Middle school math is all about the concepts and formulas of functions

Primary function, quadratic function, inverse proportional function, trigonometric function.

Primary function: In a certain change process, there are two variables x and y, if for each definite value of x, there is a uniquely definite value corresponding to it in y, then we say that y is a function of x, that is to say, x is the independent variable, and y is the dependent variable. It is expressed as y=kx+b (k≠0, k and b are constants), and when b=0 is called the proportional function of x, the proportional function is a special case in the primary function.

It can be expressed as y=kx, and the constant k is called the scale factor.

Quadratic function: Generally, a function of the form y=ax 2+bx+c(a≠0) is called a quadratic function. The independent variable (usually x) and the dependent variable (usually y).On the right is an integer, and the highest order of the independent variable is 2.

Inverse proportional function: the function y=k x (k is constant, x is not equal to 0) is called the inverse proportional function, where k is called the scale coefficient, x is the independent variable, y is the value of the function value, and the range of the value of the independent variable x is not equal to all real numbers that are not equal to 0.

Trigonometric functions: sine function = opposite edge hypotenuse tangent function = opposite edge adjacent edge cosine function = adjacent edge hypotenuse.

1. Perimeter formula:

Rectangle circumference (length and width) 2, c=2(a+b) square perimeter side length 4, c=4a

Circumference Diameter Pi , c=2 r

2. Area formula:

The area of the rectangle is length and width, s=ab

The area of the square is the length of the side and the length of the side , s=a

Induction formula formula "odd and even unchanged, symbol to see quadrant" meaning:

The trigonometric value of k 2 a(k z).

1) When k is an even number, equal to the trigonometric value of the same name, preceded by a sign that treats the value of the original trigonometric function as an acute angle;

2) When k is an odd number, it is equal to the value of the synonymous trigonometric function of , preceded by a sign that treats it as the value of the original trigonometric function when it is regarded as an acute angle.

The concept of junior high school functions is as follows:

The function represents a correspondence between each input value and a unique output value, and the standard symbol for the output value x corresponding to the input value in the function f is f(x).

The standard symbol for the output value x corresponding to the input value in the function f is f(x). The set that contains all the input values of a function is called the definition domain of the function, and the set that contains all the output values is called the value domain.

Concepts related to functions

Function: In a certain change process, if there are two variables x, y, and for each definite value of x in a certain range, y has a unique definite value corresponding to it, then y is said to be a function of x, and x is called an independent variable.

The range of values of the function argumentsThe range of the values of the function arguments should make the function analytic meaningful; In the application problem, the value range of the independent variables should also have practical significance. The process of finding the range of values of the independent variables of a function is essentially the process of solving inequalities or groups of inequalities;

The value range of common independent variables is as follows: fraction: the denominator is not 0; Quadratic radical type: the number of squares to be opened is greater than or equal to 0; Fractional and quadratic radical hybrid type: the denominator is not 0, and the number of squares is greater than or equal to 0

Function value: When the function argument x takes a certain value, the uniquely determined y value corresponding to it is called the function value when the function argument takes the value.

Connecting lines, you can make an image of a slippery branch primary function - a straight line. Therefore, an image that is a function only needs to know 2 points and connect them into a straight line. (Usually find the intersection of the function image with the x-axis and y-axis).

Any point of the property p(x,y) on a primary function satisfies the equation: y=kx+b. (2) The coordinates of the intersection point of the primary function with the y-axis are always (0,b), and the image of the proportional function that always intersects with the x-axis (-b k,0) always crosses the origin.

When k>0, a straight line must pass.

1. In the third quadrant, y increases with the increase of x; When k>0, a straight line must pass.

In the second and fourth quadrants, y decreases with the increase of x.

The concept of the junior high school function is as follows: generally, in a change process, if there are two variables x and y, and for each definite value of x, y has a unique definite value corresponding to it, we call x an independent variable, y a dependent variable, and y is a random number of x.

Three notations of junior functions:

1.Analytical: The functional relationship between two variables, sometimes expressed by an equation containing these two variables and the symbols of the number operation, this notation is called analytical.

2.List method: The method of using a list method to represent the functional relationship between two variables is called the list method.

The advantage of this method is that the value of the function corresponding to it can be read directly through the value of the known independent variable in **; The disadvantage is that only some of the corresponding values can be listed, which is difficult to reflect the whole picture of the function.

3.Image method: take the value of the independent variable x and the corresponding dependent variable y of a function as the abscissa and ordinate of the point, respectively, and trace its corresponding points in the Cartesian coordinate system, and the graph composed of all these points is called the image of the function.

This method of representing functional relationships is called the image method. The advantage of this method is that the function relationship can be intuitively and vividly represented through the function image; The disadvantage is that the quantitative relationship obtained from the image observation is approximate.

In general, the content of the ruler hall junior high school function is relatively easy and is a foundation for the high school function.

The function formula is as follows:

Induction Equation 1: The values of the same trigonometric function at the same angle with the terminal edge are equal. Let be any acute angle, the expression of the angle under the radian system:

sin（2kπ+αsinα（k∈z），cos（2kπ+αcosα（k∈z），tan（2kπ+αtanα(k∈z)，cot（2kπ+αcotα（k∈z）。

Induction Equation 2: The relationship between the trigonometric value of + and the trigonometric value of , let it be the expression of the angle under the radian system: sin( +sin ,cos( +cos ,tan( +tan ,cot( +cot.

Induction Equation 3: The relationship between the trigonometric value of the arbitrary angle and -, sin(-sin, cos(-cos, tan(-tan, cot(-cot.

Induce Equation 4: Using Equations 2 and 3, we can get the relationship between - and the trigonometric values of , sin( -sin ,cos( -cos ,tan( -tan ,cot( -cot .

Induction Equation 5: Using Equation 1 and Equation 3, we can obtain the relationship between the trigonometric values of 2 - and , sin(2 - sin, cos(2 - bend Qingwei = cos , tan(2 - tan , cot(2 - cot.

Induce the relationship between the trigonometric values of 2 and 3 and 2, sin(2+)cos, cos(2+)sin, tan(2+)cot, cot(2+)tan, sin(2-)cos, cos(2-)sin, cos(2-)sin, tan(2-)cot, cot(2-)tan.

sin（3π/2+α）cosα，cos（3π/2+α）sinα，tan（3π/2+α）cotα，cot（3π/2+α）tanα，sin(3π/2-α)cosα，cos（3π/2-α）sinα，tan（3π/2-α）cotα，cot（3π/2-α）tanα。

In its simplest definition, a function is an input-to-output mapping.

That is, a function is a rule or process that corresponds to a unique dependent variable (output) for each independent variable (input). This mapping can be represented by (x,y), and the base where x is the independent variable and y is the dependent variable. This mapping can be expressed in the form of function images, **, or formulas.

The definition of a function may be somewhat abstract, but specifically, a function is an extension of the four operations of "addition, subtraction, multiplication and division" recognized in elementary school mathematics. Unlike addition, subtraction, multiplication and division, the function requires that for each input value, there must be one and only one output value. In other words, there are no different outputs for the same input.

For example, f(x)=x is a function. where x is the independent variable and f(x) is the dependent variable, which means that for any independent variable x, the output of the function f(x) is x.

As another example, let p(x) denote the ** of an object and x denote the quantity of the object purchased. If it is calculated according to the ** of 10 yuan a piece, then there are: when x=1, p(x)=10 yuan; When x=2, p(x)=20 yuan; When x=3, p(x)=30 yuan......When x=n, p(x)=10n yuan.

p(x) in this example is also a function. It indicates that the amount you need to pay to buy an item of x is $10x. We can represent this function in various forms, such as **, formulas, images, etc.

Of course, there are other forms of function. For example, a function can be represented as a function image. where the x-coordinate represents the independent variable and the y-coordinate represents the corresponding dependent variable.

For example, the function of f(x)=x is a parabola with an opening upward. From this image, it is easy to see the nature and characteristics of the function.

When junior high school students learn functions, they also need to master the concepts of static functions and dynamic functions. A static function is a static function if the output of the function does not change if the input and output of only one of the independent variables are changed, and the other independent variable does not change.

Common examples include constant functions and primary functions, among others. The dynamic function is for two independent variables that change at the same time, such as the problem of two cars starting at the same time and moving in a straight line at a uniform speed.

Function Representation: Analytical.

List method, image method.

Proportional function: y=kx (k is constant, k≠0).

When k>0, the image passes.

In the first and second quadrants, y increases with the increase of x.

When k>0, the image passes.

In the second and fourth quadrants, y decreases with the increase of x.

Primary function: y=kx+b (k,b is constant, k≠0) When b=0, y=kx+b = y=kx, so the proportional function is a special form of the primary function.

Inverse proportional function: y=k x (k is constant, k≠0) quadratic function: y=ax+bx+c (a, b, c is constant a≠0) acute angle trigonometric function:

Sine Definition: sina = opposite side of a hypotenuse = a c cosine definition: cosa= adjacent edge of a hypotenuse = b c tangent definition: tana = opposite side of a a adjacent edge of a = a b