Definition of quadratic function
Generally, functions in the form of y=ax2+bx+c(a, b and c are constants, and a≠0) are called quadratic functions of X. For example, y=3x2, y=3x2-2, y=2x2+x- 1 are all quadratic functions.
Note: (1) quadratic function is a quadratic form about independent variables, quadratic coefficient A must be a non-zero real number, that is, a≠0, while B and C are arbitrary real numbers, and the expression of quadratic function is algebraic expression;
(2) the quadratic function y=ax2+bx+c(a, b, c are constants, a≠0), and the range of the independent variable x is all real numbers;
(3) When b=c=0, the quadratic function y=ax2 is the simplest quadratic function;
(4) Whether a function is a quadratic function can only be concluded by comparing it with the simplified definition. For example, y=x2-x(x- 1) becomes y=x after simplification, so it is not a quadratic function.
Several forms of quadratic resolution function
(1) general formula: Y = AX2+BX+C (A, b, c are constants, a≠0).
(2) Vertex: y=a(x-h)2+k(a, h, k are constants, a≠0).
(3) two expressions: y=a(x-x 1)(x-x2), where x 1, x2 is the abscissa of the intersection of parabola and x axis, that is, the two roots of quadratic equation ax2+bx+c=0, a≠0.
Description: (1) Any quadratic function can be transformed into vertex y=a(x-h)2+k by formula, and the vertex coordinate of parabola is (h, k). When h=0, the vertex of parabola y=ax2+k is on the Y axis; When k=0, the vertex of parabola a(x-h)2 is on the X axis; When h=0 and k=0, the vertex of parabola y=ax2 is at the origin.
Properties of parabola
1. Parabola is an axisymmetric figure. The axis of symmetry 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^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 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^2-4ac>； 0, parabola and x axis have two intersections.
When δ = b 2-4ac = 0, there are 1 intersections between parabola and X axis.
δ= b^2-4ac<； 0, the parabola has no intersection with the x axis.
The above are the knowledge points I summarized for you about the quadratic function of the third grade mathematics, for reference only, and I hope it will help you.