Generally speaking, we call a function in the form of y=ax2+bx+c (where A, B and C are constants and a≠0) as a quadratic function, where A is called a quadratic term coefficient, B is a linear term coefficient and C is a constant term. X is the independent variable and y is the dependent variable. The maximum number of independent variables to the right of the equal sign is 2.
Quadratic function vertex formula (1) General formula: y=ax2+bx+c (a, b, c are constants, a≠0), then y is called the quadratic function vertex coordinate of X (-b/2a, (4ac-b 2)/4a).
(2) Vertex: y=a(x-h)2+k or y = a (x+m) 2+k (a, h and k are constants, a≠0).
(3) Intersection point (with X axis): y=a(x-x 1)(x-x2)
(4) 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 the unary quadratic equation ax2+bx+c=0, a≠0.
Description:
(1) Any quadratic function can be transformed into a 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.
(2) When the parabola y=ax2+bx+c intersects with the X axis, there are real roots x 1 and x2 corresponding to the quadratic equation ax2+bx+c=0. According to the quadratic trinomial decomposition formula AX2+BX+C = A (X-X 1), 2.
The general formula for deriving the vertex coordinate formula of quadratic function is y = ax 2+bx+c (a, b, 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)