Quadratic function The basic expression of quadratic function is y=ax? The highest degree of the quadratic function of +bx+c(a≠0) must be quadratic, and the image of the quadratic function is a parabola whose symmetry axis is parallel or coincident with the Y axis. Its definition is quadratic polynomial (or monomial).
If the value of y is equal to zero, a quadratic equation can be obtained. The solution of this equation is called the root of the equation or the zero of the function.
Properties of Quadratic Function (1) The image of quadratic function is a parabola, and parabola is an axisymmetric figure. The symmetry axis is a straight line x=-b/2a.
(2) Quadratic coefficient A determines the opening direction and size of parabola. When a>0, the parabolic opening is upward; When a<0, the parabolic opening is downward. The larger |a|, the smaller the opening of parabola; The smaller the |a|, the larger the opening of the parabola.
(3) Both the linear coefficient b and the quadratic coefficient a*** determine the position of the symmetry axis.
Both linear coefficient b and quadratic coefficient a*** determine the position of the axis of symmetry. 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
(4) The constant term c determines the intersection of parabola and Y axis. The parabola intersects the y axis at (0, c).
Relationship between Quadratic Function and Image (I) Relationship between A and Image
1. Opening direction
When a>0, the opening is upward.
When a<0, the opening is downward,
2. Opening size
The larger the |a|, the smaller the image opening.
The smaller the |a|, the larger the image opening.
(B) the relationship between B and images
When b=0, the symmetry axis is the y axis.
When ab>0, the symmetry axis is on the left side of the Y axis.
When ab
(C) the relationship between C and images
When c=0, the image passes through the origin.
When c>0, the image intersects the positive semi-axis of the Y axis.
When c<0, the image intersects the negative semi-axis of the Y axis.
The formula of translation law of quadratic function is left plus right minus, addition and subtraction.
y=a(x+b)？ +c, just put y=ax? The function image of is converted according to the following rules.
( 1)b & gt； 0, the image moves to the left by b units (plus the left).
(2)b & lt； 0, the image moves b units to the right (minus the right side).
(3)c & gt； 0, the image moves up by c units (positive).
(4)c & lt； 0, the image moves down by c units (minus sign).
The vertex coordinate formula of quadratic function is quadratic function y = ax 2+bx+c
Its vertex coordinates are (-b/2a, (4ac-b 2)/4a) Intersection point: y=a(x-x? )(x-x？ ) [only when it is related to the x axis A(x? , 0) and B(x? 0) parabola]
Where x 1, 2 =-b √ b 2-4ac.
Vertex: y = a (x-h) 2+k
[Vertex P(h, k) of parabola]
General formula: y = ax 2+bx+c (a, b and c are constants, and a≠0).
Note: Among the three forms of mutual transformation, there are the following relationships: h=-b/2a= (x? +x？ ) /2 k = (4ac-b 2)/4a Intersection with X axis: X? ，x？ =(-b √b^2-4ac)/2a。