1, define and define formula:
The independent variable x and the dependent variable y have the following relationship: y=kx+b(k, b is constant, k≠0).
It is said that y is a linear function of x. In particular, when b=0, y is a proportional function of X. ..
2. The properties of linear function:
The change value of y is directly proportional to the corresponding change value of x, and the ratio is k, that is, △ y/△ x = k.
3. Images and properties of linear functions;
1) exercises and graphics: (1) list (generally find 4-6 points); (2) tracking points; (3) Connection, you can make an image of a function. (Connected by a smooth straight line)
2) Property: any point P(x, y) on the image of a linear function satisfies the equation: y = kx+b.
3) k, B and the quadrant where the function image is located.
When k > 0, the straight line must pass through the first and third quadrants, and y increases with the increase of x;
When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.
When b > 0, the straight line must pass through the first and second quadrants;
When b < 0, the straight line must pass through three or four quadrants.
When b=0, the straight line passing through the origin o (0 0,0) represents the image of the proportional function. At this time, when k > 0, the straight line only passes through one or three quadrants; When k < 0, the straight line only passes through two or four quadrants.
4. In y=kx+b, the two coordinate systems must pass through (0, b) and (-b/k, 0).
k & gt0，b & gt0k & gt； 0，b & lt0k & lt； 0，b & gt0k & lt； 0，b & lt0
inverse proportion function
1. inverse proportional function: generally speaking, if the relationship between two variables x and y can be expressed in the form of y=kx- 1(k is a constant, k≠0), then y is said to be an inverse proportional function of x.
The image of the inverse proportional function is a hyperbola.
2. The concept of inverse proportional function should pay attention to the following points: (1)(k is a constant, k ≠ 0); (2) The range of the independent variable x is all real numbers of x≠0; (3) The range of the dependent variable y is all real numbers of y≠0.
3. Because in y=k/x(k≠0), neither X nor Y can be zero, and the image of the inverse proportional function cannot intersect with the X axis or the Y axis.
4. In the inverse proportional function image, take any two points P and Q, the intersection points P and Q are parallel lines of the X axis and the Y axis respectively, the rectangular area enclosed with the coordinate axis is S 1, S2 is S 1 = S2 = | k |.
quadratic function
1. Generally speaking, the independent variable X is related to the dependent variable Y, and Y is a function of X: Y = AX 2+BX+C (A≠0) A, B and C are constants, and A ≠ 0, then Y is called a quadratic function of X.
2. Three expressions of quadratic function
General formula: y = ax 2+bx+c (a, b and c are constants, and a≠0).
Vertex: y = a(x-h)2+k[ vertex P(h, k)] For quadratic function y = ax 2+bx+c, its vertex coordinates are (-b/2a, (4ac-b 2)/(4a)).
Intersection point: y = a(X-X 1)(X-x2)[ only applicable to parabolas with intersection points a (x 1, 0) and b (x2, 0) with the x axis]
Where x 1, 2 = (-b √ (b 2-4ac))/(2a) (that is, the formula for finding the root of a quadratic equation with one variable).
Note: Among these three forms of mutual transformation, there are the following relations:
h =-b/2a
k =(4ac-b？ )/4a
x 1，x2 =(-b √b？ -4ac)/2a quadratic function image
3. Let the quadratic function y = x 2 be the image in the plane rectangular coordinate system,
Quadratic function As you can see, the image of quadratic function is a parabola.
Standard drawing steps of quadratic function
(on a plane rectangular coordinate system)
(1) list
(2) Tracking point
(3) Connection
4. The properties of parabola
1. Parabola is an axisymmetric figure. The symmetry axis is a straight line x = -b/2a. The only intersection of the symmetry axis and the parabola is the vertex p of the parabola. Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).
2. The parabola has a vertex p, and the coordinate is P (-b/2a, (4ac-b 2)/4a).
-b/2a=0, p is on the y axis; When δ = b 2-4ac = 0, p is on the x axis.
3. Quadratic coefficient A determines the opening direction and size of parabola.
When a > 0, the parabola opens upward; When a < 0, the parabola opens downward. The larger the |a|, the smaller the opening of the parabola.
4. Both the linear coefficient b and the quadratic coefficient a*** determine the position of the symmetry axis.
When the signs of A and B are the same (that is, AB > 0), the symmetry axis is left on the Y axis;
When the signs of A and B are different (that is, AB < 0), the symmetry axis is on the right side of the Y axis.
5. The constant term c determines the intersection of parabola and Y axis.
The parabola intersects the Y axis at (0, c)
6. Number of intersections between parabola and X axis
When δ = b 2-4ac > 0, the parabola has two intersections with the X axis.
When δ = b 2-4ac = 0, there are 1 intersections between parabola and X axis.
When δ = b 2-4ac < 0, the parabola has no intersection with the X axis.
When a>0, the function obtains the minimum value f (-b/2a) = 4ac-b2/4a at x= -b/2a; In {x | x-b/2a} is an increasing function; The opening of parabola is upward; The range of the function is {x | x ≥ 4ac-b 2/4a}, and vice versa.
When b=0, the axis of symmetry of parabola is the Y axis. At this point, the function is even, and the analytical expression is transformed into y = ax 2+c (a ≠ 0).
Quadratic function and unary quadratic equation
In particular, the quadratic function (hereinafter called function) y = ax 2+bx+c,
When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).
That is, ax 2+bx+c = 0. At this point, whether the function image intersects with the X axis means whether the equation has real roots.
The abscissa of the intersection of the function and the x axis is the root of the equation.