Functional knowledge structure diagram

Definition: In the process of a change, if there are two variables X and Y, and for each fixed value of X, Y has a unique fixed value corresponding to it, we say X.

The independent variable of related concepts, Y is a function of X. If x=a and y=b, then when the value of the independent variable is A, B is called the function value.

(1) analysis method

Representation Method (2) List Method

(3) Image method

Function definition: a function in the form of y = kx (k is a constant and k≠0) is called a proportional function.

Properties of the number (1) proportional function: the image is a straight line passing through the origin. When k > 0, the image passes through the first and third quadrants, and y increases with the increase of x; When k < 0, the image exceeds the first one.

In the fourth quadrant, y decreases with the increase of x 。

Definition: a function in the form of y = kx+b (k and b are constants, k≠0) is called a linear function.

(2) Linear function property: The image is a straight line passing through the point (0, b). When k > 0, y increases with the increase of x; When k < 0, y decreases with the increase of x.

The classification quadrant is determined by symbols k and B.

Definition: Functions in the form of y = (k ≠ 0) are called inverse proportional functions.

(3) The nature of inverse proportional function: the image is hyperbola. When k > 0, the image is in the first and third quadrants, and in each quadrant, y decreases with the increase of x; When k < 0, the image is in the second and fourth quadrants, and in each quadrant, y increases with the increase of x 。

Definition: a function in the form of Y = AX2+BX+C (A ≠ 0), where a, b and c are constants, is called a quadratic function.

(4) The general formula of quadratic function (1): Y = AX2+BX+C (A ≠ 0), where a, b and c are constants.

Vertex of analytical formula (2): y = a (x-h) 2+k (a ≠ 0), where (h, k) is the vertex coordinate of parabola.

(3) Intersection point: = a (x-x 1) (x-x2) (a ≠ 0), where (x 1, 0) and (x2, 0) are the coordinates of the intersection point of parabola and X axis. (This analytical formula is not universal, and the result is usually converted into a universal formula. )

① opening direction: when a > 0, the parabolic opening is upward, and when a < 0, the parabolic opening is downward.

② Symmetry axis: straight line x =.

Attribute ③ Vertex coordinates (,).

④ increase or decrease: if a > 0, x, y increases with the increase of x; If a < 0, then when x, y decreases with the increase of x 。

⑤ Maximum (minimum) value of quadratic function: (pay attention to the range of independent variables). If a > 0, then the minimum value of y = when x =.

If a < 0, the maximum value of y = when x =.

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