Basic theoretical information
Coordinates and basic concepts of functions
The length of the segment on the coordinate axis is 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, use the following formula:
The coordinates of the midpoint of the segment (only the first formula is used for the coordinate axis, the first two formulas for the coordinate plane, all three formulas for the three-dimensional coordinate system) are calculated using the formulas:
A function is a correspondence of the form y = f ( x ) between variables, by virtue of which each considered value of some variable x (argument or independent variable) corresponds to a specific value of another variable, y (dependent variable, sometimes this value is simply called the value of the function ). Note that the function implies that only one value of the dependent variable y can correspond to one value of the argument x . At the same time, the same value of ycan be obtained for different x .
The domain of the function is all the values of the independent variable (the function argument, usually x ), in which the function is defined, i.e. its meaning exists. Denotes the domain of definition D ( y ). By and large, you are already familiar with this concept. The domain of definition of a function is differently called the domain of permissible values, or DHS, which you have long been able to find.
The range of the function is all possible values of the dependent variable of the function. Denoted by E ( y ).
The function increases on the interval where the larger value of the argument corresponds to the larger value of the function. The function decreases on the interval where the larger value of the argument corresponds to the smaller value of the function.
The intervals of the sign of the constancy of a function are the intervals of the independent variable at which the dependent variable retains its positive or negative sign.
The zeros of a function are those argument values for which the value of the function is 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 having to solve the equation. Also often, the need to find gaps in the sign of constancy means simply to solve inequalities.
The function y = f ( x ) is called even if it is defined on a symmetric set and for any x from the domain of definition the equality is true:
This means that for any opposite values of the argument, the values of the even function are equal. The graph of the even function is always symmetric about the ordinate axis of the OS.
The function y = f ( x ) is called odd if it is defined on a symmetric set and for any x from the domain of definition the equality is true:
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 symmetric about the origin.
The sum of the roots of even and odd functions (the intersection points of the x-axis OX) is always zero, since for every positive root x there is a negative root – x .
It is important to note: some function does not have to be even or odd. There are many functions that are not even or odd. Such functions are called general functions , and none of the equalities or properties given above are satisfied for them.
Linear function graph
A linear function is a function that can be given by the formula:
The graph of a linear function is straight 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 of k <0, the function will be decreasing, that is, the line will be tilted in the other direction – from left to right):
Graph of the 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 1 ; 0) and ( x 2 ; 0). If there are no roots, then the quadratic function means that the OX axis does not intersect, if the root is one, then the quadratic function at this point ( x 0 ; 0) only touches the OX axis, but does not intersect it. A quadratic function always intersects the OY axis at the point with the coordinates: (0; c ). The graph of a quadratic function (parabola) may look as follows (in the figure, examples that far from exhaust all possible types of parabolas):
- 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, then the branches of the parabola are directed downwards.
The coordinates of the vertex of the parabola can be calculated by the following formulas. X vertices ( p – in the figures above) of a parabola (or the point at which the square trinomial reaches its highest or lowest value):
The vertex game ( q – in the figures above) of a parabola or maximum, if the branches of the parabola are directed down ( a <0), or minimal, if the branches of the parabola are directed up ( a > 0), the value of the square triple
Graphs of other functions
The power function is the function given by the formula:
Here are some examples of graphs of power functions:
Inversely proportional dependence is called a function defined by the formula:
Depending on the sign of the number k, a graph of inversely proportional dependence can have two principal options:
An asymptote is a line to which the line of the graph of the function approaches infinitely close, but does not intersect. The asymptotes for inversely proportional plots shown in the figure above are the coordinate axes to which the function plots infinitely close, but do not intersect them.
The exponential function with the base а is the function given by the formula:
Depending on whether the number a is greater or less than unity, a graph of an exponential function can have two principal variants (we also give examples, see below):
A logarithmic function is a function defined by the formula:
Depending on whether the number a is greater or less than one, the graph of the logarithmic function can have two principal variants:
Graph of the function y = | x | as follows:
Graphs of periodic (trigonometric) functions
The function y = f ( x ) is called periodic if there is such a non-zero, a number T such that f ( x + T ) = f ( x ), for any x in the domain of the function f ( x ). If the function f ( x ) is periodic with a period T , then the function:
where: A , k , b are constant numbers, and k is not equal to zero, also periodic with a period of T 1 , which is defined by the formula:
Most examples of periodic functions are trigonometric functions. We present the graphs of the main trigonometric functions. The following figure shows a part of the graph of the function y = sin x (the entire graph continues indefinitely left and right), the graph of the function y = sin x is called a sinusoid :
The graph of the function y = cos x is called the cosine curve . This graph is shown in the following figure. Since the sine graph, it continues indefinitely along the OX axis to the left and to the right:
The graph of the function y = tg x is called the tangenzoid . This graph is shown in the following figure. Like the graphs of other periodic functions, this graph repeats indefinitely far along the OX axis to the left and to the right.
Finally, the graph of the function y = ctg x is called the cotangent . This graph is shown in the following figure. As well as graphs of other periodic and trigonometric functions, this graph repeats indefinitely far along the OX axis to the left and to the right.