I. Definition

Definition of function: Generally speaking, in a changing 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, then we say that X is an independent variable, Y is a function of X, and the value of Y is called the function value.

Second, classification

(1), constant function: When X takes any number in the definition domain and there is y=C(C is constant), the function y=C is called a constant function, which is like a straight line or a part of a straight line parallel to the X axis.

(2) Linear function: the general form is y=kx+b(k, b is constant, k≠0), where x is independent variable and y is dependent variable. Especially when b=0, y=kx+b(k is constant, k≠0), and y is called the proportional function of x.

Third, the representation of functions.

(1), analytical method: The relationship between two variables can sometimes be expressed by an equation containing these two variables and mathematical operation symbols, which is called analytical method.

(2) List method: List a series of values of the independent variable X and the corresponding values of the function Y to represent the functional relationship. This representation is called list method.

(3) Image method: The method of expressing functional relations with images is called image method.

Iv. Images and properties of linear functions

(1), any point P(x, y) on the linear function satisfies the equation: y = kx+b.

(2) The coordinate of the intersection of the linear function and the Y axis is always (0, b), and the coordinate of the intersection of the linear function and the X axis is always (-b/k, 0).

(2) The image of the proportional function always passes through the origin.

Five, three expressions of quadratic function

(1), general formula: y = ax 2+bx+c (a, b, c are constants, a≠0).

(2) Vertex type: y = a (x-h) 2+k.

(3), intersection point: y=a(x-x? )(x-x？ ) [only when it is related to the x axis A(x? , 0) and B(x? 0) parabola].

Symmetry relation of quadratic function image

For the general formula:

① Two images, Y = AX2+BX+C and y=ax2-bx+c, are symmetrical about Y axis.

② The two images y=ax2+bx+c and y=-ax2-bx-c are symmetrical about X axis.

③ y=ax2+bx+c and y=-ax2-bx+c-b2/2a are symmetrical about the vertex.

④ y=ax2+bx+c and y=-ax2+bx-c are symmetrical about the center of the origin.