I. Definition of linear function: Generally speaking, a function whose analytical formula is y=kx+b(k and b are constants, k≠0) is called a linear function. The domains of linear functions are all real numbers. When b=0, y=kx(k≠0) is a proportional function.
Second, the image
The image of 1 and the proportional function y = kx (k ≠ 0, where k is a constant) is a straight line passing through o (0 0,0) and m (1, k).
(1) When k > 0, the image passes through the origin and the first and third image limits;
(2) When k < 0, the image passes through the origin and the second and fourth image boundaries:
2. The image of linear function y=kx+b(k is constant, k≠0) is a straight line passing through two points A(0, b) and B(-k/b, 0). When k and b≠0, the position of the image (that is, the straight line) is divided into four different situations:
When (1)k>0) k > 0 and b > 0, a straight line crosses the boundaries of the first, second and third images:
(2) When k > 0 and b < 0, the straight line passes through the boundaries of the first, third and fourth images:
(3) When k < 0 and b > 0, a straight line crosses the boundaries of the first, second and fourth images:
(4) When k < 0 and b < 0, a straight line passes through the boundaries of the second, third and fourth images:
3. Find the analytical formula of linear function.
If it is known that the image of a linear function (that is, a straight line) passes through two existing points A(x 1, y 1) and B(x 2, y 2), the method and steps are as follows:
(1) Set the analytical formula of linear function: y=kx+b(k≠0)?
(2) Substitute the coordinates of point A and point B into the analytic formula of set function, and get two equations: y 1 = kx 1+B 1? ; y ^ 2 = kx ^ 2+b②？
(3) Solve the equation simultaneously to find out the values of K and B. ..
This method of setting coefficients k and b first, and then solving the equation to get the coefficients is called undetermined coefficient method.
Inverse proportional function 1. Definition: A function with a general shape of y = k/x (where k is a constant and k≠0) is called an inverse proportional function.
(1) constant k is called the proportional coefficient, k? ≠0、x≠0、y≠0；
(2) The key to judge whether a function is an inverse proportional function is whether the product of two variables is constant;
(3) There are three common expressions of analytical formula:
(1) y? =? k/x(k≠0)； (B)xy？ =? k(k≠0)； (C)y=kx - 1 (k≠0)
Second, the image
1, k>0 o'clock
2. k<0 o'clock
Quadratic function 1. Definition: a function with a general shape of y=ax 2 +bx+c is called a quadratic function.
What needs to be emphasized here is that a, b and c are constants, a≦? 0; The maximum number of times is 2; Algebraic expression must be algebraic expression.
Second, the basic form and image
1、y=ax 2
( 1)a & gt； 0: The opening direction is upward, the vertex coordinate is (0,0), and the symmetry axis is Y axis. X>0, Y increases with the increase of X; X<0, Y decreases with the increase of X; When x=0, the minimum value of y is 0.
(2)a & lt； 0, the opening direction is downward, the vertex coordinate is (0,0), and the symmetry axis is Y axis. X>0, Y decreases with the increase of X; X<0, Y increases with the increase of X; When x=0, the maximum value of y is 0.
2、y = ax ^ 2+c
( 1)a & gt； 0: The opening direction is upward, the vertex coordinate is (0, c), and the symmetry axis is Y axis. X>0, Y increases with the increase of X; X<0, Y decreases with the increase of X; When x=0, the minimum value of y is 0.
(2)a & lt； 0, the opening direction is downward, the vertex coordinate is (0, c), and the symmetry axis is Y axis. X>0, Y decreases with the increase of X; X<0, Y increases with the increase of X; When x=0, the maximum value of y is 0.
3、y=a(x-h) 2
( 1)a & gt； 0: the opening direction is upward, the vertex coordinate is (h, 0), the symmetry axis is x = h. x>h, and y increases with the increase of x; X<h, y decrease with the increase of x; When x=h, the minimum value of y is 0.
(2) When a < 0, the opening direction is downward, and the vertex coordinate (h, 0) and the symmetry axis are x = h..x & gth, and y decreases with the increase of x; X<h, y increase with the increase of x; When x=h, the maximum value of y is 0.
4、y=a(x-h) 2 +k
( 1)a & gt； 0: the opening direction is upward, the vertex coordinate is (h, k), the symmetry axis is x = h. x>h, and y increases with the increase of x; X<h, y decrease with the increase of x; When x=h, y has a minimum value k.
(2) When a < 0, the opening direction is downward, the vertex coordinate is (h, k), the symmetry axis is x = h..x & gth, and y decreases with the increase of x; X<h, y increase with the increase of x; When x=h, y has the maximum value k.
These are the knowledge points of functions that I have compiled, and I hope they can help you.