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History of linear function mathematics
See/view/2b6808e102de2bd9605885b.html for a summary of the key knowledge of linear function, proportional function and inverse proportional function in junior high school mathematics.

Summary of key knowledge of quadratic function;

I. Definition and definition of expressions

Generally speaking, there is the following relationship between independent variable x and dependent variable y:

Y = ax 2+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

Y is called the quadratic function of X.

The right side of a quadratic function expression is usually a quadratic trinomial.

Two. Three Expressions of Quadratic Function

General formula: y = ax 2; +bx+c(a, b, c are constants, a≠0)

Vertex: y = a (x-h) 2; +k[ vertex P(h, k) of parabola]

Intersection point: y = a(X-X 1)(X-x2)[ only applicable to parabolas with intersection points a (x 1, 0) and b (x2, 0) with the x axis]

Note: Among these three forms of mutual transformation, there are the following relations:

h =-b/2a k=(4ac-b^2; )/4a x 1,x2 =(-b √b^2; -4ac)/2a

Three. Quadratic function image

Do quadratic function y=x in plane rectangular coordinate system? Images of,

It can be seen that the image of quadratic function is a parabola.

Four. 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 parabola is Y axis (that is, 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 upward; 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 the signs of A and B are the same (that is, AB > 0), the symmetry axis is left on the 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.

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

When δ = b 2-4ac > 0, the parabola has two intersections with the X axis.

When δ = b 2-4ac = 0, there are 1 intersections between parabola and X axis.

When δ = b 2-4ac < 0, the parabola has no intersection with the X axis.

Verb (abbreviation of verb) quadratic function and unary quadratic equation

Especially the quadratic function (hereinafter referred to as function) y = ax 2; +bx+c,

When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).

That's ax^2;; +bx+c=0

At this point, whether the function image intersects with the X axis means whether the equation has real roots.

The abscissa of the intersection of the function and the x axis is the root of the equation.

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.

Answer supplement

If the image passes through the origin and the axis of symmetry is the y axis, let y = ax 2; If the symmetry axis is the y axis, but not the origin, let y = ax 2+k.

Defining and defining expressions

Generally speaking, there is the following relationship between independent variable x and dependent variable y:

y=ax^2+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

Y is called the quadratic function of X.

The right side of a quadratic function expression is usually a quadratic trinomial.

X is an independent variable and y is a function of X.

Three Expressions of Quadratic Function

① general formula: y = ax 2+bx+c (a, b and c are constants, a≠0).

② Vertex [vertex P(h, k) of parabola]: y = a (x-h) 2+k.

③ Intersection point [only applicable to parabolas with intersection points A(x 1 0) and B(x2, 0) with the X axis ]: y = a (x-x 1) (x-x2).

The above three forms can be converted as follows:

Relationship between (1) General Formula and Vertex Type

For the quadratic function y = ax 2+bx+c, its vertex coordinates are (-b/2a, (4ac-b 2)/4a), that is.

h=-b/2a=(x 1+x2)/2

k=(4ac-b^2)/4a

② Relationship between general formula and intersection point

X 1, x2 = [-b √ (b 2-4ac)]/2a (that is, the formula for finding the root of a quadratic equation with one variable).