Defining and defining expressions
Generally speaking, there is a relationship between the independent variable X and the dependent variable Y: y=ax2+bx+c(a, B, C are constants, a≠0, A determines the opening direction of the function, A >；; 0, the opening direction is upward, a
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 coordinates are: P(-b/2a, (4ac-b? )/4a). When-b/2a = 0, p is on the y axis; When δ = b? When -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 upwards; 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 A and B have the same number (ab>0), the symmetry axis is on the left side of Y axis; When a and b have different numbers (i.e. AB
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:
δ= b？ -4ac & gt； 0, parabola and x axis have two intersections.
δ= b？ When -4ac=0, the parabola has 1 intersections with the X axis.
δ= b？ -4ac & lt； 0, the parabola has no intersection with the x axis. The value of x is an imaginary number (x =-b √ b? The reciprocal of the value of -4ac is multiplied by the imaginary number i, and the whole equation is divided by 2a).
Derivation of vertex coordinate formula 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) of parabola]
For quadratic function y = ax 2+bx+c
Its vertex coordinates are (-b/2a, (4ac-b 2)/4a).
Deduction:
y=ax^2+bx+c y=a(x^2+bx/a+c/a)y=a(x^2+bx/a+b^2/4a^2+c/a-b^2/4a^2)y=a(x+b/2a)^2+c-b^2/4a y=a(x+b/2a)^2+(4ac-b^2)/4a
Axis of symmetry x=-b/2a
Vertex coordinates (-b/2a, (4ac-b 2)/4a)
Test sites and requirements of mathematical quadratic function
Test center: function and function definition domain and related concepts such as function value, function representation and constant function.
Assessment requirements: (1) Understand variables, independent variables and dependent variables through examples, and understand the concept of function, its definition domain and function value; (2) Know the constant function; (3) Know the representation of functions and the meaning of symbols.
Test center: use the undetermined coefficient method to find the analytical formula of quadratic function.
Assessment requirements: (1) Master the method of finding the resolution function; (2) Using the undetermined coefficient method skillfully to find the resolution function.
Pay attention to the steps of solving the resolution function: primary design, secondary generation, three columns and four returns.
Test center: draw the image of quadratic function
Examination requirements: (1) Knowing the meaning of function image, I will draw function image by drawing points in plane rectangular coordinate system; (2) Understand the image of quadratic function and realize the idea of combining numbers with shapes; (3) Can draw an approximate image of quadratic function.
Test center: the image of quadratic function and its basic properties
Assessment requirements: (1) Establish the relationship among linear function, binary linear equation and straight line with intuitive images, and understand and master the properties of linear function; (2) The vertex coordinates of quadratic function are obtained by collocation method, and the related properties of quadratic function are described.
Note: (1) When solving problems, you should combine numbers and shapes; (2) The translation of quadratic function should be transformed into vertex.
The above is the quadratic function formula and knowledge point arrangement of junior high school mathematics that I arranged for you.