y=ax？ +bx+c(a≠0)？

1. Symmetry axis formula: straight line x =-b/2a.

2. Lowest point:

(1) When a > 0, the parabola opens upward and has the lowest point, and the coordinates of the lowest point are (-b/2a, (4ac-b? )/4a)？

(2) When a < 0, the parabola opens downwards and has no lowest point.

Extended data:

Properties of quadratic function:

The image of 1. quadratic function is a parabola, but the parabola is not necessarily a quadratic function. The parabola of the opening up or down is a quadratic function.

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 parabola is Y axis (that is, straight line x = 0).

2. A parabola has a vertex p with coordinates p (-b/2a, (4ac-b? )/4a).

-b/2a = 0, p is on the y axis; When △ = b? At-4ac, p is on the x axis.

3. 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 is, the smaller the opening of parabola is; The smaller a is, the larger the opening of parabola is.

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 on the left side of 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.

(Coincidentally, the left and right are different)

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:

△ = B2-4ac > 0, and the parabola has two intersections with the X axis.

When △ = b2-4ac = 0, there are 1 intersections between the parabola and the X axis.

△ = B2-4ac < 0, and the parabola has no intersection with the x axis.

7. When a > 0, the function obtains the minimum value f (-b/2a) = (4ac-b? )/4a；

The function is decreasing at (-∞, -b/2a,+∞) and increasing function at -b/2a; The opening of parabola is upward; The range of the function is (4ac-b? )/4a，+∞).

When a < 0, the function obtains the maximum value f (-b/2a) = (4ac-b? )/4a；

The function is a increasing function at (-∞, -b/2a,+∞) and a decreasing function at -b/2a; The opening of parabola is downward; The range of the function is (-∞, (4ac-b? )/4a .

When b = 0, the axis of symmetry of parabola is the Y axis. At this point, the function is even, and the analytic expression is transformed into y=ax2+c(a≠0).

8. domain: r

Scope: When a>0, the scope is (4ac-b? )/4a，+∞)； When a<0, the range is (-∞, (4ac-b? )/4a .

Parity: when b=0, this function is even; When b is not equal to 0, this function is a parity function.

Periodicity: None

Resources: Baidu Encyclopedia _ Quadratic Function