Linear Function (1) Knowledge Induction of 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 intersection of the straight line y = kx+b and the y axis is (0, b), and the coordinate of the intersection of the straight line y = kx+b and 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? From the perspective 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.
Summary of knowledge points of inverse proportional function (1) Inverse proportional function: If (k is a constant, k≠0), Y is called the inverse proportional function of X.
(2) Image of inverse proportional function: The image of inverse proportional function is hyperbola.
(3) Properties of inverse proportional function
① When k > 0, the two branches of the image are in the first quadrant and the third quadrant 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 and fourth quadrants respectively, and in their respective quadrants, Y increases with the increase of X. ..
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 in 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, which means that if the images of positive and negative proportional functions have intersections, then the two intersections must be symmetrical about the origin.
Knowledge point of quadratic function 1. quadratic function
If y = ax2+bx+c (a, b, c are constants, and 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 < 0, y decreases with the increase of x; When x > 0, y increases with the increase of x; When x = 0, y has a minimum value;
If a < 0, the opening of parabola y = ax2+bx+c is downward, then for any point (x, y) on the parabola, when x < 0, y increases with the increase of x; When x > 0, y decreases with the increase of x; When x = 0, y has the maximum value;
(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:
When △ = B2-4ac > 0, parabola Y = AX2+BX+C and X axis have two different common points, and their coordinates are A (X 1 0) and B (X2, 0) respectively, and the distance between these two points is AB = | x2-x1|; When △ = 0, the parabola Y = AX2+BX+C has only one common point with the X axis, which is the vertex of this parabola; When △ < 0, the parabola y = AX2+BX+C has no common point 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. Parabola y = a (x-h) 2+k can be obtained by translating parabola y = ax2 up (down) and left (right), and the direction and distance of translation should be determined according to the values of h and k.