(1) linear function

If y = kx+b (k and b are constants, k≠0), then y is called a linear function of X. 。

Especially when b = 0, the linear function y = kx+b becomes y = kx (k is constant, k≠0), and y is called the proportional function of X. 。

(2) Images of linear functions

The image with linear function y = kx+b is a straight line passing through (0, b) point and point.

In particular, the image of the proportional function is a straight line passing through the origin.

It should be noted that in the plane rectangular coordinate system, "straight line" is not equivalent to "image of linear function y = kx+b (k ≠ 0)", because there are also straight lines y = m (k = 0 at this time) and x = n (k does not exist at this time), which are not linear function images.

(3) Properties of linear functions

When k > 0, y increases with the increase of x; When k < 0, y decreases with the increase of x 。

The coordinate of the line Y = KX+B intersecting the y axis is (0, b), and the coordinate of the line intersecting the x axis is.

(4) Look at equations (groups) and inequalities from the functional point of view.

① Any one-dimensional linear equation can be transformed into the form of AX+B = 0 (A, B is constant, a≠0), so solving one-dimensional linear equation can be transformed into: linear function Y = KX+B (K, B is constant, k≠0), when Y = 0, find the value of the corresponding independent variable, from the image.

② Binary linear equations correspond to two linear functions, so they also correspond to two straight lines. From the point of view of "number", solving equations is equivalent to considering that two function values are equal when the independent variable is value. What are the two function values? In terms of shape, solving equations is equivalent to determining the coordinates of the intersection of two straight lines.

③ Any one-dimensional linear inequality can be transformed into the form of AX+B > 0 or AX+B < 0 (A and B are constants and a≠0). Solving the linear inequality of one variable can be regarded as: when the linear function value is greater than 0 or less than 0, find the value range corresponding to the independent variable.

8. Inverse proportional function

(1) inverse proportional function

If (k is a constant, k≠0), then y is called the inverse proportional function of X. 。

(2) the image of inverse proportional function

The image of the inverse proportional function is a hyperbola.

(3) Properties of inverse proportional function

① When k > 0, the two branches of the image are in the first and third quadrants respectively, and Y decreases with the increase of X in their respective quadrants.

② When k < 0, the two branches of the image are in the second quadrant and the fourth quadrant respectively, and Y increases with the increase of X in their respective quadrants.

③ The inverse proportional function image is symmetrical about the straight line Y = X and the origin.

(4) Two solutions of k

① If the point (x0, y0) is on a hyperbola, then k = x0y0.

(2) the geometric meaning of k:

If A(x, y) and AB⊥x axis are at point b, then S△AOB.

(5) The intersection of proportional function and inverse proportional function.

If the proportional function y = k 1x (k 1 ≠ 0) and the inverse proportional function, then

When K 1K2 < 0, the two function images do not intersect;

When k 1k2 > 0, the images of two functions have two intersections, and the coordinates are respectively. Therefore, if the images of positive and negative proportional functions have intersections, then the two intersections must be symmetrical about the origin.

1. Quadratic function

If y = AX2+BX+C (A, B, C are constants, a≠0), then y is called a quadratic function of X. 。

Several special quadratic functions: y = ax2 (a ≠ 0); y = ax2+c(AC≠0)； y = ax2+bx(ab≠0)； y=a(x-h)2(a≠0)。

2. Image of quadratic function

The image of quadratic function Y = AX2+BX+C is a parabola, and its symmetry axis is parallel to the Y axis.

Starting from the image of Y = AX2 (A ≠ 0), the image of Y = A (X-H) 2+K (A ≠ 0) can be obtained by translation.

3. Properties of quadratic function

The quadratic function y = ax2+bx+c has the following properties corresponding to its image:

(1) The vertex of parabola Y = AX2+BX+C is the axis of symmetry and a straight line, and the vertex must be on the axis of symmetry;

(2) if a > 0, the opening of parabola y = ax2+bx+c is upward, so for any point (x, y) on parabola, when x and y increase; When x =, y has a minimum value;

If a < 0 and the opening of parabola Y = AX2+BX+C is downward, then for any point (x, y) on the parabola, when x

(3) The intersection of parabola Y = AX2+BX+C and Y axis is (0, c);

(4) in the quadratic function y = ax2+bx+c, let y = 0 to get the intersection of the parabola y = ax2+bx+c and the x axis:

What time? = B2-4ac > 0, parabola y = ax2+bx+c and X axis have two different common * * * points, whose coordinates are sum respectively, and the distance between these two points is: What time? When = 0, the parabola Y = AX2+BX+C has only one common point with the X axis, which is the vertex of this parabola; What time? When < 0, the parabola y = AX2+BX+C has nothing in common with the x axis.

4. Translation of parabola

Parabolas y = a (x-h) 2+k and y = ax2 have the same shape but different positions. Translate the parabola y = ax2 up (down) and left (right) to get the parabola y = a (x-h) 2+k, and the direction and distance of translation should be determined according to the values of h and K. 。