move them to Word, as illustrations when solving tasks, to analyze the behavioral features of the functions of functions. The optimal browser for working with schedules on this page site is Google Chrome.. When using other browsers, the correctness of work is not guaranteed.

The function graph is a visual representation of the behavior of some function on the coordinate plane. Graphs help to understand the various aspects of the function that cannot be determined by the function itself. You can build graphs of many functions, and each of them will be specified by a certain formula. The schedule of any function is based on a specific algorithm (if you have forgotten the accurate process of building a specific function graph).

Steps

Building a linear function graphics

Determine whether the function is linear. Linear function is given by the formula of the form F (x) \u003d k x + b (\\ displaystyle f (x) \u003d kx + b) or y \u003d k x + b (\\ displaystyle y \u003d kx + b) (for example,), and its schedule is a straight line. Thus, the formula includes one variable and one constant (constant) without any indicators of degrees, root signs, and the like. If the function is given a similar species, build a graph of such a function is quite simple. Here are other examples of linear functions:

Use the constant to mark the point on the Y axis. The constant (B) is the coordinate "y" point of intersection of the graph with the Y axis. That is, this is the point, the coordinate "x" of which is 0. Thus, if in the formula to substitute x \u003d 0, then y \u003d B (constant). In our example y \u003d 2 x + 5 (\\ displaystyle y \u003d 2x + 5) The constant is 5, that is, the intersection point with the Y axis has coordinates (0.5). Apply this point on the coordinate plane.

Find the corner coefficient direct. It is equal to the multiplier with a variable. In our example y \u003d 2 x + 5 (\\ displaystyle y \u003d 2x + 5) With the variable "X" there is a multiplier 2; Thus, the angular coefficient is 2. The angular coefficient determines the angle of inclination direct to the X axis, that is, the greater the angular coefficient, the faster the function increases or decreases.

Record the angular coefficient in the form of a fraction. The angular coefficient is equal to tangent angle of inclination, that is, the ratio of the vertical distance (between two points on a straight line) to the horizontal distance (between the same points). In our example, the angular coefficient is 2, so you can declare that the vertical distance is 2, and the horizontal distance is equal to 1. Record it in the form of a fraction: 2 1 (\\ DisplayStyle (\\ FRAC (2) (1))).

If the angular coefficient is negative, the function decreases.

From the point of intersection of direct with the Y axis, apply the second point using the vertical and horizontal distance. The graph of the linear function can be built on two points. In our example, the intersection point with the y axis has coordinates (0.5); From this point, move to 2 divisions upwards, and then 1 division to the right. Mark the point; It will have coordinates (1.7). Now you can spend direct.

With the help of the line, swipe directly in two points. To avoid errors, find the third point, but in most cases the schedule can be built on two points. Thus, you built a graph of a linear function.

Application points on the coordinate plane
1. Determine the function. The function is indicated as f (x). All possible values \u200b\u200bof the variable "y" are called the function of the values \u200b\u200bof the function, and all possible values \u200b\u200bof the variable "x" are called the field definition area. For example, we consider the function y \u003d x + 2, namely f (x) \u003d x + 2.
  
  Draw two intersecting perpendicular straight lines. Horizontal straight - this is the x. Vertical straight line is the Y axis.
  
  Mark the axis of the coordinates. Spice each axis on equal segments and numb them. The intersection point of the axes is 0. For the x axis: the right (from 0) is applied positive numbers, and the left is negative. For the Y axis: the top (from 0) are made positive numbers, and the negative bottom.
  
  Find the values \u200b\u200bof the "X" values. In our example f (x) \u003d x + 2. Submold in this formula defined "x" values \u200b\u200bto calculate the corresponding values \u200b\u200bof "y". If a complex function is given, simplify it, by turning "y" on one side of the equation.
  - -1: -1 + 2 = 1
  - 0: 0 +2 = 2
  - 1: 1 + 2 = 3
2. Apply points to the coordinate plane. For each pair of coordinates, do the following: Find the corresponding value on the x axis and swipe the vertical line (dotted); Find the corresponding value on the Y axis and swipe the horizontal line (dotted line). Indicate the intersection point of two dotted lines; Thus, you have shown a point of schedule.
  
  Erase dotted lines. Do it after applying to the coordinate plane of all points of the graph. Note: The graph of the function f (x) \u003d x is a direct, passing through the center of coordinates [point with coordinates (0,0)]; The graph f (x) \u003d x + 2 is a straight line, parallel direct f (x) \u003d x, but shifted by two units up and therefore passing through a point with coordinates (0.2) (because constant is 2).
Building a chart of a complex function

Knowledge basic elementary functions, their properties and graphs No less important than knowing the multiplication table. They are as a foundation, everything is based on them, of which everything is built and everything comes down to them.

In this article, we list all the main elementary functions, we will give their schedules and we will not give and evidence. properties of basic elementary functions according to the scheme:

the behavior of the function on the boundaries of the definition area, vertical asymptotes (if necessary, see the article Classification of the function break points);
parity and oddness;
the intervals of the convexity (upwards) and concavity (convexity of the down), the inflection points (if necessary, see the case of the convexity of the function, the direction of convexity, the point of inflection, the conditions of convexity and inflection);
inclined and horizontal asymptotes;
special features of functions;
special properties of some functions (for example, the smallest positive period in trigonometric functions).

If you are interested in or, you can go to these sections of the theory.

Basic elementary features These are: a constant function (constant), root N-degree, power function, indicative, logarithmic function, trigonometric and inverse trigonometric functions.

Navigating page.

Permanent function.

The constant function is set on the set of all valid numbers by the formula where C is some valid number. The constant function puts in accordance with each valid value of an independent variable x the same value of the dependent variable y - value with. The constant function is also called the constant.

The graph of the constant function is the direct, parallel axis of the abscissa and passing through the point with coordinates (0, C). For example, we show graphs of constant functions y \u003d 5, y \u003d -2 and, in the figure below, the black, red and blue straight lines correspond to the correspondingly.

Properties of a constant function.

Definition area: all many valid numbers.
The constant function is even.
The range of values: a set consisting of a single number with.
The constant function is non-gaining and unrewining (it is constant).
It makes sense to talk about bulging and concavity constant.
Asymptot not.
The function passes through the point (0, c) of the coordinate plane.

Ni-degree root.

Consider the basic elementary function, which is defined by the formula, where N is a natural number, more units.

Ni-degree root, n is an even number.

Let's start with the function of the root N-degree at even values \u200b\u200bof the root indicator N.

For example, we give a drawing with image graphs images And, they correspond to black, red and blue lines.

A similar species have functions of the functions of an even degree at other values \u200b\u200bof the indicator.

Properties function root N-degree at even N.

Ni-degree root, N is an odd number.

The function of the root N-degree with an odd root indicator n is defined on the entire set of valid numbers. For example, give graphs of functions And, they correspond to black, red and blue curves.

With other odd values \u200b\u200bof the root rate of graphics, the functions will have a similar view.

Properties function root N-degree with odd n.

Power function.

The power function is specified by the formula of the form.

Consider the type of graphs of the power function and the properties of the power function depending on the value of the degree.

Let's start with a power function with a whole indicator a. In this case, the type of graphs of power functions and the properties of functions depend on the parity or oddness of the indicator, as well as from its sign. Therefore, we first consider the power functions at odd positive values \u200b\u200bof the indicator A, hereinafter - with even positive, further - with odd negative indicators, and, finally, with even negative a.

The properties of power functions with fractional and irrational indicators (as well as the form of graphs of such power functions) depend on the value of the indicator a. We will be considered, first, with a from zero to one, secondly, with a large units, thirdly, with A from minus units to zero, fourth, with a smaller minus one.

In conclusion of this item for the completeness of the picture, we describe the power function with the zero.

Power function with an odd positive indicator.

Consider a power function with an odd positive indicator, that is, when A \u003d 1,3,5, ....

The figure below shows the graphs of power funucles - the black line, the blue line, - the red line, is the green line. At a \u003d 1 we have linear function y \u003d x.

The properties of a power function with an odd positive indicator.

Power function with an even positive indicator.

Consider a power function with an even positive indicator, that is, at a \u003d 2,4,6, ....

As an example, we give graphs of power functions - black line, - blue line, - red line. At a \u003d 2 we have a quadratic function, the graph of which is quadratic parabala.

Properties of power functions with an even positive indicator.

Power function with an odd negative indicator.

Look at the graphs of the powerful function with the odd negative values \u200b\u200bof the indicator of the degree, that is, when and \u003d -1, -3, -5, ....

In the picture, the graphs of power functions are shown as examples - a black line - a blue line, - a red line, - a green line. When and \u003d -1 have inverse proportionalitywhose graph is hyperbola.

The properties of a power function with an odd negative indicator.

Power function with an even negative indicator.

Let us turn to the power function at a \u003d -2, -4, -6, ....

The figure shows graphs of power functions - black line - blue line, - red line.

Properties of power functions with an even negative indicator.

The power function with a rational or irrational indicator, whose value is greater than zero and less than one.

Note! If A is a positive fraction with an odd denominator, then some authors consider the area of \u200b\u200bdetermining the power of the interval. At the same time, they negotiate that the indicator of the degree A is an inconsistent fraction. Now the authors of many textbooks on algebra and the principle of analysis do not determine the power functions with an indicator in the form of a fraction with an odd denominator with negative values \u200b\u200bof the argument. We will adhere to just such a look, that is, we will consider the areas for determining the power functions with fractional positive indicators of the degree. We recommend students to find out the look of your teacher for this subtle moment to avoid disagreements.

Consider a power function with a rational or irrational indicator A, and.

We give graphs of power functions at a \u003d 11/12 (black line), and \u003d 5/7 (red line), (blue line), and \u003d 2/5 (green line).

Power function with a non-rational or irrational indicator, large units.

Consider a power function with a non-rational or irrational indicator A, and.

We give graphs of power functions specified by formulas (Black, red, blue and green lines, respectively).

With other values \u200b\u200bof the degree of degree, the graphs of the function will have a similar view.

Properties of power functions at.

The power function with a valid indicator, which is more minus one and less than zero.

Note! If a is a negative fraction with an odd denominator, then some authors consider the area of \u200b\u200bdetermining the power of the interval . At the same time, they negotiate that the indicator of the degree A is an inconsistent fraction. Now the authors of many textbooks on algebra and the principle of analysis do not determine the power functions with an indicator in the form of a fraction with an odd denominator with negative values \u200b\u200bof the argument. We will adhere to just such a look, that is, we will consider the areas for determining power functions with fractional fractional negative indicators of the degree, respectively. We recommend students to find out the look of your teacher for this subtle moment to avoid disagreements.

Go to a powerful function, kgode.

To prevent a good form of graphs of power functions, we give examples of graphs of functions (black, red, blue and green curves, respectively).

The properties of the power function with the indicator A.

The powerful function with a non-effective indicator, which is less than minus one.

We give examples of graphs of power functions when They are depicted black, red, blue and green lines, respectively.

The properties of the power function with a non-target negative indicator, less minus one.

When a \u003d 0 and we have a function - this is a direct point of which is excluded (0; 1) (expression 0 0, it was not possible to give any value).

Exponential function.

One of the main elementary functions is the indicative function.

The graph of the indicative function, where it takes a different form depending on the value of the base a. We will figure it out in it.

First, consider the case when the base of the indicative function takes the value from zero to one, that is,.

For example, we give graphs of the indicative function at a \u003d 1/2 - the blue line, a \u003d 5/6 - the red line. A similar species have graphs of an indicative function with other base values \u200b\u200bfrom the interval.

Properties of an indicative function based on a smaller unit.

Go to the case when the basis of the indicative function is greater than the unit, that is,.

As an illustration, we give graphics of the indicative functions - the blue line and the red line. With other values \u200b\u200bof the base, large units, graphics of the indicative function will have a similar look.

Properties of the indicative function with the basis of a large unit.

Logarithmic function.

The next main elementary function is the logarithmic function, where,. The logarithmic function is defined only for positive values \u200b\u200bof the argument, that is, when.

The logarithmic function graph takes a different form depending on the value of the base a.

Let's start with the case when.

For example, give graphs of logarithmic function at a \u003d 1/2 - blue line, a \u003d 5/6 - red line. With other base values \u200b\u200bthat do not exceed units, graphs of logarithmic function will have a similar view.

The properties of the logarithmic function with the base of a smaller unit.

We turn to the case when the basis of the logarithmic function is greater than one ().

Let's show graphs of logarithmic functions - blue line, - red line. With other values \u200b\u200bof the base, large units, graphs of logarithmic functions will have a similar look.

Properties of logarithmic function with the basis of a large unit.

Trigonometric functions, their properties and graphics.

All trigonometric functions (sinus, cosine, tangent and catangenes) refer to the main elementary functions. Now we will look at their graphics and list properties.

Trigonometric functions inherent concept periodicity (Repeatability of functions of functions with different values \u200b\u200bof argument, different from each other by the amount of the period where T is a period), therefore, the item added to the list of properties of trigonometric functions "The smallest positive period". Also for each trigonometric function, we indicate the values \u200b\u200bof the argument in which the corresponding function is drawn to zero.

Now we will deal with all trigonometric functions in order.

Function sinus y \u003d sin (x).

I will depict a graph of the sinus function, it is called "sinusoid".

Properties function sinus y \u003d sinx.

The cosine function y \u003d cos (x).

The graph of the cosine function (it is called "cosineida") has the form:

Properties function cosine Y \u003d COSX.

Tangent function Y \u003d TG (X).

The schedule of the Tangent function (it is called "TangentSoid") has the form:

Properties of the Tangent function Y \u003d TGX.

Cotangent function Y \u003d CTG (X).

I will depict the schedule of the Kotangent function (it is called "Kothangensoid"):

Properties of the function of the Cotangent Y \u003d CTGX.

Inverse trigonometric functions, their properties and graphics.

Inverse trigonometric functions (Arksinus, Arkskosinus, Arctangent and Arkotanent) are the main elementary functions. Often due to the prefix "Ark" inverse trigonometric functions are called arcfunctions. Now we will look at their graphics and list properties.

Arxinus function Y \u003d arcsin (x).

I will depict the schedule of the Arksinus function:

Properties of the function of Arkkothangence Y \u003d ArcCTG (X).

Bibliography.

Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginning of the analysis: studies. for 10-11 cl. General institutions.
Profitable M.Ya. Handbook of elementary mathematics.
Novoselov S.I. Algebra and elementary functions.
Tumanov S.I. Elementary algebra. Manual for self-education.

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